Demonstrations in Chaos

by V. Balakrishnan and Suresh Govindarajan

Demos written by Phani Kiran and Prabhu Ramachandran

A dynamical system is a set of variables whose evolution in time is specified by a

specific rule. As an example, consider the growth of the population of bacteria in a

colony. Let the total number of bacteria at time t be denoted by X(t). The simplest

evolution rule is given by a differential equation

dX/dt = F(X)

Consider the simple case when

F(X) = A X .

where A is a constant. This is the linear case. If A > 0 , the population of the system

increases without limit as t increases, while for A < 0 the population simply dies out

eventually. Clearly this is too simplistic a model of the actual growth of a bacterial

population. A more realistic model would include the effect of competition between

the bacteria for nutrients. This can be done by including a term on the right-hand side

that has a negative sign, and is quadratic in X. To do this we choose

F(X) = A (X - B X^2) .

where B is a positive constant (this is called the logistic equation). This modification

immediately brings in the possibility of a population stabilizing at the value B: it is

easy to see that dX/dt = 0 in equilibrium, and hence x is either equal to 0 (the

uninteresting case) or equal to B in equilibrium. The nonlinearity of the equation is

what has led to this more complicated behaviour of the solutions. Nonlinear

dynamical systems behave quite differently from linear dynamical systems.

The example above involved a single independent variable, x. If the system is

described by 2 independent variables (X and Y, say) satisfying a pair coupled

nonlinear differential equations, many more possibilities open up for the kinds of

equilibrium solutions (e.g., fixed points of various kinds, as well as isolated periodic

solutions called limit cycles). If the system is described by 3 or more independent

variables satisfying coupled nonlinear differential equations, the possibilities naturally

become even more diverse. Most surprisingly, however, a totally new kind of

possibility arises for the equilibrium solution, that is neither a fixed point nor a limit

cycle. This is called a strange attractor . It is a sort of tangled structure in the space of

the variables representing a region in which the system is not periodic, but at the same

time is a region to which the system is confined for all time once it falls into the

attractor. Moreover, the attractor has a fractal structure. The famous Lorenz

model (which involves 3 variables) has a strange attractor with two "wings", shaped

somewhat like a butterfly, with a fractal dimension of about 2.06 (for suitable values

of the parameters of the model). The dynamical system can then exhibit chaos.

When the evolution of a dynamical system is considered in discrete time rather than

continuous time, the system gets described by difference equations rather than

differential equations. (In other words, it is a "map" rather than a "flow".) In this case

even a system described by a single independent variable can exhibit chaos, provided

the map is a suitably nonlinear one. For example, the discrete version of the logistic

equation introduced above becomes the logistic map

X(n+1) = A (X(n) - B X(n)^2 )

where n indicates the time step. In contrast to its differential equation counterpart, this

map has an astonishingly rich variety of dynamical behaviour. For suitable values of

A and B, it exhibits "period-doubling cascades", "intermittency", "crises", and of

course chaos.

We illustrate some features of chaotic dynamics with the help of some two-

dimensional maps and flows in the demos that follow.

1. Chaotic systems are "extremely sensitive to the choice of initial conditions". This is illustrated using three different two- dimensional maps, i.e., maps

involving two variables. Click here for demo on Extreme Sensitivity to Initial

Conditions

2. A chaotic system not periodic, and hence does not precisely return to its initial state at any time. However, there is a certain sense in the which the system does

return again and again to states close to its initial state! This is a consequence

of a property called ergodicity , and such returns to the "neighbourhood" of the

intial state are called Poincare recurrences. This phenomenon is illustrated here

in the case of a two dimensional map. Click here for demo on Recurrence in

Chaotic Systems (In actuality, what the demo above shows is periodicity rather

than Poincare recurrence (which would generally occur irregularly). The

reasons are technical, and have to do with the very gross "coarse-graining"

done on the system for the purpose of the demo.)

3. Another feature of chaotic sytems is called mixing . Mixing refers literally to the scrambling up of a set of initial conditions that start out near each other, but

which end up thoroughly "mixed" all over the available "phase space" of the

dynamical system - very much like the thorough mixing of two shades of paint

to make a uniform intermediate shade. Click here for demo on Mixing