Chaos Theory

A. BRIEF HISTORY

1. Father of chaos Theory – Henri Poincaré

Poincaré is regarded as the last "universalist" capable of understanding and contributing in virtually all parts of mathematics. He is considered to be the father of chaos theory, although the term ‘chaos’ was not coined by him. He was working to the famous three-body problem. Due to the efforts of Poincaré, today we know that the problem is not solvable analytically. But in 1889, this was not known. The three-body problem is a specific case of n-body problem (n>=3). A system consisting of two bodies which are interacting under gravitation forces can be expressed as a differential equations. Using Newton’s laws, the motions of two masses can be expressed as a differential equation. In mathematical parlance, it is said that the system is “analytically solvable” or the set of differential equation don’t have a “closed form solution”. Intuitively this indicates that long range prediction of orbits is not possible. Poincaré showed that a system of three bodies interacting under gravitational forces cannot be solved analytically. The solutions in such cases can be approximated by using computational methods.

2. Edward Lorentz

Poincaré’s monumental discovery of what is now called deterministic chaos was neglected. Probably, the time at which it came, scientists were primarily interested in Relativity theory and Quantum Physics. Also, without the computer it is very difficult to show the behavior of chaotic deterministic system.

If we are allowed to say that Poincaré gave birth to chaos theory, then it can be safely said that Lorentz gave it a rebirth. Edward Lorentz was a meteorologist working at MIT on long range prediction of weather using computer models. The models consisted of differential equations, no more complicated than Newton’s laws of motion. He created a basic computer program using mathematical equations which could theoretically predict what the weather might be. One day he wanted to run a particular sequence again, and to save time he started it from the middle of the sequence. After letting the sequence run he returned to find that the sequence had evolved completely different from the original. At first he couldn't comprehend such different results but then realized that he had started the sequence with his recorded results to 3 decimal places, whereas the computer had recorded them to 6 decimal places. As this program was theoretically deterministic we would expect a sequence very close to the original, however this tiny difference in initial conditions had given him completely different results.

This sensitivity to initial conditions became known as The Butterfly Effect. This is because it compares to the idea that a butterfly flapping its wings could produce a tiny change in the atmosphere, and trigger a series of changes which could eventually lead to a hurricane on the other side of the world. The Butterfly Effect is a key characteristic to a chaotic system.

B. LOGISTIC FUNCTION

In this section, I will show the behaviour of a very simple function known as Logistic function. The use of this function is fairly consistent in literature for the introduction to chaos theory. The reason for this is that it is the simplest one dimensional, nonlinear (x squared term), single parameter (b) model that shows an amazing variety of dynamical response. This function can be intuitively seen as a simulation of the population dynamics of some species over the time.

Logistic Function: f(x) = b*x*(1-x) = b*x – b*x^2, where b is a constant known as the effective growth rate.

The population size, ( f^n(x), composition of the function f ‘n’ times) at the nth year, is defined relative to the maximum population size the ecosystem can sustain and is therefore a number between 0 and 1. The parameter b is also restricted between 0 and 4 to keep the system bounded and therefore the model to make physical sense.

The aim is to know, what happens to the long term behavior of the system, for fixed value of the parameter b and any given initial condition, say, x0. So, I change the value of ‘b’ and show the various kinds of dynamical responses of the function.

(a) 0 <= b <= 1
It is clear that for values of b between [0,1], if we start iterating the equation with any value of x the value of x will settle down to 0. The point zero is called the fixed point of the system and is stable for b = [0,1]. A stable fixed point has the property that the points near it are moved even closer to the fixed point under the iterations. For an unstable fixed point the points move away as time progresses. The behaviour of logistic function for b=0.9 is shown on the graph below:

(b) 1 < b < 3

For values of b between (1,3) the iterates instead of being attracted to zero, get attracted to a different fixed point other than zero. The fixed point x = 0 is unstable for b between (1,3) and a new fixed point exists for b >= 1. The new fixed point is that x = 1 - 1/b. The behaviour of logistic function for b=2.8 is shown on the graph below:






(c) 3 <= b <~ 3.5

Apart from 0 and (1 - 1/b) , for all other initial values, the logistic equation will never converge to any fixed point. Instead the system settles down to a period 2 limit cycle. That is, the iterates of the logistic equation oscillate between two values. The fixed points 0 and 1 - 1/b still exist but they are unstable. The behaviour of logistic function for b=3.2 is shown on the graph below:






(d) 3.5 <~ b <~ 3.65

On further increasing b the period 2 limit cycle becomes unstable and a period 4 limit cycle is created. The behaviour of logistic function for b=3.52 is shown on the graph below:











(e) b=4

For b=4, it is found that there are orbits of all periods. This is known as chaotic behaviour. The behaviour of logistic function for b=4 is shown on the graph below:










C. BIFURCATION DIAGRAM

The behaviour of logistic function with different values of ‘b’ can be shown very elegantly on a diagram known as bifurcation diagram. On this diagram, the limiting value of the logistic function is plotted against the value of ‘b’.









D. FORMAL DEFINITION OF CHAOS

Before I give the formal definition of chaos, I will define three terms.

Definition 1 (Density) – A set A is dense in another set B, if for all ε >0 and b Є B there is an a Є A such that | a-b | < ε.

Intuitively, Definition 2 (Sensitive dependence on initial conditions) – We say that the one dimensional map f: D -->D has a sensitive dependence on initial conditions if there exists M>0 such that, for any x Є D and any ε>0 there exists y ЄD and n ЄN such that |x-y| < ε and |f^n(x) – f^n(y)| >M.

Intuitively, Definition 3 (Topological transitivity) - f: D --> D is said to be topologically transitive if for any ε>0 and x,y Є D, there exists z Є D and k Є N such that |x-z| < ε and |y – f^k(z)| < ε. Intuitively, Defination of Chaos:

Let f: D --> D be a one dimensional map. We say that f is chaotic if
1. The set P(f) of periodic points is dense in D.
2. f has sensitive dependence on initial conditions.
3. f is topologically transitive.