NON-AUTONOMOUS SYSTEMS

Thus far we have examined the occurence of chaos in autonomous systems of differential equations of dimension greater than two. Although the Poincaré-Bendixson Theorem guarantees that chaos cannot occur in autonomous systems of dimension two this is not necessarily true for non-autonomous systems. To investigate the occurence of chaos in such systems consider the equation below

This equation can be rewritten as an autonomous first order system by using the transformations resulting in the autonomous system of three equations

Notice that the system contains all the features necessary for the occurrence of chaotic behaviour i.e. it is a nonlinear autonomous system in three dimensions. The system has no equilibria in three dimensions since is never equal to zero. In plotting the trajectories of the system it is only necessary to consider the projection onto the x-y plane since the variable z has only been introduced as a device to make the system autonomous. Note that the resulting plot is not a true phase portrait of the system.

The system exhibits extremely complex behaviour and some features will be investigated in the exercises. Here we consider the behaviour of the system for and increasing values of B.

Limit Cycle

Periodic Attractor

Two Periodic Attractors

Single Periodic Attractor

Strange Attractor

Maple Simulation