Thoughts on the Edge of Chaos
M.M.Taylor and R.A. Pigeau
2. Basic Ideas: Information and structure, Attractors and Repellors
3. Basic Ideas: Catastrophe
Structure and chaos in non-equilibrium conditions
Several very general insights have transformed our understanding of structure and order in the world. Nicolis and Prigogine (1977) demonstrated that stucture tends to develop in the strong energy flows of systems far from equilibrium. This structure feeds on itself: small disturbances grow into larger disturbances, and spawn subordinate disturbances. The development of structure seems to be a very general aspect of non-equilibrium thermodynamics; it is seen in fluid mechanics, chemical self-catalytic reactions, and in general in situations where the result of an action tends to enhance the probability that the action will recur. Such a situation is known as auto-catalytic.
Approaches to Chaos
In fluid flow, a single number--the Reynolds number--determines whether the fluid is stationary (in equilibrium), smoothly flowing (near equilibrium), or turbulent (far from equilibrium). Feigenbaum ( M J Feigenbaum, Universal behavior in nonlinear systems, in Order in chaos, Los Alamos, N.M., 1982, Phys. D 7 (1-3) (1983), 16-39) found that for a large class of recursive (feedback) functions of the kind xn = f( xn-1) there was a single controlling number that determined whether x would approach a stable value (equilibrium), a cyclic attractor (near equilibrium) or a chaotic attractor of infinite period (far from equilibrium). Insofar as f(x) can be approximated by a polynomial with a non-zero quadratic term, the approach to chaos is controlled largely by its quadratic term (i.e. The controlling function is approximated by xn = ax2n-1 + k, and the shift from a single attractor to an attractor of successively doubling cycle length is described by a single number, Feigenbaum's constant, 4.6692016....
In the case of a complex variable, the function zn = z2n-1 + c is the limiting case of a polynomial function. For the case z0 = 0, variation in the value of c determines whether z approaches a single stable value, a finite cycle of fixed values, or diverges to infinity. In this case, the bounding values of the different regions of c in the complex plane do not form a simple series as they do on the real line. Rather, they describe an increasingly complex set of curves as the attractor cycle increases in length, describing a fractal curve in the limit, on which successive values of z follow a chaotic orbit. The region of values of c bounded by this fractal is called the Mandelbrot set. For values of c outside the Mandelbrot set, z diverges to infinity.
For any value of c in the Mandelbrot set, different values of z 0 lead to different behaviours of z n as n approaches infinity. Some values of z 0 may lead to a single stable attractor, some to a finite attractive cycle, and some may diverge to infinity. The boundaries between these regions form a (usually) fractal curve, which is a repellor called the Julia set of the function for the particular value of c. One intriguing and important feature of the Julia set is that every point on it is a boundary point of every region of different behaviours possible for that value of c. If z 0 is in the Julia set, then so is z n, for every n, and the values of zn follow a chaotic orbit.
It is the Julia sets that form the basis for our discussion of cognitive evolution. We will make the analogy between z and an evolving thought element (a meme), and between c and the environment within which this meme may or may not replicate. The argument is that a replication with an attractive cycle of length 1 is both unobservable and uninteresting, in that the result never changes. Longer attractor cycles imply that the meme is regenerated through a kind of life-cycle, much as a gene replicates by building a baby body, which grows to adulthood and then generates new copies of the gene. But stable attractor cycles again do not present much interest; the meme may be conserved, but its changes are repetitive. If the initial meme is such that successive cycles diverge to infinity, the meme is, in effect, lost, and is again uninteresting. The interesting condition is when the initial meme is near the Julia set, for then the attractive orbit is long, and though near-recurrences happen often, there is always the possibility of change to a new nearly stable cycle.
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