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)
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 X2) .
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 (Xn - B Xn2)
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.
