> {|}~q hbjbjt+t+ MAAeV]tNB $UI
y
pjt ]CHAPTER 2
THE MATHEMATICS OF DISCONTINUITY
AOn the plane of philosophy properly speaking, of metaphysics, catastrophe theory cannot, to be sure, supply any answer to the great problems which torment mankind. But it favors a dialectical, Heraclitean view of the universe, of a world which is the continual theatre of the battle between >logoi,= between archetypes.@
Ren Thom, 1975
ACatastrophe Theory: Its Present State and Future Perspectives,@ p. 382
AClouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.@
Benoit B. Mandelbrot, 1983
The Fractal Geometry of Nature, p. 1
2.1. General Overview
Somehow it is appropriate if ironic that sharply divergent opinions exist in the mathematical House of Discontinuity with respect to the appropriate method for analyzing discontinuous phenomena. Different methods include catastrophe theory, chaos theory, fractal geometry, synergetics theory, self-organizing criticality, spin glass theory, and emergent complexity. All have been applied to economics in one way or another.
What these approaches have in common is more important than what divides them. They all see discontinuities as fundamental to the nature of nonlinear dynamical reality. In the broadest sense discontinuity theory is bifurcation theory of which all of these are subsets. Ironically then we must consider the bifurcation of bifurcation theory into competing schools. After examining the historical origins of this bifurcation of bifurcation theory, we shall consider the possibility of a reconciliation and synthesis within the House of Discontinuity between these fractious factions.
2.2. The Founding Fathers
The conflict over continuity versus discontinuity can be traced deep into a variety of disputes among the ancient Greek philosophers. The most clearly mathematical was the controversy over Zeno=s paradox, the argument that motion is unreal because of the alleged impossibility of an infinite sequence of discrete events (locations) occurring in finite time (Russell, 1945, pp. 804-806). Arguably Newton and Leibniz independently invented the infinitesimal calculus at least partly to resolve once and for all this rather annoying paradox.
Newton=s explanation of planetary motion by the law of gravitation and the infinitesimal calculus was one of the most important intellectual revolutions in the history of human thought. Although Leibniz=s (1695) version contained hints of doubt because he recognized the possibility of fractional derivatives, probably the earliest conceptualization of fractals after some of Leonardo da Vinci=s drawings of turbulent water flow patterns, the Newtonian revolution represented the triumph of the view of reality as fundamentally continuous rather than discontinuous.
The high water mark of this simplistic perspective came with Laplace (1814) who presented a completely deterministic, continuous, general equilibrium view of celestial mechanics. Laplace went so far as to posit the possible existence of a demon who could know from any given set of initial conditions the position and velocity of any particle in the universe at any succeeding point in time. Needless to say quantum mechanics and general relativity completely retired Laplace=s demon from science even before chaos theory appeared. Ironically enough the first incarnation of Laplacian economics came with Walras= model of general equilibrium in 1874 just when the first cracks in the Laplacian mathematical apparatus were about to appear.
In the late nineteenth century two lines of assault emerged upon the Newtonian-Laplacian superstructure. The first came from pure mathematics with the invention (discovery) of Amonstrous@ functions or sets. Initially viewed as irrelevant curiosa, many of these have since become foundations of chaos theory and fractal geometry. The second line of assault arose from unresolved issues in celestial mechanics and led to bifurcation theory.
The opening shot came in 1875 when duBois Reymond publicly reported the discovery by Weierstrass in 1872 of a continuous but non-differentiable function (Mandelbrot, 1983, p. 4), namely
4
W0(t) = (1-W2)-23Wnexp(2pbnt) (2.1)
T=0
with b > 1 and W = bh, 0 < h < 1. This function is discontinuous in its first derivative everywhere. Lord Rayleigh (1880) used a Weierstrass-like function to study the frequency band spectrum of blackbody radiation, the lack of finite derivatives in certain bands implying infinite energy, the Aultraviolet catastrophe.@ Max Planck resolved this difficulty by inventing quantum mechanics whose stochastic view of wave/particle motion destroyed the deterministic Laplacian vision, although some observers see chaos theory as holding out a deeper affirmation of determinism, if not of the Laplacian sort (Stewart, 1989; Ruelle, 1991).
Sir Arthur Cayley (1879) suggested an iteration using Newton=s method on the simple cubic equation
z3 - 1 = 0. (2.2)
The question he asked was which of the three roots the iteration would converge to from an arbitrary starting point. He answered that this case Aappears to present considerable difficulty.@ Peitgen, Jrgens, and Saupe (1992, pp. 774-775) argue that these iterations from many starting points will generate something like the fractal Julia set (1918) and is somewhat like the problem of the dynamics of a pendulum over three magnets, which becomes a problem of fractal basin boundaries.
Georg Cantor (1883) discovered the most important and influential of these monsters, the Cantor set or Cantor dust or Cantor discontinuum. Because he spent time in mental institutions it was tempting to dismiss as Apathological@ his discoveries, which included transfinite numbers and set theory. But the Cantor set is a fundamental concept for discontinuous mathematics. It can be constructed by taking the closed interval [0,1] and iteratively removing the middle third, leaving the endpoints, then removing the middle thirds of the remaining segments, and so forth to infinity. What is left behind is the Cantor set, partially illustrated in Figure 2.1.
Figure 2.1: Cantor Set on Unit Interval
The paradoxical Cantor set is infinitely subdividable, but is completely discontinuous, nowhere dense. Although it contains a continuum of points it has zero length (Lebesgue measure zero) as all that has been removed adds up to one. A two dimensional version is the Sierpinski (1915, 1916) carpet which is constructed by iteratively removing the central open ninths of a square and its subsquares. The remaining set has zero area but infinite length. A three dimensional version is the Menger sponge which is constructed by iteratively removing the twenty-sevenths of a cube and its subcubes (Menger, 1926; Blumenthal and Menger, 1970). The remaining set has zero volume and infinite surface area.
Other ghastly offspring of Cantor=s monster include the Aspace-filling@ curves of Peano (1890) and Hilbert (1891) and the Asnowflake@ curve of von Koch (1904). This latter imitates the Cantor set in exhibiting Aself-similarity@ wherein a pattern at one level is repeated at smaller levels. Cascades of self-similar bifurcations frequently constitute the transition to chaos and chaotic strange attractors often have a Cantor set like character.
Henri Poincar (1890) developed the second line of assault while trying to resolve an unresolved problem of Newtonian-Laplacian celestial mechanics, the three-body problem (more generally, the n-body problem, n > 2). The motion of n bodies in a gravitational system can be given by a system of differential equations:
xi = fi(x1,x2,...,xn) (2.3)
where the motion of the ith body depends on the positions of the other bodies. For n = 2 such a system can be easily solved with the future motions of the bodies based on their current positions and motions, the foundation of Laplacian naivet. For n > 2 the solutions become very complicated and depend on further specifications. Facing the extreme difficulty and complexity of calculating precise solutions, Poincar (1880-1890, 1899) developed the Aqualitative theory of differential equations@ to understand aspects of the solutions. He was particularly concerned with the asymptotic or long-run stability of the solutions. Would the bodies escape to infinity, remain within given distances, or collide with each other?
Given the asymptotic behavior of a given dynamical system, Poincar then posed the question of structural stability. If the system were slightly perturbed would its long run behavior remain approximately the same or would a significant change occur? The latter indicates the system is structurally unstable and has encountered a bifurcation point. A simple example for the two-body case is that of the escape velocity of a rocket traveling away from earth. This is about 6.9 miles per second which is a bifurcation value for this system. At a speed less than that the rocket will fall back to earth, while at one greater than that it will escape into space. It is no exaggeration to say that bifurcation theory is the mathematics of discontinuity.
Poincar=s concept of bifurcation concept is fundamental to all that follows. Consider a general family of n differential equation whose behavior is determined by a k-dimensional control parameter, m:
.
x = fm(x); x 0 Rn, m 0 Rk. (2.4)
The equilibrium solutions of (2.4) are given by fm(x) = 0. This set of equilibrium solutions will bifurcate into separate branches at a singularity, or degenerate critical point, that is where the Jacobian matrix Dxfm (the derivative of fm(x) with respect to x) has a zero real part for one of its eigenvalues.
An example of (2.4) is
fm(x) = mx - x3. (2.5)
for which
Dxfm = m - 3x2. (2.6)
The bifurcation point is at (x,m) = (0,0). The equilibria and bifurcation point are depicted in Figure 2.2.
Figure 2.2: Supercritical Pitchfork Bifurcation
In this figure the middle branch to the right of (0,0) is unstable locally whereas the outer two are stable, corresponding with the index property that stable and unstable equilibria alternate when they are sufficiently distinct. This particular bifurcation is called the supercritical pitchfork bifurcation.
The supercritical pitchfork is one of several different kinds of bifurcations (Sotomayer, 1973; Guckenheimer and Holmes, 1983; Thompson and Stewart, 1986). Table 2.1 illustrates several major kinds of local bifurcations that occur in economic models of the one and two dimensional type. The prototype equations are in discrete map form (compare the supercritical pitchfork equation with (2.4) which is continuous), except for the Hopf bifurcations which are in continuous form.
The bifurcations can be continuous (subtle) or discontinuous (catastrophic). A bifurcation is continuous if a path in (x,m) can pass across the bifurcation point without leaving the ensemble of the sets of points to which the system can asymptotically converge. Such sets are called attractors and the ensemble of such sets is the attractrix.
Name Type Prototype Map
________________________________________________________
Pitchfork (Supercritical) Continuous xt+1 = xt + mxt - xt3
Pitchfork (Subcritical) Discontinuous xt+1 = xt + mxt + xt3
Flip (Supercritical) Continuous xt+1 = -xt - mxt + xt3
Flip (Subcritical) Discontinuous xt+1 = -xt - mxt - xt3
Fold (Saddle-Node) Discontinuous xt+1 = xt + m - xt2
Transcritical Continuous xt+1 = xt + mxt - xt2
.
Hopf (Supercritical) Continuous x = -y + x[m-(x2+y2)]
.
y = x + y[m -(x2+y2)]
.
Hopf (Subcritical) Discontinuous x = -y + x[m+(x2+y2)]
.
y = x + y[m +(x2+y2)]
_______________________________________________________
Table 2.1: Main One and Two Dimensional Bifurcation Forms
In the fold and transcritical bifurcations exchanges of stability occur across the bifurcations. Flip bifurcations can involve period doubling and can thus be associated with transitions to chaos. The Hopf (1942) bifurcation involves the emergence of cyclical behavior out of a steady state and exhibits an eigenvalue with zero real parts and imaginary parts that are complex conjugates. As we shall see later, this bifurcation is important in the theory of oscillators and business cycle theory.
The degeneracy of (0,0) in the supercritical pitchfork example in Figure 2.2 can be seen by examining the original function (2.4) closely. The first derivative at (0,0) is zero but it is not an extremum. Such degeneracies, or singularities, play a fundamental role in understanding structural stability more broadly.
The connection between the eigenvalues of the Jacobian matrix of partial derivatives of a dynamical system at an equilibrium point and the local stability of that equilibrium was first explicated by the Russian mathematician, Alexander M. Lyapunov (1892). He showed that a sufficient condition for local stability was that the real parts of the eigenvalues for this matrix be negative. It was a small step from this theorem to understanding that such an eigenvalue possessing a zero real part indicated a point where a system could shift from stability to instability, in short a bifurcation point.
Lyapunov was the clear founder of the most creative and prolific strand of thought in the analysis of dynamic discontinuities, the Russian School. Significant successors to Lyapunov include A.A. Andronov (1928) who made important discoveries in bifurcation theory and who with L.S. Pontryagin (1937) dramatically advanced the theory of structural stability, Andrei N. Kolmogorov (1941, 1954) who significantly developed the theory of turbulence and stochastic perturbations, A.N. Sharkovsky (1964) who uncovered the structure of periodicity in dynamic systems, V.I. Oseledec (1968) who developed the theory of Lyapunov characteristic exponents that are central to identifying the Abutterfly effect@ in chaotic dynamics, and Vladimir I. Arnol=d (1968) who most completely classified singularities. This exhausts neither the contributions of these individuals nor the list of those making important contributions to studying dynamic discontinuities from the Russian School.
Meanwhile returning to Poincar=s pathbreaking work, several other new concepts and methods emerged. He analyzed the behavior of orbits of bodies in a Newtonian system by examining crossections of the orbits in spaces of one dimension less than where they actually happen. Such Poincar maps illustrate the long-run limit set, or Aattractor set,@ of the system. Poincar used these maps to study the three-body problem, coming to both Aoptimistic@ and Apessimistic@ conclusions.
An Aoptimistic@ conclusion is the Poincar-Bendixson Theorem for planar motions (Bendixson, 1901) which states that a nonempty limit set of planar flow which contains its boundary (is compact) and which contains no fixed point is a closed orbit. Andronov, Leontovich, and Gordon (1966) used this theorem to show that all nonwandering planar flows fall into three classes: fixed points, closed orbits, and the unions of fixed points and trajectories connecting them. The latter are heteroclinic orbits when they connect distinct points and homoclinic orbits when they connect a point with itself, the latter being useful to understanding chaotic strange attractors, discussed later in this chapter. The Poincar-Bendixson results were extended in a rigorous set theoretic context by George D. Birkhoff (1927), who, among other things, showed the impossibility of three bodies colliding in the case of the three-body problem.
A Apessimistic@ conclusion of Poincar=s studies is much more interesting from our perspective. For the three-body problem he saw the possibility of a nonwandering solution of extreme complexity, an Ainfinitely tight grid@ that could in certain cases be analogous to a Cantor dust. It can be argued that Poincar here discovered the first known strange attractor of a dynamical system.
2.3. The Bifurcation of Bifurcation Theory
2.3.1. The Road to Catastrophe
2.3.1.1. The Theory
The principal promulgators and protagonists of catastrophe theory have been Ren Thom and E. Christopher Zeeman. Thom was strongly motivated by the question of qualitative structural change in developmental biology, especially influenced by the work of D=Arcy Thompson (1917) and C.H. Waddington (1940). Thom=s work was summarized in his highly influential Stabilit Structurelle et Morphogense (1972) from which Zeeman and his associates at Warwick University drew much of their inspiration. But Thom=s Classification Theorem culminates a long line of work in singularity theory, and the crucial theorems rigorously establishing his conjecture were proven by Bernard Malgrange (1966) and John N. Mather (1968).
This strand of thought developed from the work of Afounding father@ Poincar and his follower, George Birkhoff. Following Birkhoff (1927), Marston Morse (1931) distinguished critical points of functions between non-degenerate (maxima or minima) and degenerate (singular, non-extremal). He showed that a function with a degenerate singularity could be slightly perturbed to a new function exhibiting two distinct non-degenerate critical points in place of the singularity, a bifurcation of the degenerate equilibrium. This is depicted in Figure 2.3 and indicates sharply the link between the singularity of a mapping and its structural instability.
Figure 2.3: Bifurcation at a Singularity
Hassler Whitney (1955) advanced the work of Morse by examining different kinds of singularities and their stabilities. He showed that for differentiable mappings between planar surfaces (two-dimensional manifolds) there are exactly two kinds of structurally stable singularities, the fold and the cusp. In fact these are the two simplest elementary catastrophes and the only ones that are stable in all their forms (Trotman and Zeeman, 1976). Thus it can be argued that Whitney was the real founder of catastrophe theory.
Thom (1956) discovered the concept of transversality, widely used in catastrophe theory, chaos theory, and dynamic economics. Two linear sub-spaces are transversal if the sum of their dimensions equals the dimension of the linear space containing them. Thus, they either do not intersect or they intersect in an non-degenerate way such that at the intersections none of their derivatives are equal. They Acut@ each other Acleanly.@ Following this, Thom (1972) developed the classification of the elementary catastrophes.
Consider a dynamical system given by n functions on r control variables, ci. The n equations determine n state variables, xj:
xj = fj(c1,...cr). (2.7)
Let V be a potential function on the set of control and state variables:
V = V(ci,xj) (2.8)
such that for all xj
MV/Mxj = 0. (2.9)
This set of stationary points constitutes the equilibrium manifold, M. We further assume that the potential function possesses a gradient dynamic governed by some convention that tends to move the system to this manifold. Sometimes the control variables are called Aslow@ and the state variables Afast,@ as the latter presumably adjust quickly to be on M, whereas the former control the movements along M. These are not trivial assumptions and have been the basis for serious mathematical criticisms of catastrophe theory, as we shall see later.
Let Cat(f) be the map induced by the projection of the equilibrium manifold, M, into the r-dimensional control parameter space. This is the catastrophe function whose singularities are the focus of catastrophe theory.
Thom=s theorem states that if the underlying functions, fj, are generic (qualitatively stable under slight perturbations), if r # 5, and if n is finite with all but two state variables being representable by linear and non-degenerate quadratic terms, then any singularity of a catastrophe function of the system will be one of eleven types and that these singularities will be structurally stable (generic) under slight perturbations. These eleven types constitute the elementary catastrophes, usable for topologically characterizing discontinuities appearing in a wide variety of phenomena in many different disciplines and contexts.
The canonical forms of these elementary catastrophes can be derived from the germ at the singularity plus a perturbation function derived from the eigenvalues of the Jacobian matrix at the singularity. This Acatastrophe germ@ represents the first non-zero components of the Taylor expansion about the singularity (the polynomial expansion using the set of ever higher order derivatives of the function). Thom (1972) specifically named the seven forms for which r # 4, and described them and their characteristics at great length. Table 2.2 contains a list of these seven with some of their characteristics.
Name dim X dim C germ perturbation
________________________________________________________
fold 1 1 x3 cx
cusp 1 2 x4 c1x + c2x2
swallowtail 1 3 x5 c1x + c2x2 + c3x3
butterfly 1 4 x6 c1x + c2x2 + c3x3 + c4x4
hyperbolic
umbilic 2 3 x13 + x23 c1x1 + c2x22 + c3x1x2
elliptic
umbilic 2 3 x13x23 c1x1 + c2x2 + c3(x12 + x22)
parabolic
umbilic 2 4 x12x2 + x24 c1x1 + c2x2 + c3x12 + c4x22
Table 2.2: Seven Elementary Catastrophes
For dimensionalities greater than five in the control space and two in the state space the number of catastrophe forms is infinite. However up to where the sum of the control and state dimensionalities equals eleven it is possible to classify families of catastrophes to some degree. Beyond this level of dimensionality even the categories of families of catastrophes apparently become infinite and hence very difficult to classify (Arnol=d, Gusein-Zade, and Varchenko, 1985).
Thom (1972, pp. 103-108) labels such higher dimensional catastrophes as Anon-elementary@ or Ageneralized catastrophes.@ He recognizes that such events may have applications to problems of fluid turbulence but finds them uninteresting due to their extreme topological complexity. Such events constitute central topics of chaos theory and fractal geometry.
Among the elementary catastrophes the two simplest, the fold and the cusp, have been applied the most to economic problems. Figure 2.4 depicts the fold catastrophe with one control variable and one state variable. Two values of the control variable constitute the catastrophe or bifurcation set, the points where discontinuous behavior in the state variable can occur, even though the control variable may be smoothly varying. Figure 2.4 also shows a hysteresis cycle as the control variable oscillates and discontinuous jumps and drops of the state variable occur at the bifurcation points.
The dynamics presented in Figure 2.4 use the delay convention that assumes a minimizing potential determined by local conditions (Gilmore, 1981, Chap. 8). The most sharply contrasting convention is the Maxwell convention in which the state variable would drop to the lower branch as soon as it lies under the upper branch and vice versa. The middle branch represents an unstable equilibrium and hence is unattainable except by infinitesimal accident.
Figure 2.5 depicts the cusp catastrophe with two control variables and one state variable. C1 is the normal factor and C2 is the splitting factor. Continuous oscillations of the normal factor will not cause discontinuous changes in the state variable if the value of the splitting factor is less than a critical value given by the cusp point. Above this value of the splitting variable a pleat appears in the manifold and continuous variation of the normal factor can now cause discontinuous behavior in the state variable.
Figure 2.4: Fold Catastrophe
Figure 2.5: Cusp Catastrophe
Zeeman (1977, p. 18) argues that a dynamic system containing a cusp catastrophe can exhibit any of five different behavioral patterns, four of which also can occur with fold catastrophes. These are bimodality, inaccessibility, sudden jumps (catastrophes), hysteresis, and divergence, the latter not occurring in fold catastrophe structures.
Bimodality can occur if a system spends most of its time on either of two widely separated sheets. The intermediate values between the sheets are inaccessible. Sudden jumps occur if the system jumps from one sheet to another. Hysteresis occurs if there is a cycle of jumping back and forth due to oscillations of the normal factor, but with the jumps not happening at the same point. Divergence arises from increases in the splitting factor with two parallel paths initially near one another moving apart if they end up on different sheets after the splitting factor passes beyond the cusp point.
A third form sometimes applied in economics is the butterfly catastrophe with four control variables and one state variable. Zeeman (1977, pp. 29-52) argues that this form is appropriate to situations where there are two sharply conflicting alternatives with an intermediate alternative accessible in some regions. Examples include a bulimic-anorexic who normally alternates between fasting and binging and achieves a normal diet and compromises achieved in war/peace negotiations.
Figure 2.6 displays a cross-section of the bifurcation set of this five-dimensional structure for certain control variable values. It shows the Apocket of compromise,@ bounded by three distinct cusp points. As with the cusp catastrophe, C1 and C2 are normal and splitting factors respectively. Zeeman labels C3 the bias factor which tilts the initial cusp surface one way or another. C4 is the butterfly factor (not to be confused with the Abutterfly effect@ of chaos theory) that generates the pocket of compromise zone for certain of its values. Zeeman=s advocacy of the significance and wide applicability of this particular catastrophe form became a focus of the controversy discussed in the next section.
Figure 2.6: Butterfly Catastrophe
The hyperbolic umbilic and elliptic umbilic catastrophes both have three control variables and two state variables, their canonical forms listed in Table 2.2. Figures 2.7 and 2.8 show their respective three-dimensional bifurcation sets.
Figure 2.7: Hyperbolic Umbilic Catastrophe
Figure 2.8: Elliptic Umbilic Catastrophe
Thom argues that an archetypal example of the hyperbolic umbilic is the breaking of the crest of a wave, and that an archetypal example of the elliptic umbilic is the extremity of a pointed organ such as a hair. The six-dimensional parabolic umbilic (or Amushroom@) is difficult to depict except in very limited subsections. Guastello (1995) applies it to analyzing human creativity.
2.3.1.2. The Controversy
Thom (1972) argued that catastrophe theory is a method of analyzing structural and qualitative changes in a wide variety of phenomena. Besides his extended discussion of embryology and biological morphology, his major focus and inspiration, he argued for its applicability to the study of light caustics, the hydrodynamics of waves breaking, the formation of geological structures, models of quantum mechanics, and structural linguistics. The latter represents one of the most qualitative such applications and Thom seems motivated to link the structuralism of Claude Lvi-Strauss with the semiotics of Ferdinand de Saussure. This is an example of what Arnol=d (1992) labels Athe mysticism of catastrophe theory,@ another example of which can be found in parts of Abraham (1985b).
Zeeman (1977) additionally suggested applications in economics, the formation of public opinion, Abrain modeling,@ the physiology of heartbeat and nerve impulses, stress, prison disturbances, the stability of ships, and structural mechanics, especially the phenomenon of Euler buckling. Discussions of applications in aerodynamics, climatology, more of economics, and other areas can be found in Poston and Stewart (1978, Woodcock and Davis (1978), Gilmore (1981), and Thompson (1982).
In response to these claims and arguments a strong reaction developed, culminating in a series of articles by Kolata (1977), Zahler and Sussman (1977), and Sussman and Zahler (1978a, 1978b). Formidable responses appeared as correspondence in Science (June 17 and August 26, 1977) and in Nature (December 1, 1977), as well as articles in Behavioral Science (Oliva and Capdevielle, 1980; Guastello, 1981), with insightful and balanced overviews in Guckenheimer (1978) and Arnol=d (1992), as well as an acute satire of the critics in Fussbudget and Snarler (1979). That the general outcome of this controversy was to leave catastrophe theory in somewhat bad repute can be seen by the continuing ubiquity of dismissive remarks regarding it (Horgan, 1995, 1997) and the dearth of articles using it, despite occasional suggestions of its appropriate applicability (Gennotte and Leland, 1990), thus suggesting that Oliva and Capdevielle=s (1980) complaint came true, that Athe baby was thrown out with the bathwater.@
Although some of the original criticism was overdrawn and inappropriate, e.g. snide remarks that many of the original papers appeared in unedited Conference Proceedings, a number of the criticisms remained either unanswered or unresolved. These include: excessive reliance on qualitative methods, inappropriate quantization in some applications, and the use of excessively restrictive and narrow mathematical assumptions. Let us consider these in turn with regard especially to their relevance to economics.
A simple response to the first point is that although the theory was developed in a qualitative framework as was the work of Poincar, Andronov, and others, this in no way excludes the possibility of constructing or estimating specific quantitative models within the qualitative framework. Nevertheless, this issue is relevant to the division in economics between qualitative and quantitative approaches and also divides Thom and Zeeman themselves, a bifurcation of the bifurcation of bifurcation theory, so to speak.
Even scathing critics such as Zahler and Sussman (1977) admit that catastrophe theory may be applicable to certain areas of physics and engineering such as structural mechanics, where specific quantifiable models derived from well-established physical theories can be constructed. Much of the criticism focused on Zeeman=s efforts to extend such specific model building and estimation into Asofter@ sciences, thus essentially agreeing that the proof is in the pudding of such specifically quantized model building.
Thom (1983) responded to this controversy by defending a hard-line qualitative approach. Criticizing what he labels Aneo-positive epistemology,@ he argues that science constitutes a continuum between two poles: Aunderstanding reality@ and Aacting effectively on reality.@ The latter requires quantified locally specific models whereas the former is the domain of the qualitative, of heuristic Aclassification of analogous situations@ by means of geometrization. He argues that Ageometrization promotes a global view while the inherent fragmentation of verbal conceptualization permits only a limited grasp@ (1983, Chap. 7). Thus, he sides with the critics of some of Zeeman=s efforts declaring, AThere is little doubt that the main criticism of the pragmatic inadequacy of C.T. [catastrophe theory] models has been in essence well founded@ (ibid.). This does not disturb Thom who sees qualitative understanding as at least as philosophically valuable as quantitative model building.
Although the long-term trend has been to favor Aneo-positive@ quantitative model building, this division between qualitative and quantitative approaches continues to cut across economics as one of its most heated ongoing fundamental controversies. Most defenders of qualitative approaches tend to reject all mathematical methods and prefer institutional-historical-literary approaches. Thus, Thom=s method offers an intriguing alternative for the analysis qualitative change in institutional structures in historical frameworks.
Compared with other disciplines for which catastrophe theoretic models have been constructed, economics more clearly straddles the qualitative-quantitative divide, residing in both the Ahard@ and Asoft@ camps. Catastrophe theory models in economics range from specifically empirical ones through specifically theoretical ones to ones of a more mixed character to highly qualitative ones with hard-to-quantify variables and largely ad hoc relationships between the variables.
Given this diversity of approaches in economics, it may well be best for catastrophe theoretic models in economics if they are clearly in one camp or the other, either based on a solid theoretical foundation with well-defined and specified variables, or fully qualitative. Models mixing quantitative variables with qualitative variables, or questionably quantifiable variables, are likely to be open to the charge of Aspurious quantization@ or other methodological or philosophical sins.
Which brings us to the charge of Aspurious quantization.@ Perhaps the most widely and fiercely criticized such example was Zeeman=s (1977, Chaps. 13, 14) model of prison riots using institutional disturbances as a state variable in a cusp catastrophe model with Aalienation@ and Atension@ as control variables. The former was measured by Apunishment plus segregation@ and the latter by Asickness plus governor=s applications plus welfare visits@ for Gartree prison in 1972, a period of escalating disturbances there. Two separate cusp structures were imputed to the scattering of points generated by this data. Quite aside from issues of statistical significance, this model was subjected to a storm of criticism for the arbitrariness and alleged spuriousness of the measures for these variables. These criticisms seem reasonable. Thus this case would seem to be an example for which this charge is relevant.
For most economists this simply will boil down to insuring that proper econometric practices are carried out for any cases in which catastrophe theory models are empirically estimated, and that half-baked such efforts should not be made for purely heuristic qualitative models. It is the case that Sussman and Zahler (1978a,b) went further and argued that any surface could be fit to a set of points and thus one could never verify that a global form was correct from a local structural estimate. This would seem indeed to be Athrowing the baby out with the bathwater@ by denying the use of significance testing or other methods such as out-of-sample prediction tests for any econometric model, including the most garden variety of linear ones. Of course there are many critics of econometric testing who agree with these arguments, but it is a bit contradictory of Sussman and Zahler on the one hand to denounce catastrophe theory for its alleged excessive qualitativeness and then to turn around and denounce it again using arguments that effectively deny the possibility of fully testing any quantitative model.
We note that although these have only been sparingly used in economics, there is a well developed literature on using multimodal probability density functions based on exponential transformations of data for estimating catastrophe theoretic models (Cobb, 1978, 1981; Cobb, Koppstein and Chen, 1983; Cobb and Zacks, 1985, 1988, Guastello, 1995). Crucial to these techniques are data adjustments for location (often using deviations from the sample mean) and for scale that use some variability from a mode rather than the mean. There are difficulties with this approach, such as the assumption of a perfect Markov process in dynamic situations, but they are not insurmountable in many cases.
With respect to the argument that catastrophe theory involves restrictive mathematical assumptions three different points have been raised. The first is that a potential function must be assumed to exist. Balasko (1978) argued that true potential functions rarely exist in economics, although Lorenz (1989) responded by suggesting that the existence of a stable Lyapunov function may be sufficient. Of course most qualitative models have no such functions.
The second restrictive mathematical assumption is that gradient dynamics do not explicitly allow time to be a variable, something one finds in quite a few catastrophe theory models. However, Thom (1983, pp. 107-108) responds that an elementary catastrophe form may be embedded in a larger system with a time variable, if the larger system is transversal to a catastrophe set in the enlarged space. Thom admits that this may not be the case and will be difficult to determine. Guckenheimer (1973) especially notes this as a serious problem for many catastrophe theory models.
The third critique is that the elementary catastrophes are only a limited subset of the possible range of bifurcations and discontinuities. The work of Arnol=d (1968, 1992) demonstrates this quite clearly and the fractal geometers and chaos theorists would also agree. Clearly the House of Discontinuity has many rooms.
Thus elementary catastrophe theory is a fairly limited subspecies of bifurcation theory, while nevertheless suggesting potentially useful interpretations of economic discontinuities and occasionally more specific models. But is this why it has apparently been in such disfavor among economists?
An ironic reason may have to do with Zahler and Sussman=s (1977) original attack on Zeeman=s work in economics. In particular, the first catastrophe theory model in economics was Zeeman=s (1974) model of the stock exchange in which he allowed heterogeneous agents, rational Afundamentalists@ and irrational Achartists.@ Zahler and Sussman ridiculed this model on theoretical grounds arguing that economics cannot allow irrational agents, 1977 being a time of high belief in rational expectations among economists. Today this criticism looks ridiculous as there is now a vast literature (see Chapters 4 and 5 in this book) on heterogeneous agents in financial markets. The irrelevance of this criticism of Zeeman=s work has largely been forgotten, but the fact that a criticism was made has long been remembered. Indeed, the baby did get thrown out with the bathwater.
Clearly catastrophe theory has numerous serious limits. Indeed it may be more useful as a mode of thought about problems than for its classification of the elementary catastrophes per se. Nevertheless, economists should no longer feel frightened of using it or thinking about it because of the residual memory of attacks upon past applications in economics. Some of those applications (e.g. the 1974 Zeeman stock market model) now look very up-to-date and more useful than the models to which they were disparagingly compared.
2.3.2. The Road to Chaos
2.3.2.1. Preliminary Theoretical Developments
2.3.2.1.1. General Remarks
Despite being plagued by philosophical controversies and disputes over applications, the basic mathematical foundation and apparatus of catastrophe theory are well established and understood. The same cannot be said for chaos theory where there remains controversy, dispute, and loose ends over both definitions and certain basic mathematical questions, notably the definitions of both chaotic dynamics and strange attractor and the question of the structural stability of strange attractors in general (Guckenheimer and Holmes, 1983; Smale, 1991; Viana, 1996). In any case, chaos theory emerged in the 1970s out of several distinct streams of bifurcation theory and related topics that developed from the work of Poincar.
2.3.2.1.2. Attractors, Repellors, and Saddles
Having just noted that there are definitional problems with some terms, let us try to pin down some basic concepts in dynamic systems, namely attractor, repellor, and saddle. These are important because any fixed point of a dynamic systems must be one of these. Unfortunately there is not precise agreement about these, but we can give a reasonable definitions that will be sufficient for our purposes.
For a mapping G in n-dimensional real number space, Rn, with time (t) as one dimension, the closed and bounded (compact) set A is an attracting set if for all x 0 A, G(x) 0 A (this property is known as invariance), and if there exists a neighborhood U of A such that if G(x) in U for t $ 0 then G(x) ---> A as t---> 4. The union of all such neighborhoods of A is called its basin of attraction (or domain of attraction) and is the stable manifold of A within which A will eventually capture any orbit occurring there.
A repelling set is defined analogously but by replacing t with -t. If an attracting set is a distinct fixed point it is called a sink. If a repelling set is a distinct fixed point it is called a source. A distinct fixed point that is neither of these is a saddle. Basins of attraction of disjoint attracting sets will be separated by stable manifolds of non-attracting sets (separatrix).
A physical example of the above is the system of hydrologic watersheds on the earth=s surface. A watershed constitutes a basin of attraction with the mouth of the system as the attractor set, a sink if it is a single point rather than a delta. The separatrix will be the divide between watersheds. Along such divides will be both sources (peaks) and saddles (passes). In a gradient potential system the separatrices constitute the bifurcations or catastrophe sets of the system in catastrophe theory terms.
Although many observers identify attractor with attracting set (and repellor with repelling set), Eckmann and Ruelle (1985) argue that an attractor is an irreducible subset of an attracting set. Such a subset cannot be made into disjoint sets. Irreducibility is also known as indecomposability and as topological transitivity. Most attracting sets are also attractors, but Eckmann and Ruelle (1985, p. 623) provide an example of an exception, even as they eschew providing a precise definition of an attractor.
2.3.2.1.3. The Theory of Oscillations
After celestial mechanics the category of models first studied capable of generating chaotic behavior came from the theory of oscillations, the first general version of which was developed by the Russian School in the context of radio-engineering problems (Mandel=shtam, Papaleski, Andronov, Vitt, Gorelik, and Khaikin, 1936). This was not merely theoretical as it is now clear that this study generated the first experimentally observed example of chaotic dynamics (van der Pol and van der Mark, 1927). In adjusting the frequency ratios in telephone receivers they noted zones where Aan irregular noise is heard in the telephone receivers before the frequency jumps to the next lower value...[that] strongly reminds one of the tunes of a bagpipe@ (van der Pol and van der Mark, 1927, p. 364).
Indeed prior to the generalizations of the Russian School, two examples of nonlinear forced oscillators were studied, the Duffing (1918) model of an electro-magnetized vibrating beam and the van der Pol (1927) model of an electrical circuit with a triode valve whose resistance changes with the current. Both of these models have been shown to exhibit cusp catastrophe behavior for certain variables in certain forms (Zeeman, 1977, Chap. 9).
Moon and Holmes (1979) showed that the Duffing oscillator could generate chaotic dynamics. Holmes (1979) showed that as a crucial parameter is varied the oscillator can exhibit a sequence of period-doubling bifurcations in the transition to chaotic dynamics, the AFeigenbaum cascade@ (Feigenbaum, 1978), although period-doubling cascades were first studied by Myrberg (1958, 1959, 1963). Early work on complex aspects of the Duffing oscillator was done by Cartwright and Littlewood (1945) which underlies the detailed study of the strange attractor driving the Duffing oscillator carried out by Ueda (1980, 1991).
As noted above, van der Pol and van der Mark were already aware of the chaotic potential of van der Pol=s forced oscillator model, a result proven rigorously by Levi (1981). The unforced van der Pol oscillator inspired the Hopf bifurcation (Hopf, 1942) which has been much used in macroeconomic business cycle theory and which sometimes occurs in transitions to chaotic dynamics. It happens when the vanishing of the real part of an eigenvalue coincides with conjugate imaginary roots. This indicates the emergence of limit cycle behavior out of non-cyclical dynamics. The simple unforced van der Pol equation is
.. .
x + e(x2 -b)x + x = 0 (2.10)
where e > 0 and b is a control variable. For b < 0 the flow has an attractor point at the origin. At b = 0 the Hopf bifurcation occurs and for b > 0 the attracting set is a paraboloid of radius = 2%b which determines the limit cycles while the axis is now a repelling set. This is depicted in Figure 2.9.
Figure 2.9: Hopf Bifurcation
2.3.2.1.4. Non-Integer (Fractal) Dimension
Another important development was an extension by Felix Hausdorff (1918) of the concept of dimensionality beyond the standard Euclidean, or topological. This effort was largely inspired by contemplation of the previously discussed Cantor set and Koch curve. Hausdorff understood that for such highly irregular sets another concept of dimension was more useful than the traditional Euclidean or topological concept, a concept that could indicate the degree of irregularity of the set. The measure involves estimating the rate at which the set of clusters or kinks increases as the scale of measurement (a gauge) decreases. This depends on a cover of a set of balls of decreasing size. Thus, the von Koch snowflake has an infinite length even though it surrounds only a finite area. The Hausdorff dimension captures the ratio of the logarithms of the length of the curve to the decrease in the scale of the measure of the curve.
The precise definition of the Hausdorff dimension is quite complicated and is given in Guckenheimer and Holmes, 1990, p. 285 and Peitgen, Jrgens, and Saupe, 1992, pp. 216-218. Farmer, Ott, and Yorke (1983), Edgar (1990), and Falconer (1990) discuss relations between different dimension measures.
Perhaps the most widely used empirical dimension measure has been the correlation dimension of Grassberger and Procaccia (1983a). To obtain this dimension one must first estimate the correlation integral. This is defined for a trajectory in an m-dimensional space known as the embedding dimension. The embedding theorem of Floris Takens (1981) states that under appropriate conditions this dimension must be at least twice as great as that of the attractor being estimated (Areconstructed@). Such reconstruction is done by estimating a set of delay coordinates. For a given radius, e, the correlation integral will be the probability that there will be two randomly chosen points of the trajectory within e of each other and is denoted as Cm(e). The correlation dimension for embedding dimension m will then be given by
Dm = lim [ln(Cm(e))/ln(e)]. (2.11)
e->0
The correlation dimension is the value of Dm as m-->4 and is less than or equal to the Hausdorff dimension. It can be viewed as measuring the degree of fine structure in the attractor. It also can be interpreted as the minimum number of parameters necessary to describe the attractor and its dynamics, and thus is an index of the difficulty of forecasting from estimates of the system. A dimension of zero indicates completely regular structure and full forecastibility whereas a dimension of 4 indicates pure randomness and inability to forecast. Investigators hoping to find some usable deterministic fractal structure search for some positive but low correlation dimension.
Much controversy has accompanied the use of this measure, especially in regard to measures of alleged climatic attractors (Nicolis and Nicolis, 1984, 1987; Grassberger, 1986, 1987; Ruelle, 1990) as well as in economics where biases due to insufficient data sets are serious (Ramsey and Yuan, 1989; Ramsey, Sayers, and Rothman, 1990). Ruelle (1990, p. 247) is especially scathing, comparing some of these dimension
estimates to the episode in D. Adams= The hitchhiker=s guide to the galaxy wherein Aa huge supercomputer has answered >the great problem of life, the universe, and everything=. The answer obtained after many years of computation is 42.@
For smooth manifolds and Euclidean spaces these measures will be the same as the standard Euclidean (topological) dimension, which always has an integer value. But for sufficiently irregular sets they will diverge, with the Hausdorff and other related measures generating non-integer values and the degree of divergence from the standard Euclidean measure providing an index of the degree of irregularity of the set. A specific example is the original triadic Cantor set on the unit interval discussed earlier in this chapter. Its Euclidean dimension is zero (same as a point), but its Hausdorff (and also correlation) dimension equals ln2/ln3.
The Hausdorff measure of dimension has become the central focus of the fractal geometry approach of Benoit Mandelbrot (1983). He has redefined the Hausdorff dimension to be fractal dimension and has labeled as fractal any set whose fractal dimension does not equal its Euclidean dimension. Unsurprisingly, given the multiplicity of dimension measures there are oddball cases that do not easily fit in. Thus, Afat fractals@ of integer dimension have been identified (Farmer, 1986; Umberger, Mayer-Kress, and Jen, 1986) that must be estimated by using Ametadimensional@ methods. Also, Mandelbrot himself has recognized that some sets must be characterized by a spectrum of fractal numbers known as multifractals (Mandelbrot, 1988) or even in some cases by negative fractal dimensions (Mandelbrot, 1990a,b).
As already noted, the use of such fractal or non-integer measures of dimension has been popular for estimating the Astrangeness of strange attractors,@ which are argued to be of fractal dimensionality, among other things. Such a phenomenon implies that a dynamical system tracking such an attractor will exhibit irregular behavior, albeit deterministically driven, this irregular behavior reflecting the fundamental irregularity of the attractor itself.
Yet another application of the concept that has shown up in economics has been to cases where the boundaries of basins of attraction are fractal, even when the attractors themselves might be quite simple (McDonald, Grebogi, Ott, and Yorke, 1985; Lorenz, 1992). In such cases, extreme difficulties in forecasting can arise without any other forms of complexity being involved. Figure 2.10 shows a case of fractal basin boundaries arising from a situation where a pendulum is held over three magnets, whose locations constitute the three simple point attractors.
Figure 2.10: Fractal Basin Boundaries, Three Magnets
Both attractors with fractal dimension and fractal basin boundaries can occur even when a dynamical system may not exhibit sensitive dependence on initial conditions, widely argued to be the sine qua non of truly chaotic dynamics.
2.3.2.2. The Emergence of the Chaos Concept
2.3.2.2.1. The Lorenz Model
As with the emergence of discontinuous mathematics in the late nineteenth century, the chaos concept emerged from the separate lines of actual physical models and of theoretical mathematical developments. Indeed, the basic elements had already been observed by the early twentieth century along both lines, but had simply been ignored as anomalies. Thus Poincar and Hadamard had understood the possibility of sensitive dependence on initial conditions during the late nineteenth century; Poincar had understood the possibility of deterministic but irregular dynamic trajectories; Cantor had understood the possibility of irregular sets in the 1880s while Hausdorff had defined non-integer dimension for describing such sets in 1918, and in 1927 van der Pol and van der Mark had even heard the Atunes of bagpipes@ on their telephone receivers. But nobody paid any attention.
Although nobody would initially pay attention, in 1963 Edward Lorenz published results about a three equation model of atmospheric flow that contains most of the elements of what has since come to be called chaos. They would pay attention soon enough.
It has been reported that Lorenz discovered chaos accidentally while he was on a coffee break (Gleick, 1987; Stewart, 1989; E.N. Lorenz, 1993). He let his computer simulate the model with a starting value of a variable different by 0.000127 from what had been generated in a previous run, this starting point being partway through the original run. When he returned from his coffee break, the model was showing significantly different behavior as shown in Figure 2.11, where (a) shows the original run, (b) shows the run beginning partway with the rounded-off starting value, and 8 shows the difference between the two. This was sensitive dependence on initial conditions (SDIC), viewed widely as the essential sign of chaotic dynamics (Eckmann and Ruelle, 1985).
Later Lorenz would call this the butterfly effect for the idea that a butterfly flapping its wings in Brazil could cause a hurricane in Texas. The immediate implication for Lorenz was that long-term weather forecasting is essentially impossible. Butterflies are everywhere.
Figure 2.11: The Butterfly Effect in the Lorenz Model
The model consists of three differential equations, two for temperature and one for velocity. They are
.
x = s(y - x) (2.12)
.
y = rx - y - xy (2.13)
.
z = -bz + xy (2.14)
where s is the so-called Prandtl number, r is the Rayleigh number (Rayleigh, 1916), and b an aspect ratio. The usual approach is to set s and b at fixed values (Lorenz set them at 10 and 8/3 respectively) and then vary r, the Rayleigh number. The system describes a two-layered fluid heated from above. For r < 1 the origin (no convection) is the only sink and is nondegenerate.
At r = 1 the system experiences a cusp catastrophe. The origin now becomes a saddle point with a one-dimensional unstable manifold while two stable attractors, C and C=, emerge on either side of the origin, each representing convective behavior. This bifurcation recalls the discontinuous emergence of hexagonal ABnard cells@ of convection in heated fluids that Rayleigh (1916) had studied both theoretically and experimentally. As r passes through 13.26 the locally unstable trajectories return to the origin, while C and C= lose their global stability and become surrounded by local basins of attraction, N and N=. Trajectories outside these basins go back and forth chaotically. This is a zone of Ametastable chaos.@ As r increases further, the basins N and N= shrink and the zone of metastable chaos expands as infinitely many unstable turbulent orbits appear. At r = 24.74 an unstable Hopf bifurcation occurs (Marsden and McCracken, 1976, Chap. 4). C and C= become unstable saddle points and a zone of universal chaos has been reached.
Curiously enough, as r increases further to greater than about 100 order, begins to reemerge. A sequence of period-halving bifurcations happens until at around r = 313 a single stable periodic orbit emerges that then remains as r goes to infinity. All of this is summarized by Figure 2.12 drawn from Robbins (1979). This represents the case with s = 10 and b = 8/3 that Lorenz studied, but the bifurcation values of r would vary with different values of these parameters, these variations having been intensively studied by Sparrow (1982).
Figure 2.12: Bifurcation Structure of the Lorenz Model
In his original paper Lorenz studied the behavior of the system in the chaotic zone by iterating 3000 times for r = 28. He found that fairly quickly the trajectories moved along a branched, S-shaped manifold that has a fine fractal structure. This has been identified as the Lorenz attractor and a very strange attractor it is. It has been and continues to be one of the most intensively studied of all attractors (Guckenheimer and Williams, 1979; Smale, 1991; Viana, 1996). Generally trajectories initially approach one of the formerly stable foci and then spiral around and away from the focus on one half of the attracting set until jumping back to the other half of the attractor fairly near the other focus and then repeat the pattern again. This behavior and the outline of the Lorenz attractor are depicted in Figure 2.13.
Figure 2.13: Lorenz Attractor
2.3.2.2.2. Structural Stability and the Smale Horseshoe
It took nearly ten years before mathematicians became aware of Lorenz=s results and began to study his model. But at the same time that he was doing his work, Steve Smale (1963, 1967) was expanding the mathematical understanding of chaotic dynamics from research on structural stability of planar flows, following work by Peixoto (1962) that summarized a long strand of thought running from Poincar through Andronov and Pontryagin.
In particular, he discovered that many differential equations systems contain a horseshoe map which has a non-wandering Cantor set containing a countably infinite set of periodic orbits of arbitrarily long periods, an uncountable set of bounded nonperiodic flows, and a dense orbit. These phenomena in conjunction with SDIC, are thought by many to fully characterize chaotic dynamics, and indeed orbits near a horseshoe will exhibit SDIC (Guckenheimer and Holmes, 1990, p. 110).
The Smale horseshoe is the largest invariant set of a dynamical system; an orbit will stay inside the set once there. It can be found by considering all the backward and forward iterates of a control function on a Poincar map of the orbits of a dynamical system which remain fixed. Let the Poincar map be a planar unit square S and let f be the control function generating the set of orbits in S. The forward iterates will be given by f(S)1S for the first iterate and f(f(S)1S)1S for the second iterate and so forth. The backward iterates can be generated from f-1(S1f-1(S)) and so forth. This process of Astretching and folding@ of a bounded set lies at the heart of chaos as the stretching in effect generates the local instability of SDIC as nearby trajectories diverge while the folding generates the return towards each other of distant trajectories that also characterizes chaos.
As depicted in Figures 2.14-2.16 the set of forward iterates will be a countably infinite set of infinitesimally thick vertical strips while the backward iterates will be a similar set of horizontal strips, both of these being Cantor sets. The entire set will be the intersection of these two sets which will in turn be a Cantor set of infinitesimal rectangles, L. This set is structurally stable in that slight perturbations of f will only slightly perturb L. Thus, the monster set was discovered to be sitting in the living room.
Figure 2.14: Forward Iterates of Smale Horseshoe
Figure 2.15: Backward Iterates of Smale Horseshoe
Figure 2.16: Combined Iterates of Smale Horseshoe
Many systems can be shown to possess a horseshoe, including the Duffing and van der Pol oscillators (Guckenheimer and Holmes, 1990, Chap. 2). Smale horseshoes arise when a system has a transversal homoclinic orbit, one that contains intersecting stable and unstable manifolds. Guckenheimer and Holmes (1990, p. 256) define a strange attractor as one that contains such a transversal homoclinic orbit and thus a Smale horseshoe. However, we must note that this does not necessarily imply that the horseshoe is itself an attractor, only that its presence in an attractor will make that attractor a Astrange@ one. If orbits remain outside the horseshoe they may remain periodic and well-behaved, there being nothing necessarily to attract orbits into the horseshoe that are not already there. Nevertheless, the Smale horseshoe provided one of the first clear mathematical handles on chaotic dynamics, and incidentally brought the Cantor set permanent respectability in the set of sets.
2.3.2.2.3. Turbulence and Strange Attractors
If the early intimations of chaotic dynamics came from studying multibody problems in celestial mechanics and nonlinear oscillatory systems, the explicit understanding of chaos came from studying fluid dynamics, the Lorenz model being an example of this. Further development of this understanding came from contemplating the emergence of turbulence in fluids (Ruelle and Takens, 1971).
This was not a new problem and certain complexity ideas had arisen earlier from contemplating it. The earliest models of turbulence with emphasis on wind were due to Lewis Fry Richardson (1922, 1926). One idea widely used in chaos theory, especially since Feigenbaum (1978) is that of a hierarchy of self-similar eddies linked by a cascade, a highly fractal concept. Richardson proposed such a view, declaring (Richardson, 1922, p. 66):
ABig whorls have little whorls,
Which feed on their velocity;
And little whorls have lesser whorls,
And so on to viscosity
(in the molecular sense).@
In his 1926 paper Richardson questioned whether wind can be said to have a definable velocity because of its gusty and turbulent nature and invoked the above Weierstrass function as part of this argument.
Another phenomenon associated with fluid turbulence that appears more generally in chaotic systems is that of intermittency. Turbulence (and chaos) is not universal but comes and goes, as in the Lorenz model above. Chaos may emerge from order, but order may emerge from chaos, an argument especially emphasized by the Brussels School (Prigogine and Stengers, 1984). Intermittency of turbulence was first analyzed by Batchelor and Townsend (1949) and more formally by Kolmogorov (1962) and Obukhov (1962).
A major advance came with Ruelle and Takens (1971) who introduced the term strange attractor under the influence of Smale and Thom, although without knowing of Lorenz=s work. Their model was an alternative to the accepted view of Lev Landau (Landau, 1944; Landau and Lifshitz, 1959) that turbulence represents the excitation of many independent modes of oscillation with some having periodicities not in rational number ratios of each other, a pattern known as quasi-periodicity (Medio, 1998). Rather they argued that the modes interact with each other and that a sequence of Hopf bifurcations culminates in the system tracking a set with Cantor set Smale horseshoes in it, a strange attractor, although they did not formally define this term at this time.
Figure 2.17 shows this sequence of bifurcations. The n-dimensional systems has a unique stable point at C=0 which persists up to the first Hopf bifurcation at C=C1 after which there is a limit cycle with a stable periodic orbit of angular frequency w1. Then at C=C2 another Hopf bifurcation occurs followed by a limit torus (doughnut) with quasiperiodic flow governed by (w1,w2) as frequency components. As C increases and the ratio of the frequencies varies the flow may vary rapidly between periodic and nonperiodic. At C=C3 there is a third Hopf bifurcation followed by motion on a stable three-torus governed by quasiperiodic frequencies (w1,w2,w3). After C=C4 and its fourth Hopf bifurcation, flow is quasiperiodic with frequencies (w1,w2,w3,w4) on a structurally unstable four-torus with an open set of perturbations containing strange attractors with Smale horseshoes and thus Aturbulence.@
Figure 2.17: Ruelle-Takens Transition to Chaos
In 1975 Gollub and Sweeny experimentally demonstrated a transition to turbulence of the sort predicted by Ruelle and Takens for a rotating fluid. This experiment did much to change the attitude of the scientific community toward the ideas of strange attractors and chaos.
A major open question is the extent of structural stability among such strange attractors. Collett and Eckmann (1980) could not determine the structural stability of the attractors underlying the Duffing and van der Pol oscillators. Hnon (1976) numerically estimated a much-studied attractor that is a structurally unstable Cantor set. The first structurally stable planar strange attractor to be discovered resembles a disc with three holes in it (Plykin, 1974).
However considerable controversy exists over the definition of the term Astrange attractor.@ Ruelle (1980, p. 131) defines it for a map F as being a bounded m-dimensional set A for which there exists a set U such that: 1) U is an m-dimensional neighborhood of A containing A, 2) any trajectory starting in U remains in U and approaches A as t-->4, 3) there is sensitive dependence on initial conditions (SDIC) for any point in U, and 4) A is indecomposable (same as irreducible), that is any two trajectories starting in A will eventually become arbitrarily close to each other.
Although accepted by many, this definition has come under criticism from two different directions. One argues that it is missing a condition, namely that the attractor have a fractal dimensionality. This is implied by the original Ruelle-Takens examples which possess Smale horseshoes and thus have Cantor sets or fractal dimensionality to them. But Ruelle=s definition above does not require this. Others who insist on fractal dimensionality as well as the above characteristics include Guckenheimer and Holmes (1990, p. 256) who define it as being a closed invariant attractor that contains a transversal homoclinic orbit (and therefore a Smale horseshoe). Peitgen, Jrgens, and Saupe (1992, p. 671) simply add to Ruelle=s definition that it have fractal dimension.
In the other direction is a large group that argues that it is the fractal nature of the attractor that makes it strange, not SDIC (Grebogi, Ott, Pelikan, and Yorke, 1984; Brindley and Kapitaniak, 1991). They call an attractor possessing SDIC a chaotic attractor and call those with fractal dimension but no SDIC strange nonchaotic attractors. It may well be that the best way out of this is to call the strange nonchaotic attractors Afractal attractors@ and to call those with SDIC Achaotic attractors.@ The term Astrange attractor@ could either be reserved for those which are both, or simply eliminated. But this latter is unlikely as the term seems to have a strange attraction for many.
2.3.2.2.4. APeriod Three Equals Chaos@ and Transitions to Chaos
Although a paper by the mathematical ecologist, Robert M. May (1974), had used the term earlier, it is widely claimed that the term chaos was introduced by Tien-Yien Li and James A. Yorke in their 1975 paper, APeriod Three Equals Chaos.@ Without doubt this paper did much to spread the concept. They proved a narrower version of a theorem established earlier by A.N. Sharkovsky (1964) of Kiev. But their deceptively simple version highlighted certain important aspects of chaotic systems, even as their definition of chaos, known as topological chaos, has come to be viewed as too narrow and missing crucial elements, notably sensitive dependence on initial conditions (SDIC) in the multi-dimensional case.
Essentially their theorem states that if f continuously maps an interval on the real number line into itself and it exhibits a three-period cycle (or, more generally a cycle wherein the fourth iteration is not on the same side of the first iteration as are the second and third), then: 1) cycles of every possible period will exist, 2) there will be an uncountably infinite set of aperiodic cycles which will both diverge to some extent from every other one, and also become arbitrarily close to every other one, and 3) that every aperiodic cycle will diverge to some extent from every periodic cycle. Thus, Aperiod three equals chaos.@
The importance of period three cycles was becoming clearer through work on transitions to chaos as a control (Atuning@) parameter is varied. We have seen above that Ruelle and Takens (1971) observed a pattern of transition involving a sequence of period-doubling bifurcations. This was extended by May (1974, 1976) who examined in more detail the sequence of period-doubling pitchfork bifurcations in the transition to chaos arising from varying the parameter, a, in the logistic equation
xt+1 = axt(1-xt), (2.15)
arguably the most studied equation in chaotic economic dynamics models.
Figure 2.18 shows the transition to chaos for the logistic equation as a varies. At a = 3.00 the single fixed point attractor bifurcates to a two-period cycle, followed by more period-doubling bifurcations as a increases with an accumulation point at a = 3.570 for the cycles of 2n as n-->4, beyond which is the chaotic regime which contains non-zero measure segments with SDIC. The first odd-period cycle appears at a = 3.6786 and the first three-period cycle appears at a = 3.8284, thus indicating the presence of every integer-period cycle according to the Li-Yorke Theorem. The Li-Yorke Theorem emphasizes that three-period cycles appear in chaotic zones after period-doubling bifurcation sequences have ended. Interestingly the three-period cycle appears in a Awindow@ in which the period-doubling sequence is reproduced on a smaller scale with the periods following the sequence, 3x2n. This window is shown in more detail in Figure 2.19.
Figure 2.18: Logistic Equation Transition to Chaos
Figure 2.19: Three-Period Window in Transition to Chaos
Inspired by Ruelle and Takens (1971) and work by Metropolis, Stein, and Stein (1973), the nature of these period-doubling cascades was more formally analyzed by Mitchell J. Feigenbaum (1978, 1980) who discovered the phenomenon of universality. In particular, during such a sequence of bifurcations, there is a definite rate at which the subsequent bifurcations come more quickly as they accumulate to the transition to chaos point. Thus, if Ln is the value of the tuning parameter at which the period doubles for the nth time, then
dn = (Ln+1-Ln)/(Ln+2-Ln+1). (2.16)
Feigenbaum shows that as n increases, dn very rapidly converges to a universal constant, d = 4.6692016... .
Closely related to this he also discovered another universal constant for period-doubling transitional systems, a, a scaling adjustment factor for the process. Let dn be the algebraic distance from x = 2 to the nearest element of the attractor cycle of period 2n in the 2n cycle at ln. This distance scales down for the 2n+1 cycle at ln+1 according to
dn/dn+1 -a. (2.17)
Feigenbaum discovered that a = 2.502907875... universally. Unsurprisingly, cascades of period-doubling bifurcations are called Feigenbaum cascades.
There are other possible transitions to chaos besides period-doubling. Another that can occur for functions mapping intervals into themselves involves intermittency and is associated with the tangent or saddle-node bifurcation (Pomeau and Manneville, 1980), a phenomenon experimentally demonstrated for a nonlinearly forced oscilloscope by Perez and Jeffries (1982). As the bifurcation is approached, the dynamics exhibit zones of long period cycles separated by bursts of aperiodic behavior, hence the term Aintermittency@ (Thompson and Stewart, 1986, pp. 170-173; Medio with Gallo, 1993, pp. 165-169).
Yet another one-dimensional case is that of the hysteretic chaotic blue-sky catastrophe (or chaostrophe) initially proposed by Abraham (1972; 1985a) in which a variation of a control parameter brings about a homoclinic orbit that destroys an attractor as its basin of attraction suddenly goes to infinity, the Ablue sky.@ This has been shown for the van der Pol oscillator (Thompson and Stewart, 1986, pp. 268-284) and is illustrated in Figure 2.20. A more general version of this is the chaotic contact bifurcation when a chaotic attractor contacts its basin boundary (Abraham, Gardini, and Mira, 1997).
Figure 2.20: Blue-Sky Catastrophe (Chaostrophe)
The theory of multidimensional transitions is less well understood, but it is thought based on experimental evidence that transitions through quasi-periodic cycles may be possible in this case (Thompson and Stewart, 1986, pp. 284-288). In the case of two interacting frequencies on the unit circle mapping into itself, such a transition would involve avoiding zones of mode-locking known as Arnol=d tongues (Arnol=d, 1965). It remains uncertain mathematically whether such a transition is possible with the experimental evidence possibly having been contaminated by noise (Thompson and Stewart, 1986, p. 288).
2.3.2.2.5. The Chaos of Definitions of Chaos
We have been gradually building up our picture of chaos. Chaotic dynamics are deterministic but seem random, lacking any periodicity. They are locally unstable in the sense of the butterfly effect (or SDIC), but are bounded. Initially adjacent trajectories can diverge, but will also eventually become arbitrarily close again. These are among the characteristics observed in the Lorenz (1963) model as well as those studied by May (1974, 1976) and in the theorem of Li and Yorke (1975) for the one-dimensional case.
However, despite the widespread agreement that these are core characteristics of chaotic dynamics, it has proven very difficult to come up with a universally accepted definition of chaos, with some surprisingly intense emotions erupting in the debates over this matter (observed personally by this author on more than one occasion). Some (Day, 1994) have stuck with the characteristics of the Li-Yorke Theorem given above as defining chaotic dynamics, perhaps in honor of their alleged coining of the term,@chaos.@ But the problem with this is that their theorem does not include SDIC as a characteristic and indeed does not guarantee the existence of SDIC beyond the one-dimensional case. And of all the characteristics identified with chaos, the butterfly effect is perhaps the most widely accepted and understood.
Yet another group, led by Mandelbrot (1983), insists that chaotic dynamics must involve some kind of fractal dimensionality of an attractor. And while many agree that fractality is a necessary component of being a strange attractor, most of these also accept that SDIC is the central key to chaotic dynamics, per se. Thus, Mandelbrot=s is a distinctly minority view.
Perhaps the most widely publicized definition of chaotic dynamics is due to Robert Devaney (1989, p. 50) and involves three parts. A map of a set into itself, f:V-->V, is chaotic if 1) it exhibits sensitive dependence on initial conditions (SDIC, the Abutterfly effect@), 2) is topologically transitive (same as indecomposable or irreducible), and 3) periodic points are dense in V (an element of regularity or Aorder out of chaos@).
A formal definition for a map f:V-->V of sensitive dependence on initial conditions is that depending on f and V there exists a d > 0 such that in every non-empty open subset of V there are two points whose eventual iterates under f will be separated by at least d. This does not say that such separation will occur between any two points, neither does it say that such a separation must occur exponentially, although some economists argue that this should be a condition for chaos, as they define chaos solely by the presence of a positive real part of the largest Lyapunov characteristic exponent which indicates exponential divergence (Brock, 1986; Brock, Hsieh, and LeBaron, 1991; Brock and Potter, 1993).
A formal definition of topological transitivity is that for f:V-->V if for any pair of open subsets of V, U and W, there exists a k > 0 such that the kth iterate, fk(U)1 W i. In effect this says that the map wanders throughout the set and is the essence of the indecomposability that many claim is a necessary condition for a set to be an attractor.
A formal definition of denseness is that a subset U of V is dense if the closure of U = V (closure means union of set with its limit points). Thus, for Devaney, the closure of the set of periodic points of the map f:V-->V must equal V. This implies that, much as in the Li-Yorke Theorem, there must be at least a countably infinite set of such trajectories and that they just about fill the set. Of the three conditions proposed by Devaney, this latter has been perhaps the most controversial. Indeed, it is not used by Wiggins (1990) who defines chaos only by SDIC and topological transitivity.
A serious problem is that denseness does not guarantee that the periodic points (much less those exhibiting SDIC) constitute a set of positive Lebesgue measure, that is are observable in any empirical sense. An example of a dense set of zero Lebesgue measure is the rational numbers. Their total length in the real number line is zero, implying a zero probability of randomly selecting a rational number out of the real number line.
This view of Devaney=s is more topological and contrasts with a more metrical view by those such as Eckmann and Ruelle (1985) who insist that one must not bother with situations in which Lebesgue measure is zero (Athin chaos@) and in which one cannot observe anything. There has been much discussion of whether given models exhibit positive Lebesgue measure for the sets of points for which chaotic dynamics can occur (Athick chaos@), a discussion affected by what one means by chaotic dynamics. Nusse (1987) insists that chaotic dynamics are strictly those with aperiodic flow rather than flows of arbitrary length, and Melse and Transue (1986) argue that for many systems the points for these constitute measure zero, although Lasota and Mackey (1985) present a counterexample. Day (1986) and Lorenz (1989) argue that arbitrary periodicities may behave like chaos for all practical purposes. Drawing on work of Sinai (1972) and Bowen and Ruelle (1975), Eckmann and Ruelle (1985) present a theory of ergodic chaos in which the observability of chaos is given by the existence of invariant ergodic SRB (Sinai-Ruelle-Bowen) measures that are absolutely continuous with respect to Lebesgue measure along the unstable manifolds of the system, drawing on the earlier work of Sinai (1972) and Bowen and Ruelle (1975). One reason for the widespread use of the piecewise-linear tent map in models of chaotic economic dynamics has been that it generates ergodic chaotic outcomes.
Yet another source of controversy surrounding Devaney=s definition involves the possible redundancy of some of the conditions, especially in the one-dimensional case. Thus, Banks, Brooks, Cairns, Davis, and Stacey (1992) show that topological transitivity and dense periodic points guarantee SDIC, thus making the most famous characteristic of chaos a redundant one, not a fundamental one. For the one-dimensional case of intervals on the real number line, Vellekoop and Berglund (1994) show that topological transitivity implies dense periodic points. Thus both SDIC and dense periodic points are redundant in that case, which underlies why the Li-Yorke Theorem can imply that the broader conditions of chaos hold for the one-dimensional case. In the multidimensional case, Wiggins (1990, pp. 608-611) lays out various possibilities with an exponential example that shows SDIC and topological transitivity but no periodic points on a noncompact set, a sine function on a torus example with SDIC and countably infinite periodic points but only limited topological transitivity, and an integrable twist map example that shows SDIC and dense periodic points but no topological transitivity at all. Clearly there is still a lot of Achaos@ in chaos theory.
2.3.2.2.6. The Empirical Estimation of Chaos
Despite its deductive redundance, sensitive dependence on initial conditions remains the centerpiece of chaos theory in most peoples= eyes. If it wasn=t the bestselling account by James Gleick (1987) of Edward Lorenz=s now immortal coffee break that brought the butterfly effect to the attention of the masses, it was Achaotician@ Jeff Goldblum=s showing water drops diverging on his hand in the movie version of Jurassic Park. Thus it is unsurprising that most economists simply focus directly on SDIC in its exponential form as the single defining element of chaos, although it is generally also assumed that chaotic dynamics are bounded.
For observable dynamical systems with invariant SRB measures, among the most important of these are the Lyapunov characteristic exponents (LCEs, also known as AFloquet multipliers@). The general existence and character of these was established by Oseledec (1968) and their link with chaotic dynamics was more fully developed by Pesin (1977) and Ruelle (1979). In particular, if the largest real part of a dynamical system=s LCEs is positive, then that system exhibits SDIC, the butterfly effect. Thus Lyapunov characteristic exponents (or more commonly just ALyapunov exponents@) are the Holy Grail of chao-econometricians.
Let ft(x) be the t-th iterate of f starting at initial condition x, D be the derivative, v be a direction vector,
2 2 be the Euclidean distance norm, and ln the natural logarithm, then the largest Lyapunov characteristic exponent of f is
l1 = lim [ln(2Dft(x)Av2)/t]. (2.18)
t->4
This largest real part of the LCEs represents the exponential rate of divergence or convergence of nearby points in the system. If all the real parts of the LCEs are negative the system will be convergent. If l1 is zero, there may be a limit cycle, although convergence can occur in some cases. A l1 > 0 indicates divergence and thus SDIC. If more than one l has a real part that is positive the system is hyperchaotic (Rssler and Hudson, 1989; Thomsen, Mosekilde, and Sterman, 1991) and if there are many such positive l=s but they are all near zero, this is homeochaos (Kaneko, 1995).
An important interpretation of a positive l1 is that it represents the rate at which information or forecastibility is lost by the system (Sugihara and May, 1990; Wales, 1991), the idea being that short-term forecasting may be possible with deterministically chaotic systems, even if long-term is not. This suggests a deeper connection with measures of information. In particular, Kolmogorov (1958) and Sinai (1959) formalized a link between information and entropy initially proposed by Claude Shannon in 1948 (Peitgen, Jrgens, and Saupe, 1992, p. 730). This Kolmogorov-Sinai entropy is rarely exactly computable but is approximated by
K = lim lim {(1/t)[ln(Cm(e))/(Cm+1(e))]}, (2.19)
m->4 e->0
where t is the observation interval and Cm(e) is the correlation integral defined above in 2.3.2.1.4. as being the probability that for a radius e two randomly chosen points on a trajectory will be within e of each other (Grassberger and Procaccia, 1983b).
This entropy measure indicates the gain in information from having a finer partition of a set of data. Thus it is not surprising that it (the exact measure of K) can be related to the LCEs. In particular, the sum of the positive Lyapunov exponents will be less than or equal to the Kolmogorov-Sinai entropy. If there is absolute continuity on the unstable manifolds, and thus a unique invariant SRB measure, this relationship becomes an equality known as the Pesin (1977) equality. For two-dimensional mappings Young (1982) has shown that the Hausdorff dimension equals the entropy measure times the difference between the reciprocals of the two largest Lyapunov exponents.
Unsurprisingly, a major cottage industry has grown up in searching for the best algorithms and methods of statistical inference for estimating Lyapunov exponents. Broadly there have been two competing strands. One is the direct method, originally due to Wolf, Swift, Swinney, and Vastano (1985), which has undergone numerous refinements (Rosenstein, Collins, and de Luca, 1993; Bask, 1998) and which focuses on estimating just the maximum LCE. Its main rival is the Jacobian method, due originally to Eckmann, Kamphorst, Ruelle, and Ciliberto (1986), which uses the Jacobian matrix of partial derivatives which can estimate the full spectrum of the Lyapunov exponents and which has been improved by Genay and Dechert (1992) and McCaffrey, Ellner, Gallant, and Nychka (1992), although the latter focus only on estimating the dominant exponent, l1. However this method is subject to generating spurious LCEs associated with the embedding dimension being larger than the attractor=s dimension, although there are ways of partially dealing with this problem (Dechert and Genay, 1996).
The problem of the distributional theory for statistical inference for LCE estimates has been one of the most difficult in empirical chaos theory, but now may have been partialy solved. That it is a problem is seen by Brock and Sayers (1988) showing that many random series appear to have positive Lyapunov exponents according to some of the existing estimation methods. An asymptotic theory that establishes normality of the smoothing-based estimators of LCEs of the Jacobian-type approaches is due to Whang and Linton (1999), although it does not hold for all cases, is much weaker in the multidimensional case, and has very high data requirements.
A somewhat more ad hoc, although perhaps more practical procedure involves the use of the moving blocks version of Efron=s (1979) bootstrap technique (Genay, 1996). In effect this allows one to create a set of distributional statistics from an artificially created sample generated from successive blocks within the data series. Bask (1998) provides a clear description of how to use this approach, focusing on the direct method for estimating l1, and Bask and Genay (1998) apply it to the Hnon map.
We shall not review the multitude of econometric estimates of Lyapunov exponents at this point, as we shall be referring to these throughout this book. We note, however, that beginning with Barnett and Chen (1988) many researchers have found positive real parts for Lyapunov exponents in various economic time series, although until recently there were no confidence estimates for most of these. Critics have argued, however, that what is required (aside from overcoming biases due to inadequate data in many cases) is to find low-dimensional chaos that can allow one to make accurate out-of-sample forecasts. Critics who claim that this has yet to be achieved include Jaditz and Sayers (1993) and LeBaron (1994).
One response by some of those who argue for the presence of chaotic dynamics in economic time series has been to use alternative methods of measuring chaos. Some of these have attempted to directly estimate the topological structure of attractors by looking at close returns (Mindlin, Hou, Solari, Gilmore, and Tufilaro, 1990; C. Gilmore, 1993, R. Gilmore, 1998). Another approach is to estimate continuous chaos models that require an extra dimension, the first model of continuous chaos being due to Otto Rssler (1976). Wen (1996) argues that this avoids biases that appear due to the arbitrariness of time periods that can allow noise to enter into difference equation model approaches. An earlier method was to examine spectral densities (Bunow and Weiss, 1979). Pueyo (1997) proposes a randomization technique to study SDIC in small data series as found in ecology.
Finally we note that a whole battery of related techniques are used in the preliminary stages by researchers searching for chaos to show that linear or other nonlinear but nonchaotic specifications are inadequate. A variety of tests have been compared by Barnett, Gallant, Hinich, Jungeilges, Kaplan, and Jensen (1994, 1998), including the Hinich bispectrum test (1982), the BDS test, the Lyapunov estimator of Nychka, Ellner, Gallant, and McCaffrey (1992), White=s neural net estimator (1989), and Kaplan=s (1994) test, not a full set. One found to have considerable power by these researchers and one of the most widely used is the BDS test originally due to Brock, Dechert, and Scheinkman (1987), which tests against a null hypothesis that series is i.i.d., that is it is independently and identically distributed. The statistic uses the correlation integral, with n being the length of the data series and is
BDSm,n(e) = n1/2{[Cm,n(e)-Cn(e)m]/sm,n(e)}, (2.20)
with Cn(e)m being the asymptotic value of Cm,n(e) as n-->4 and s being the standard deviation.
Practical use of this statistic is discussed in Brock, Hsieh, and LeBaron (1991) and Brock, Dechert, LeBaron, and Scheinkman (1996). It can be used successively to test various transformations to see if there is remaining unexplained dependence in the series. But it is not itself directly a test for chaotic dynamics or any specific nonlinear form, despite using the correlation integral.
2.3.2.2.7. Controlling Chaos
It is a small step from learning to estimate chaos to wanting to control it if one can. Studying how to control chaos and actually doing it in some cases has been a major research area in chaos theory in the 1990s. This wave was set off by a paper on local control of chaos by Ott, Grebogi, and Yorke (1990), one by Shinbrot, Ott, Grebogi, and Yorke (1990) on a global targeting method of control using SDIC, and a paper showing experimentally the control of chaos by the local control method in an externally forced, vibrating magnetoelastic ribbon (Ditto, Rauseo, and Spano, 1990). Shinbrot, Ditto, Grebogi, Ott, Spano, and Yorke (1992) experimentally demonstrated the global SDIC method with the same magnetoelastic ribbon. Control of chaos has since been demonstrated in a wide variety of areas including mechanics, electronics, lasers, biology, and chemistry (Ditto, Spano, and Lindner, 1995).
One can argue that the control of chaos had been discussed earlier, that it is implicit in the idea of changing a control parameter in a major way to move a system out of a chaotic zone, as suggested by Grandmont (1985) in the context of a rational expectations macroeconomic model. But these methods all involve small perturbations of a control parameter that somehow stabilize the system while not moving it out of the chaotic zone.
The local control method due to Ott, Grebogi, and Yorke (1990), often called the OGY method, relies upon the fact that chaotic systems are dense in periodic orbits, and even contain fixed saddle points that have both stable and unstable manifolds going into them. There are three steps in this method, which has been extended to the multidimensional case by Romeiras, Ott, Grebogi, and Dayawansa (1992). The first is to identify an unstable periodic point by examining close returns in a Poincar section. The second is to identify the local structure of the attractor using the embedding and reconstruction techniques described above, with particular emphasis on locating the stable and unstable manifolds. The final part is to determine the response of the attractor to an external stimulus on a control parameter, which is the most difficult step. The ultimate goal is to determine the location of a stable manifold near where the system is and
then to slightly perturb the system so that it moves on to the stable manifold and approaches the periodic point. Ott, Grebogi, and Yorke (1990) provide a formula for the amount of parameter change needed that depends on the parallels and perpendicular eigenvalues of the unstable manifold about the fixed point, the distance of the system from the fixed point, and the responsiveness of the fixed point itself to changes in the control parameter.
The main problems with this method are that once on the stable manifold it can take a long time to get to the periodic point during when it can go through a variety of complex transients. Also, given this long approach, it can be disturbed by noise and knocked off the stable manifold. Dressler and Nitsche (1991) stress the need for constant readjustment of the control parameter to keep it on the stable manifold and Aston and Bird (1997) show how the basin of attraction can be expanded for the OGY technique.
The global targeting method of Shinbrot, Ott, Grebogi, and Yorke (1990) avoids the problem of transients during a long delay of approach. It starts by considering possible next step iterates of the system as a set of points and then looks at further iterates of this set which will diverge from each other and begin wandering all over the space because of SDIC. The goal is to find an iterate that will put the system within a particular small neighborhood. If the observer has sufficiently precise knowledge of the system, this can then be achieved usually with a fairly small number of iterations.
The main problems with this method are that it requires a much greater degree of knowledge about the global dynamics of the system than does the OGY method and that it will only get one to a neighborhood rather than a particular point. Thus, one gains speed but loses precision. But in a noisy environment such as the economy, speed may equal precision.
There have now been several applications in economics. Holyst, Hagel, Haag, and Weidlich (1996) apply the OGY method to a case of two competing firms with asymmetric investment strategies and Haag, Hagel, and Sigg (1997) apply the OGY method to stabilizing a chaotic urban system, while Kopel (1997) applies the global targeting method to a model of disequilibrium dynamics with financial feedbacks as do Bala, Majumdar, and Mitra (1998) to a model of ttonnement adjustment. Kaas (1998) suggests the application of both in a macroeconomic stabilization context. The global method is used to get the system within the neighborhood of a stable manifold that will take the system to a desirable location, and then the OGY method is used to actually get it there by local perturbations.
Unsurprisingly, all of these observers are very conscious of the difficulty of obtaining sufficient data for actually using either of these methods and of the severe problems that noise can create in trying to do so. Given that we are still debating whether or not there actually even is deterministic economic chaos of whatever dimension, we are certainly rather far from actually controlling any that does exist.
2.4. The Special Path to Fractal Geometry
Fractal geometry is the brainchild of the idiosyncratic genius, Benoit Mandelbrot. Many of its ideas have been enumerated in earlier sections, and there is clearly a connection with chaos theory. Certainly the notions of Afractal dimension@ and Afractal set@ are important in the concept of strange attractors, or Afractal attractors@ as Mandelbrot prefers to call them. The very idea of attempting to measure the Aregularity of irregularity@ or the Aorder in chaos@ is a central theme of Mandelbrot=s work.
Like Ren Thom, Mandelbrot claims to have discovered an all-embracing world-explaining theory. This has gotten him in trouble with other mathematicians. Just as Thom claims that chaos theory is an extension of catastrophe theory, so Mandelbrot claims that it is an extension of fractal geometry. The reaction of many mainstream bifurcation theorists is to pretend that he is not there. Thus, neither he nor the word Afractal@ appear in Guckenheimer and Holmes=s comprehensive Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1990). At least Thom and Zeeman rate brief mentions in that book, even though Mandelbrot is probably closer in spirit to Guckenheimer and Holmes than are Thom or Zeeman.
But then Mandelbrot ignores Thom and Zeeman, not mentioning either or catastrophe theory in his magisterial Fractal Geometry of Nature (1983). At least Thom at one point (1983, p. 107) explicitly recognizes that Mandelbrot has presented what Thom labels Ageneralized catastrophes@ and mentions him by name in this context.
But there is a deep philosophical divide between Thom and Mandelbrot that goes beyond the appropriate labels for mathematical objects or who had the prettiest pictures in Scientific American, a contest easily won by Mandelbrot with his justly famous Mandelbrot Set (see especially Peitgen, Jrgens, and Saupe, 1992). It is also despite Thom and Mandelbrot both favoring geometric over algebraic or metric approaches, as well as less formal notions of proof, in contrast with the Russian School and mainstream bifurcation theorists such as Guckenheimer and Holmes. The latter are intermediate between Thom and Mandelbrot who divide on whether the world is fundamentally stable and well-ordered or fundamentally irregular, with Thom holding the former position and Mandelbrot holding the latter. Mandelbrot is the vanguard of the radical chaos position, as Mirowski (1990) argues, in sharp contrast with the relative continuity and order of Thom=s position. Both see reality as a balance of order and chaos, of continuity and discontinuity, but with the two sides operating at different levels and relating in different ways.
Although Thom parades as the prophet of discontinuity, he is fixated on structural stability, part of the title of his most famous book. Between catastrophe points, Thom sees dynamic systems evolving continuously and smoothly. His major innovation is the concept of transversality, central to proving the structural stability of the elementary catastrophes, the basis for their claimed universal significance and applicability. For Thom to carry out his wide ranging qualitative analysis of linguistic and other structures, he must believe in the underlying well-ordered nature of the universe, even if the order is determined by the pattern of its stable discontinuities.
But Mandelbrot will have none of this. For him, the closer one looks at reality, the more irregular and fragmented it becomes. In the second chapter of Fractal Geometry of Nature he quotes at length from Jean Perrin (1906) who won a Nobel Prize for studying Brownian motion. Perrin invokes a vision of matter possessing Ainfinitely granular structure.@ At a sufficiently small scale finite volumes and densities and smooth surfaces vanish into the Aemptiness of intra-atomic space@ where Atrue density vanishes almost everywhere, except at an infinite number of isolated points where it reaches an infinite value.@ Ultimate reality is an infinitely discontinuous Cantor set.
Mandelbrot=s vision of ultimate discontinuity carries over to his view of economics. Mandelbrot claims that the original inspiration for his notions of fractal measures and self-similar structures came from work he did on random walk theory of stock prices and cotton prices (Mandelbrot, 1963).It has often been thought that the random walk theory was inspired by the Brownian motion theory. But Mandelbrot (1983) argues persuasively that Brownian motion theory was preceded, if not directly inspired, by the random walk theory of speculative prices due to Louis Bachelier (1900).
Thus Mandelbrot=s view of price movements in competitive markets is one of profound and extreme discontinuity, in sharp contrast with most views in economics. His radically discontinuous view is exemplified by the following quotation (Mandelbrot, 1983, pp. 334-335):
ABut prices on competitive markets need not be,
continuous, and they are conspicuously discontinuous.
The only reason for assuming continuity is that many
sciences tend, knowingly or not, to copy the procedures
that prove successful in Newtonian physics. Continuity
should prove a reasonable assumption for diverse
Aexogenous@ quantities and rates that enter economics
but are defined in purely physical terms. But prices
are different: mechanics involves nothing comparable,
and gives no guidance on this account.
The typical mechanism of price formation involves
both knowledge of the present and anticipation of the
future. Even where the exogenous physical determinism
of a price vary continuously, anticipations change
drastically, Ain a flash.@ When a physical signal of
negligible energy and duration, Athe stroke of a pen,@
provokes a brutal change of anticipations, and when no
institution injects inertia to complicate matters, a
price determined on the basis of anticipation can crash
to zero, soar out of sight, do anything.@
We note again that despite his assertions of universal irregularity, Mandelbrot constantly seeks the hidden order in the apparent chaos.
2.5. The Complexity of Other Forms of Complexity
2.5.1. What is Complexity?
John Horgan (1995, 1997) has made much in a negative light of a claimed succession from cybernetics to catastrophe to chaos to complexity theory, labeling the practitioners of the latter two, Achaoplexologists.@ Certainly such a succession can be identified through key individuals in various disciplines, but the issue arises as to what is the relationship between these? One approach is to allocate to the last of them the most general nature and view the others as subcategories of it. This then puts the burden squarely on how we define complexity. As Horgan has pointed out, there are numerous definitions of complexity around, more than 45 by the latest count of Seth Lloyd of MIT, so many that we have gone Afrom complexity to perplexity@ according to Horgan.
Although some of our discussions above of entropy and dimension measures point us towards some alternative definitions of complexity, we shall stick with one tied more clearly to nonlinear dynamics and which can encompass both catastrophic and chaotic dynamics, as well as the earlier cybernetics of Norbert Wiener (1948) that Jay Forrester (1961) first applied to economics. Due to Richard Day (1994), this definition calls a nonlinear dynamical system complex if for nonstochastic reasons it does not go to either a fixed point, to a limit cycle, or explode. This implies that it must be a nonlinear system, although not all nonlinear systems are complex, e.g. the exponential function. It also implies that the dynamics are bounded and endogenously generated. All of this easily allows for the earlier three of the Afour C=s.@
But then we need to know what distinguishes Apure complexity@ from these earlier three. Arthur, Durlauf, and Lane (1997), speaking for the ASanta Fe perspective,@ identify six characteristics associated with Athe complexity approach@: 1) dispersed interaction among heterogeneous agents, 2) no global controller, 3) cross-cutting hierarchical organization, 4) continual adaptation, 5) perpetual novelty (4 and 5 guaranteeing an evolutionary perspective), and 6) out-of-equilibrium dynamics. Of these the first may be the most important and underpins another idea often associated with complex dynamics, namely emergent structure, that higher-order patterns or entities emerge from the interactions of lower-order entities (Baas, 1997). These ideas are reasonably consistent with those of older centers than Santa Fe of what is now called complexity research, namely Brussels where Ilya Prigogine (1980) has been the key figure and Stuttgart where Hermann Haken (1977) has been the key figure.
Many of these characteristics also apply to cybernetics as well, but a notable contrast appears with both catastrophe and chaos theory. These two are often stated in terms of a small number of agents, possibly as few as one; there might be a global controller; there may be no hierarchy, much less a cross-cutting one; neither adaptation nor novelty are guaranteed, although they might happen, and equilibrium is not out of the question. So, there are some real conceptual differences, even though the complexity approaches contain and use many ideas from catastrophe and chaos theory. One implication of these differences is that it is much easier to achieve analytical results with catastrophe and chaos theory, whereas in the complexity models one is more likely to see the use of computer simulations to demonstrate results, whether in Brussels, Stuttgart, Santa Fe, or elsewhere.
2.5.2. Discontinuity and Statistical Mechanics
An approach borrowed from physics in economics that has become very popular among the Santa Fe Institute (SFI) complexologists is that of statistical mechanics, the study of the interaction of particles. Such systems are known as interacting particle systems (IPS) models, as spin-glass models, or as Ising models, although the term Aspin glass@ properly only applies when negative interactions are allowed (Durlauf, 1997). The original use of these models was to model phase transitions in matter, spontaneous magnetizations or changes from solid to liquid states, and so forth. Kac (1968), Spitzer (1971), Sherrington and Kirkpatrick (1975), Liggett (1985), and Ellis (1985) present mathematical and physical foundations of these models and the conditions in them under which discontinuous phase transitions will occur.
Their first application in economics was by Hans Fllmer (1974) in a model of local interaction with a conditional probability structure on agent characteristics. Idiosyncratic shocks can generate aggregate consequences, a result further developed in Durlauf (1991) for business cycles. A major development was the introduction into this model by Brock (1993), Blume (1993), and Brock and Durlauf (1995) of the discrete choice theory by agents of Manski and McFadden (1981) and Anderson, de Palma, and Thisse (1992). A particularly influential version of this was the mean field approach introduced by Brock (1993).
Let there be n individuals who can choose from a discrete choice set {1,-1} with m representing the average of the choices by the agents, J a strength of interaction between them, an intensity of choice parameter b (interpreted as Ainverse temperature@ in the physics models of material phase transitions), a parameter describing the probabilistic state of the system, h, which shows the utility gain from switching to a positive attitude, and an independent and identically distributed extreme value exogenous stochastic process. In this simple model, utility maximization leads to the Curie-Weiss mean field equation with tanh being the hypertangent:
m = tanh(bJm + bh). (2.21)
This equation admits of a bifurcation at bJ = 1 at which a phase transition occurs. Below this value m = 0 if h = 0, but above this value there will be two solutions with m- = -m+. If h 0 then for bJ > 1 there will be a threshold H such that if bh exceeds it there will be unique solution, but if bh < 0 then there will be three solutions, one with the same sign as h and the other two with opposite sign (Durlauf, 1997, p. 88). Brock (1993) and Durlauf (1997) review numerous applications, some of which we shall see later in this book. Brock and Durlauf (1999) discuss ways of dealing with the deep identification problems associated with econometrically estimating such models as noted by Manski (1993, 1995). A general weakness of this approach is its emphasis on binary choices, although Yeomans (1992) offers an alternative to this. In any case this approach exhibits discontinuous emergence, how interactions among agents can lead to a discontinuous phase transition in which the nature of the system suddenly changes.
Figure 2.21 shows the bifurcation for this mean field equation (from Rosser, 1999b).
Figure 2.21: Interacting Particle Systems Mean Field Solutions
A curious link between the mean field version of interacting particle systems models and chaotic dynamics has been studied by Shibata and Kaneko (1998). Kaneko (1990) initiated the study of globally coupled logistic map systems. Shibata and Kaneko consider the emergence of self-coherent collective behavior in zones of such systems with networks of entities that are behaving chaotically independently (from logistic equations). In the windows of periodicity within the chaotic zones, tongue-like structures can emerge within which this coherent collective behavior can occur. Within these structures internal bifurcations can occur even as the basic control parameters remain constant as the mean field interacting particle system dynamics accumulate to critical points. This model has not been applied to economics yet, but one possibility for a modified version might be in providing mechanisms for finding coherences that would allow overcoming the Manski (1993, 1995) identification problems in such systems as Brock and Durlauf (1999) note that nonlinear models actually allow possible solutions not available to linear models because of the additional information that they can provide.
2.5.3. Self-Organized Criticality and the >Edge of Chaos=
Another approach that is popular with the Santa Fe Institute complexity crowd is that of self-organized criticality, due to Per Bak and others (Bak, Tang, and Wiesenfeld, 1987; Bak and Chen, 1991) with extended development by Bak (1996). This approach shares with the Brussels School approach of Prigogine an emphasis on out-of-equilibrium states and processes. Agents are arrayed in a lattice that determines the structure of their interactions. In a macroeconomics example due to Bak, Chen, Scheinkman, and Woodford (1993) this reflects a demand-supply structure of an economy. There is a Gaussian random exogenous bombardment of the system with demand shocks that trigger responses throughout the system as it tries to maintain minimum inventories. The system evolves to a state of self-organized criticality where these bombardments sometimes trigger chain reactions throughout the system that are much larger than the original shock.
A widely used metaphor for these is sandpile models. Sand is dropped from above in a random way. The long-run equilibrium is for it to be flat on the ground, but it builds up into a sandpile. At certain critical points a drop of sand will trigger an avalanche that restructures the sandpile. The distribution of these avalanches follows a power law that generates a skewed distribution with a long tail out towards the avalanches, in comparison with the normal distribution of the exogenous shocks.
More formally for the Bak, Chen, Scheinkman, Woodford (1993) macro model, letting y = aggregate output, n = the number of final buyers (which is large), t be a parameter tied to the dimensionality of the lattice, and G be the gamma function of probability theory, then the asymptotic distribution of y will be given by
Q(y) = 1/n1/t{sin(pt/2)G(1+t)/[(y/n1/t)1+t]}. (2.22)
The borderline instability character of these models has led them to be associated with another class of models that have been associated with the Santa Fe group, although more controversially. This is the edge of chaos concept associated with modelers of artificial life such as Chris Langton (1990) and of biological evolution such as Stuart Kauffman (1993, 1995, and originally Kauffman and Johnsen, 1991). This idea has been identified by some popularizers (Waldrop, 1992) as the central concept of the Santa Fe Institute=s complexity approach, although that has been disputed by some associated with the SFI (see Horgan, 1995, 1997).
Generally the concept of chaos used by this group is not the same as that we have been using, although there are exceptions such as Kaneko=s (1995) use of homeochaos to generate edge of chaos self-organization. Rather it is a condition of complete disorder as defined in informational terms. The edge of chaos modelers simulate systems of many interacting agents through cellular automata models or genetic algorithms such as those of Holland (1992) and observe that in many cases there will be a large zone of complete order and a large zone of complete disorder. In neither of these does much of interest happen. But on their borderline, the proverbial Aedge of chaos,@ self-organization happens and structures emerge. Kauffman has gone so far as to see this as the model for the origins of life.
Although we shall see considerable use of the self-organization concept in economics, rarely has it directly followed the lines of the edge of chaos theorists, despite a few efforts by Kauffman in particular (Darley and Kauffman, 1997) and Kauffman=s work exerting a more general influence.
2.5.4. A Synergetics Synthesis
Finally we contemplate how the Stuttgart School approach of synergetics (Haken, 1977) might offer a possible synthesis that can debifurcate bifurcation theory and give a semblance of order to the House of Discontinuity. Reasonable continuity and stability can exist for many processes and structures at certain scales of perception and analysis while at other scales quantum chaotic discontinuity reigns. It may be God or the Law of Large Numbers which accounts for this seemingly paradoxical coexistence.
At the large scale where many processes and structures appear continuous and stable much of the time, important changes may occur discontinuously, perhaps as the result of complex emergent processes or phase transitions bubbling up from below, perhaps as high level catastrophic bifurcations. In turn, chaotic oscillations can arise out of the fractal process of a cascade of period-doubling bifurcations, with discontinuities appearing at the bifurcation points and most dramatically at the accumulation point where chaos emerges.
These can be subsumed under the synergetics perspective which operates on the principle of adiabatic approximation, which in the hands of Wolfgang Weidlich (Weidlich, 1991; Weidlich and Braun, 1992) and his master equation approach can admit of numerous interacting agents making probabilistic transitions within economic models. In the version of Haken (1983, 1996), a complex system is divided into Aorder parameters@ that change slowly and Aslave@ fast moving variables or subsystems, usually defined as linear combinations of underlying variables, which can create difficulties for interpretation in economic models. This division corresponds to the division in catastrophe theory between Aslow dynamics@ (control variables) and Afast dynamics@ (state variables). Nevertheless stochastic perturbations are constantly occurring leading to structural change when these occur near bifurcation points of the order parameters.
Let the slow dynamics be generated by a vector F and the fast dynamics by a vector q. Let A,B, and C be matrices and e be a stochastic noise vector. A general locally linearized model is given by
.
q = Aq + B(F)q C(F) + e. (2.23)
Adiabatic approximation allows this to be transformed into
.
q = -(A + B(F))-1C(F). (2.24)
Thus the fast variable dependence on the slow variables is determined by A + B(F). Order parameters will those of the least absolute value, a hierarchy existing of these.
A curious aspect of this is that the Aorder parameters@ are dynamically unstable in possessing positive real parts of their eigenvalues while the Aslave variables@ will exhibit the opposite. Haken (1983, Chap. 12) argues that chaos occurs when there is a destabilization of a formerly Aslaved@ mode which may then Arevolt@ (Diener and Poston, 1984) and become a control parameter. Thus, chaotic dynamics may be associated with a deeper catastrophe or restructuring and the emergence of a whole new order, an idea very much in tune with the Aorder out of chaos@ notions of the Brussels School (Prigogine and Stengers, 1984). And so, perhaps, a synergetics synthesis can pave the way to peace in the much-bifurcated House of Discontinuity.
NOTES
. There are cases where discontinuity replaces nonlinearity. Thus piecewise linear models of the variance of stock returns using regime switching models may outperform models with nonlinear ARCH/GARCH specifications regarding volatility clustering, especially when the crash of 1987 is viewed as a regime switching point (de Lima, 1996). Such models are of course nonlinear, strictly speaking.
. Smale (1990) argues that this problem reappears in computer science in that modern computers are digital and discrete and thus face fundamental problems in representing the continuous nature of real numbers. This issue shows up in its most practical form as the roundoff problem, which played an important role in Edward Lorenz=s (1963) observation of chaos.
. Despite his Laplacian image, Walras was aware of the possibility of multiple equilibria and of various kinds of complex economic dynamics (see Day and Pianigiani, 1991; Bala and Majumdar, 1992; Day, 1994; Rosser, 1999a for further discussion). After all, Attonnement@ means Agroping.@
. It was actually from this case that Ren Thom adopted the term Acatastrophe@ for the current theory of that name.
. Otto Rssler (1997, Chap. 1) argues that the idea of fractal self-similarity to smaller and smaller scales can be discerned in the writings of the pre-Socratic philosopher, Anaxagoras.
. That the Cantor set is in some sense a zero set while not being an empty set has led El Naschie (1994) to distinguish between zero sets, almost empty sets, and totally empty sets. Mandelbrot (1990a) has developed the notion of negative fractal dimensions to measure Ahow empty is an empty set.@
The idea that something can be Aalmost zero@ lay behind the idea of infinitesimals when calculus was originally invented, with Newton=s Afluxions@ and Leibniz=s Amonads.@ This was rejected later, especially by Weierstrass, but has been revived with non-standard analysis that allows for infinite real numbers and their reciprocals, infinitesimal real numbers, not equal to but closer to zero than any finite real number (Robinson, 1966).
. Mandelbrot (1983) and Peitgen, Jrgens, and Saupe (1992) present detailed discussions and vivid illustrations of these and other such sets.
. Poincar (1908) also addressed the question of shorter run divergences, what is now called sensitive dependence on initial conditions, the generally accepted sine qua non of chaotic dynamics. But he was preceded in his recognition of the possibility of this by Hadamard=s (1898) study of flows on negatively curved geodesic surfaces, despite the concept being implicit in Poincar=s earlier work, according to Ruelle (1991). Loua (1997, p. 216) credits James Clerk Maxwell as preceding both Hadamard and Poincar with his discussion in 1876 (p. 443) of Athat class of phenomena such that a spark kindles a forest, a rock creates an avalanche or a word prevents an action.@
. Although Poincar was the main formal developer of bifurcation theory, it had numerous precursors. Arnol=d (1992, Appendix) credits Huygens in 1654 with discovering the stability of cusp points in caustics and on wave fronts, Hamilton in 1837-1838 with studying critical points in geometrical optics, and numerous algebraic geometers of the late nineteenth century, including Cayley, Kronecker, and Bertini, among others, with understanding the typical singularities of curves and smooth surfaces, to the point that discussions of these were in some algebraic geometry textbooks by the end of the century.
. A local bifurcation involves qualitative dynamical changes near the equilibria. Global bifurcations are such changes that do not involve changes in fixed point equilibria, the first known example being the blue-sky catastrophe (Abraham, 1972). The attractor discontinuously disappears into the Ablue sky.@ Examples of these latter can occur in transitions to chaos, to be discussed later in this chapter. If the blue-sky catastrophe is achieved by a perturbed forced oscillation that leads to homoclinic transversal intersections during the bifurcation event, this is known as a blue-bagel chaostrophe (Abraham, 1985a). In three-dimensional maps such an event is a fractal torus crisis (Grebogi, Ott, and Yorke, 1983). For more discussion of global bifurcations see Thompson (1992) on indeterminate bifurcations, Palis and Takens (1993) on homoclinic tangent bifurcations, and Mira (1987) and Abraham, Gardini, and Mira (1997) on chaotic contact bifurcations.
. Although we shall not generally do so, some observers distinguish attractors from attracting sets (Eckmann and Ruelle, 1985). The former are subsets of the latter that are indecomposable (topologically transitive).
. For a complete classification and analysis of singularities see Arnol=d, Gusein-Zade, and Varchenko (1985).
. For the link between transversality and the classification of singularities see Golubitsky and Guillemin (1973).
. Apparently this number is only true for the case of generic metrics and potentials. The number of locally topologically distinct bifurcations in more generalized gradient dynamical systems depending on three parameters is much greater than conjectured by Thom and may even be infinite for the case of such systems depending on four parameters (Arnol=d, 1992, Preface).
. Although it has been rather sparsely applied in economics since the end of the 1970s, catastrophe theory has been much more widely applied in the psychology literature (Guastello, 1995).
. Arnol=d (1992, Appendix) suggests that this application was implicit in some of Leonardo da Vinci=s work who studied light caustics.
. M.V. Berry (1976) presents rigorous applications to the hydrodynamics of waves breaking. Arnol=d (1976) deals with both waves and caustics.
. See Gilmore (1981) for discussion of quantum mechanics applications of catastrophe theory.
. See Guastello (1995) for studies of stress using catastrophe theory, along with a variety of other psychology and organization theory applications.
. Thompson and Hunt (1973, 1975) use catastrophe theory to analyze Euler buckling.
. Horgan=s jibes are part of a broader blast at Athe four C=s,@ the allegedly overhyped Acybernetics, catastrophe, chaos, and complexity.@ However, he uses the supposedly total ill-repute of the first two to bash the second two, which he sardonically conflates as Achaoplexity.@ See Rosser (1999b) for further discussion of Horgan=s ideas along these lines.
. One such, advocated by Thom himself, is the dialectical approach that sees qualitative change arising from quantitative change. This Hegelian perspective is easily put into a catastrophe theory framework where the quantitative change is the slow change of a control variable that at a bifurcation point triggers a discontinuous change in a state variable (Rosser, 1999c).
. Oliva, Desarbo, Day, and Jedidi (1987) use a generalized multivariate method (GEMCAT) for estimating cusp catastrophe models. Guastello (1995, p. 70) argues that this technique is subject to Type I errors due to the large number of models and parameters estimated although it may be useful as an atheoretic exploratory technique.
. This is clearly idealistic. I am not so naive as to advise junior faculty in economics attempting to obtain tenure to spend lots of time now writing and submitting to leading economics journals papers based on catastrophe theory.
. For more detailed analysis of problems in defining attractors, see Milnor (1985).
. An apparent case of observed chaotic dynamics in celestial mechanics is the unpredictable Atumbling@ rotation of Saturn=s irregularly shaped moon, Hyperion (Stewart, 1989, pp. 248-252).
. For generalizations of the Takens approach see Sauer, Yorke, and Casdagli (1991). For problems with attractor reconstruction in the presence of noise see Casdagli, Eubank, Farmer, and Gibson (1991). For dealing with small sample problems see Brock and Dechert (1991). For an overview of embedding issues see Ott, Sauer, and Yorke (1994, Chapter 5).
. A related concept is that of order (Savit and Green, 1991; Cheng and Tong, 1992), roughly the number of successive elements in a time series which determine the state of the underlying system. Order is at least as great as the correlation dimension (Takens, 1996).
. For more general limits to estimating the correlation dimension, see Eckmann and Ruelle (1991) and Stefanovska, Strle, and Kroelj (1997). In some cases these limits are related to the Takens embedding theorem. Brock and Sayers (1988), Frank and Stengos (1988a), and Scheinkman and LeBaron (1989) suggest for doubtful cases fitting an AR model and then estimating the dimension which should be the same. This is the residual diagnostic test.
. One point of contention involves whether or not a Afractal set@ must have the Aself-similarity@ aspect of smaller scale versions reproducing larger scale versions, as one sees in the original Cantor set and the Koch curve. Some insist on this aspect for true fractality, but the more general view is the one given in this book that does not require this.
. For applications of multifractals in financial economics, see Mandelbrot (1997) and Mandelbrot, Fisher, and Calvet (1997).
. Even more topologically complicated are situations where sections of boundaries may be in three or more basins of attraction simultaneously, a situation known as basins of Wada (Kennedy and Yorke, 1991), first observed by Yoneyama (1917). This can occur in the Hnon attractor (Nusse and Yorke, 1996).
. Crannell (1995) speculates that the phrase dates to a 1953 Ray Bradbury story, AA Sound of Thunder,@ in which a time traveler changes the course of history by stepping on a prehistoric butterfly.
Edward Lorenz (1993) reports that he was unaware of this story when he coined the phrase in a talk he gave in 1972. This talk appears as an appendix in E.N. Lorenz (1993). Lorenz (1993) also speculates that part of the popularity of the phrase, which he attributes to the popularity of Gleick=s (1987) book, came from the butterfly appearance of the Lorenz attractor. He originally thought of using a sea gull instead of a butterfly in his 1972 talk and reports (1993, p. 15) that it was an old line among meteorologists that a man sneezing in China could set people in New York to shoveling snow.
. The Russian School was close on Smale=s heels as Shilnikov (1965) showed under certain conditions near a three-dimensional homoclinic orbit to a saddle point that a countably infinite set of horseshoes will exist.
. A curious fact about Richardson is that measurements he made of the length of Britain=s coastline using different scales of measurement inspired Mandelbrot=s concept of fractal dimension (Mandelbrot, 1983, Chap. 5).
. Ulam and von Neumann (1947) studied the logistic equation as a possible deterministic random number generator.
. May (1976) was the first to consciously suggest the application of chaos theory to economics and proposed a number of possible such applications that were later carried out by economists, generally with no recognition of May=s earlier suggestions, although his paper has been widely cited by economists. Ironically it was originally submitted to Econometrica which rejected it before it was accepted by Nature.
There was a much earlier paper in economics by Strotz, McAnulty, and Naines (1953) that discovered the possibility of business cycles of an infinite number of periods as well as completely wild orbits depending on initial conditions in a version of the Goodwin (1951) nonlinear accelerator model. But they did not fully appreciate the mathematical implications of what they had found, arguably the first demonstration of chaotic dynamics in economics.
. Shibata and Kaneko (1998) show for globally coupled logistic maps, tongue-like structures can arise from these windows of periodic behavior in the chaotic zone of the logistic map in which self-consistent coherent collective behavior can arise. Kaneko (1990) initiated the study of such globally coupled maps.
. See Cvitanovic (1984) for more thorough discussion of universality and related issues.
. For more detailed classification of period-doubling sequences, see Kuznetsov, Kuznetsov, and Sataev (1997).
. That a property that brings trajectories back toward each other rather than mere boundedness is a part of chaos can be seen by considering the case of path dependence, the idea that at a crucial point random perturbations can push a system toward one or another path that then maintains itself through some kind of increasing returns, a case where Ahistory matters@ with distinct multiple equilibria (Arthur, 1988, 1989, 1990, 1994). In this case there is a butterfly effect of sorts, but there is not the sort of irregularity of the trajectories that we identify with chaotic dynamics. The trajectories simply move apart and stay apart. The same can be said for the sort of crucial historical accidents exemplified by: AFor want of a nail the shoe was lost; for want of the shoe the horse was lost; for want of the horse the battle was lost; for want of the battle the kingdom was lost@ (McCloskey, 1991).
. Although only briefly dealing with the definition-of-chaos issue, the intensity of polemics sometimes surrounding this topic can be seen in print in the exchange between Helena Nusse (1994a,b) and Alfredo Medio (1994).
. An example would be a map of the unit circle onto itself consisting of a one-third rotation. This would generate a three-period cycle but would certainly not exhibit SDIC or any other accepted characteristic of chaotic dynamics (I thank Cars Hommes for this example).
. Even James Yorke of the Li-Yorke Theorem would appear to have accepted this centrality of SDIC for chaos given that he was a co-coiner of the term Anonchaotic strange attractors,@ for attractors with fractal dimension but without SDIC (Grebogi, Ott, Pelikan, and Yorke, 1984).
. Another difference between Devaney and Wiggins is that the latter imposes a condition that the set V must be compact (closed and bounded in real number space) whereas Devaney leaves the nature of V open. Many observers take an intermediate position by requiring V to be a subset of n-dimensional real number space, Rn.
. A time series is Aergodic@ if its time average equals its space average (Arnol=d and Avez, 1968). Many Post Keynesian economists object to this assumption as being ontologically unsound in a fundamentally uncertain world (Davidson, 1991). Davidson (1994, 1996) carries this further to criticize the relevance of ergodic chaos theory in particular. He is joined in this by Mirowski (1990) and Carrier (1993) who argue that such approaches merely lay the groundwork for a reaffirmation of standard neoclassical economic theory. Mirowski sees the approach of Mandelbrot as more fundamentally critical.
. Crannell (1995) proposes an alternative to topological transitivity in the form of blending.
. For discussions of measuring chaos in the absence of invariant SRB measures see El-Gamal (1991), Domowitz and El-Gamal (1993), and Geweke (1993).
. This equation gives the global maximum LCE defined asymptotically. There has also been interest in local Lyapunov exponents defined out to n time periods with the idea of varying degrees of local predictability (Kosloff and Rice, 1981; Abarbanel, Brown, and Kennel, 1991, 1992; Bailey, 1996). See Abarbanel (1996) for a more general discussion.
. Nusse (1994a, p. 109) provides an example from the logistic map where one of the LCEs = -1 and there is convergence even though another LCE = 0.
. The Kaplan-Yorke conjecture (Frederickson, Kaplan, Yorke, and Yorke, 1983) posits a higher dimensional analogue of Young=s result with a relationship between Kolmogorov-Sinai entropy and a quantity known as the Lyapunov dimension. See Eckmann and Ruelle (1985) or Peitgen, Jrgens, and Saupe (1992, pp. 738-742) for more detailed discussion.
. Andrews (1997) warns that bootstrapping can generate asymptotically incorrect answers when the true parameter is near the boundary of the parameter space. Ziehmann, Smith, and Kurths (1999) show that bootstrapping can be inappropriate for quantifying the confidence boundaries of multiplicative ergodic statistics in chaotic dynamics due to problems arising from the inability to invert the necessary matrices. Blake LeBaron in a personal communication argues that the problems identified by them arise ultimately from an inability of bootstrapping to deal with the long memory components in chaotic dynamics.
Despite the problems associated with bootstrapping, LeBaron (personal communication) argues that it avoids certain limitations facing related techniques such as the method of surrogate data, favored by some physicists (Theiler, Eubank, Longtin, Galdrikan, and Farmer, 1992, which assume Gaussian disturbances, whereas bootstrapping simply uses the estimated residuals. Li and Maddala (1996) provide an excellent review of bootstrapping methods.
. We note a view that argues that nonlinear estimation is unnecessary because Wold (1938) showed that any stationary process can be expressed as a linear system generating uncorrelated impulses known as a Wold representation. But such representations may be as complicated as a proper nonlinear formulation, will not capture higher moment effects, and will fail to capture interesting qualitative dynamics. Finally, not all time series are stationary.
. Brock and Baek (1991) study multiparameter bifurcation theory of BDS and its relation to Kolmogorov-Sinai entropy using U-statistics. Golubitsky and Guckenheimer (1986) approach multiparameter bifurcations more theoretically.
. One loose end with using BDS is determining s, which might be dealt with via bootstrapping (I thank Dee Dechert for this observation).
Another possible complication involves when data changes discretely, as with US stock market prices which used to change in $ 1/8 increments (a Abit@ or Apiece of eight,@ reflecting the origin of the New York stock market as dating from the Spanish dollar period), and now change in $ 1/16 (a Apicayune@) increments. Krmer and Rose (1997) argue that this discrete data change induces a Acompass rose@ pattern that can lead the BDS technique to falsely reject an i.i.d. null hypothesis, although the BDS test was verified on indexes which vary continuously (I thank Blake LeBaron for this observation). Ironically this counteracts the argument of Wen (1996) that discreteness of data leads to false rejections of chaos in comparison with continuous processes.
. In the same year Pecora and Carroll (1990) developed the theory of synchronization of chaos. Astakhov, Shabunin, Kapitaniak, and Anishchenko (1997) show how such synchronization can break down through saddle periodic bifurcations and Ding, Ding, Ditto, Gluckman, In, Peng, Spano, and Yang (1997) show deep links between the synchronization and the control theory of chaos. Drawing on earlier work of Lorenz (1987c) and Puu (1987) on coupled oscillators in international trade and regional models, Lorenz (1993b) showed such a process of saddle periodic bifurcations in a model of Metzlerian inventory dynamics in a macroeconomic model.
. There are situations with lasers, mechanics of coupled pendulums, optics, and biology, and other areas where chaos is a Agood thing@ and one wishes to control to maintain it. One technique is a kind of mirror-image of OGY, moving the system onto the unstable manifolds of basin boundary saddles using small perturbations (Schwartz and Triandof, 1996).
. Another method slightly resembling global targeting involves using a piecewise linear controller to put the system on a new chaotic attractor that has a specific mean and a small maximal error, thus keeping it within a specified neighborhood (Pan and Yin, 1997). This is Ausing chaos to control chaos.@
. In a personal communication to this author, Mandelbrot dismissed catastrophe theory as being only useful for the study of light caustics and criticized Thom for his metaphysical stance.
. Blum, Cucker, Shub, and Smale (1998) demonstrate that the Mandelbrot Set is unsolvable in the sense that there is no Ahalting set@ for a Turing machine trying to describe it. This is one way of defining Acomputational complexity@ and is linked to logical Gdelian undecidability. See Albin (1982), Albin with Foley (1998), Binmore (1987) and Koppl and Rosser (1998) for discussions of such problems in terms of interacting economic agents.
. For Mandelbrot=s most recent work on price dynamics see Mandelbrot (1997) and Mandelbrot, Fisher, and Calvet (1997).
. A widely used approach for multiple agent simulations with local interactions is that of cellular automata (von Neumann, 1966), especially in its AGame of Life@ version due to John Conway. Wolfram (1986) provides a four-level hierarchical scheme for analyzing the complexity of such systems. Albin with Foley (1998) links this with Chomsky=s hierarchy of formal grammars (1959) and uses it to analyze complex economic dynamics. In this view the highest level of complexity involves self-referential Gdelian problems of undecidability and halting (Blum, Cucker, Shub, and Smale, 1998; Rssler, 1998).
. I thank Steve Durlauf for bringing this point to my attention.
. Of course Hegel=s (1842) favorite example of a dialectical change of quantity into quality was that of the freezing or melting of water, an example picked up by Engels (1940). See Rosser (1999c) for further discussion.
. Deriving from the work on genetic algorithms is that on articial life (Langton, 1989). Epstein and Axtell (1996) provide general social science applications and Tesfatsion (1997) provides economics applications. The work of Albin with Foley (1998) is closely related.
. For applications of the master equation in demography and migration models see Weidlich and Haag (1983). Zhang (1991) provides a broader overview of synergetics applications in economics.
Vdf24b KL
*+
&(8: "68:<>@BDNPRT!!L!N!P!$$v$x$%CJOJQJCJH*OJQJCJH*CJH*CJOJQJj0JCJH*U5CJ6CJCJOJQJCJCJOJQJKXZ2L
,-.1:XZ2L
,-.1:"N$P$$$Q(R(,,,,,,/233#4$488;===4>6>@@@`AbAvAxAAABBBRCTCDFII4JJDKKKHLJLLLFMHMMNNNdOO4PPQ~QQQR d"N$P$$$Q(R(,,,,,,/233#4$488;===4>6>%%J&L&O(P(****--00.000J0L0000000f1h111333333444 466Z6\699::<<<<=========>>>>
>>>????@@A
AAA"A$AAACJH*CJOJQJCJH*OJQJCJOJQJ6CJCJH*j0JCJH*UCJCJOJQJO6>@@@`AbAvAxAAABBBRCTCDFII4JJDKKKHLJLLLFMHMAAAAABBDBDDEEFFGGHHFIZIII
KKKK K"K$K(K.K0K2KKKKKKKKKKKKL"L,L.L4L6L8L:LBLDLFLLLLLLLLLLLL M&M.M0M6M8M@MBMDMMMMMMMMMMMM~NNNNCJH*j0JCJH*U6CJCJH*CJOJQJCJXHMMNNNdOO4PPQ~QQQRS=UWW\`dhhhhhhhhhNNNLONOVOXO\O^OP P&P(P,P.PPPPPPPTTZZZZ[[&](]_2______V`X`l`n`x`z```````?cRcyccccffffHfJfgggghhjjnjpjjkkknn n"nrrrrss5CJj0JCJH*UCJOJQJ6CJCJOJQJCJH*CJURS=UWW\`dhhhhhhhhhlYqZq[q\qqqsvwwxxzxyyyyyy"z#z$z~~xWhňƈABR89:qrsgEhlYqZq[q\qqqsvwwxxzxyyyyyy"z#z$z~~xWsssvvv
v"v$v4v6vvvwwww x"x*x,x0x2xxPyRyVyXyyyyyyyyy||
||>|@|J|L|jl҂ʃ "ÈĈ CJH*CJOJQJCJH*j0JCJH*UCJOJQJ6CJCJUWhňƈABR89:qrs !()./015678<=>?pqsvwx{|}~ÉȉɉʉˉωЉщ҉։ىډۉ߉ $%&'+,-./34567`bHJfhprCJOJQJCJH*CJCJH*^K[EFN[abjz!J[ekuŚƚ,.̜ΜNPZpVXxz<>£[sҥӥ"$ln©ĩ*+ahҬ46رڱ025CJCJOJQJj0JCJH*UCJH*6CJCJX%&'(TU:;<mnabcdҡӡZ[tuuvi4)df(ZSLm2KLef2jlxzZ\DFPRԽֽ0202NPVXjlDFrt&(24BD
8:~RT$*
Fj0JCJH*U6CJCJOJQJCJ[FH(*L9IN_RgkqRf68n{17ou&@H#+D%EFhj24,.RTCJOJQJCJH*6CJ5CJj0JCJH*UCJCJOJQJTYfg&' z|
FH"
Hh68468:"$z|
"L^`
##/#6#~##$$$$%CJOJQJCJH*j0JCJH*UCJOJQJ6CJ5CJCJOJQJCJH*CJCJOJQJNz^vaW"&V*W*,,,,,,----..2T4%%<%>%!&.&A&B&((F(H(`+a+---.22$2&2&4+4>7@7778888::;; <"<<<&=(=2=4=@@$A&A>A@ABBCC4D6DXDZDDDFFHFIIIIKLNOOOQQUU V"V&V(VVVVCJH*CJOJQJCJOJQJ5CJj0JCJH*U6CJCJCJOJQJRT4Y8Z8t9u9v9w999V:X:p:::;~;;;<<>FJJJJJJJNLNLNNNNNOOGQ,SXX[[[\\\>\?\@\w\x\Za[aaaceee6fVVVV*W,WXWZWBZDZ[[]]____aa[aaeeggXhYhjjll(n9noopppppqqqqqqrrsssssssssss|t~tttttttttttuuuuzz&z(z$|&|CJH*5CJj0JCJH*U6CJCJOJQJCJOJQJCJH*CJCJOJQJQ6fffggZh[h_jRouuuuvv.wyy}< :<P&||}݄&(bdƆȆ 68jlNPn68"$02
"'(,-pr'()֜ڜCJOJQJCJOJQJCJH*CJH*j0JCJH*U5CJCJOJQJ6CJCJQPQjlnp/0@BprOd
tv46HJ^`bȠʠР"$&*,9LMN +¤ڧܧ«̫ΫǬѬҬCDUV5CJj0JCJH*U6CJCJOJQJCJH*CJOJQJCJOJQJCJCJH*PSTǬ;D
kJL24pr9:FHhj|6@ºĺлһij~&yzXZ!"4LN.0LN>_(*.05CJCJOJQJCJH*CJOJQJ6CJCJOJQJj0JCJH*UCJR&(LN8:tZ\^>@BAH ,.02<BDF.06CJj0JCJH*UCJH*CJOJQJCJOJQJCJH*CJOJQJCJR<>hST aTV"$&(*,lrtv ghJK $ & 0 2 H J !!!!!!!!."0"%%5CJCJOJQJCJH*CJOJQJj0JCJH*UCJH*CJOJQJCJRab!N@"&(.._16X9;;r<<^==R>>4?%%%%%&P&n''''((6(8()
***-.102N3P333P4R4444444556686::<<X<h<<<>>>>BBBBBBBBDDDEFFHFGGGGKIUIJJJNNNNNNZO\OzO|OPPFPHPP5CJj0JCJH*U6CJCJOJQJCJZ4??@=@y@@@(ARBBBCC0DDDEEFFJFLFjJNgThTWWXX\PPPPRRWWWXY!Y3Y=YLYQYZZZZZZ[[-_@_n`p`````ddddddeeeeeeeeJfLfff3mHmopppppqttxxxxlypyry|yyyyyyyyyyyCJH*OJQJCJH*CJOJQJCJH*CJOJQJ5CJj0JCJH*U6CJCJCJOJQJN\]_bbJdLdNd7jjjjjjjjoo p"ptvvLyNyyy|"rtyyzz||~~<= t&VmʊيDE"$TV`bJLfh"$*.Zd:<
DFRT
02MRUVOJQJ
j0JUCJH*CJOJQJ5CJj0JCJH*UCJOJQJ6CJCJCJH*OJQJMtۇؑRTʒ̒jlĔLMSUDvx:٦DFJLbdrt(*vxxy68NPTVfh٦ڦhiƧЩҩ.0 "st<>PRӲ(=>NP68"$fh6OJQJ
j0JU_٦h s=N6"νEܾ0'(g2V
rtνнEFܾݾ01PRfhln()gh:<v8:XZ23Uchj
bd6OJQJ
j0JU_
bijoh"&ygX>_D
\hjkop ":<hj"#>@24&'|yzghXY>? "_`OJQJ5H*5OJQJ
j0JU6[D
E
<>@B
XZ46VX$&NPhjfhghCJ6
j0JUOJQJ2D
@
4$ffgh+0P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%. 00P/ =!"#$%
[<@<Normal1$CJOJQJhmH nH <A@<Default Paragraph Font4&@4Footnote ReferenceK*
d'!#l%*C<JLcUgo6o|oqruw>&}8^_%'L))39D=xCSFGWOWO_a@cf,osldȑAѢ@UZ$Q8o
x[Y ` M"#2$d%(e)@**.Z//#034568`::U;<G=>BDxEIKLNNPQoSSTUdVgV- . g""S-7ZBMFXcm|V1U
n&I5BZO[ixZhZ$-]M}ݱ
=mͲ-]M}ݴ
=m͵-]%ANs 2F%V&|%Pyh'+-/24589;=@BDFGIKMOQSUV6>HMhWuT4LN6fPa4?\t٦
D
h(*,.1367:<>?ACEHJLNPRTWRg)0College of BusinessC:\My Documents\CHAP2.CCG.doc@HP LaserJet 4M Plus\\JMU\.ZSH_436_HP.COB.J3.JMUHPPCL5MSHP LaserJet 4M PlusHP LaserJet 4M Plus@w XX@MSUDHP LaserJet 4M Plus<d
HP LaserJet 4M Plus@w XX@MSUDHP LaserJet 4M Plus<d
t9 9
99
99CC$$!"'%'&'''(')s,s-s.s/s0s1s9s:s;s<s=u@uA^C^DbEbFKLMNOPQSTUYZdhdidjdldmwpwqtuvw{|}~kkkkBBBBAAAAAAAAAAAAAAAAAAzz&&&&&&&\\\\\\\\\\\\\\\\\\ffffff%%% |
|||
"#()#+#,#-#.#0#1#5#7#;#<?@AaCaDaEaFaGaHaIaKaLaMRSVWXK`KaKbKcz{|}܀܂܃܇܈܉܊HHHH{{{{{SSSSSSSSvvvvvvvvv...... !Z$PPPP@PP @PPP,@PPP8@PP PD@P$P&PP@P*PX@P0Pd@P6Pp@P<P>P@PBP@PHPJPLPNPPP@PZP\P^P`P@PfP@PjP@PnP@PvPxP@P|P@PP@PPP@PP@PPP@@PPH@PPT@PPPPh@PPPPPPPPP@PPPP@PP@PP@PPPP@PP@PP@PPPP@PPP@PP
P@PPPP0@PP8@PP@@P$PL@P(PT@P2Ph@P6Pp@P:P<P|@P@PBPDPFPHP@PNP@PTPVP@PZP@P^P@PdPfP@PlP@PpPrPtP@PzP|P@PPP@PPPP@PP$@PP,@PP4@PPPPD@PPPP@PPX@PPh@PPPPx@PP@PP@PP@PPP@PPPPPPP@PPP@PP@PPP@PPPP@P P"P$PL@P(PT@P,P\@P2P4P6Pp@P:P<P>P@PBPDPFP@PJP@PNPPP@PZP@P`P@PdPfP@PpP@PxP@P|P~P@PPP@PPPPPPPP4@PPPD@PPL@PPPP\@PPd@PPPPP|@PPP@PP@PP@PPP@PPPP@PPPP@PPPP@PP@PPP@PP@PPPPPPP0@GTimes New Roman5Symbol3&ArialSWP TypographicSymbols9WP MathAIWP MathExtendedA?5 Courier New" hك8&ك8&B|,$ CHAPTER 2College of BusinessCollege of Business
Oh+'0 $
@L
Xdlt|
CHAPTER 2 HAPCollege of BusinessollollNormal College of Business2llMicrosoft Word 8.0@@N@NB|
՜.+,D՜.+,Lhp
James Madison UniversityD,1
CHAPTER 2Title 6>
_PID_GUIDAN{BEFF4A81-53EE-11D3-8FFD-00500414593B}
!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~
!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~
!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[]^_`abcdefghijlmnopqrtuvwxyzRoot Entry`CQCjC[CtCeC~CoC F,/C1TableظCɸCCӸCCݸCCCCCCCC(CC2C#C<\FCWordDocumentdCUCnC_CxCiCCsCCCCCCCMCSummaryInformationCCCCC(C C"CC,CC6C'k1CDocumentSummaryInformationrChC8CCCCCCCsCompObjѺCֺCۺCCCCCCCCCCC!jCObjectPoolCXC]CbCgClCqCvC{C//CCCCƻC˻CлCջCڻCCCCCCCCCC C%C*C/C4C9C
FMicrosoft Word Document
MSWordDocWord.Document.89q