INTRODUCTION

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Recall that for autonomous systems in the plane the Poincaré-Bendixson theorem essentially states that the only attractors which can exist are point attractors (stable nodes and foci) or stable limit cycles. This is no longer true if the system is either an autonomous system in three (or more) dimensions or a non-autonomous system in the plane..

Before beginning our invesigation it is important to note some of the results which carry over from planar autonomous systems and those which do not:

- The Hartman-Grobman theorem carries over;
- The method of examining dynamical behaviour using Lyapunov functions carries over with minor modifications;

but

- The classification of equilibrium points into nodes, foci, centres and saddle points does not carry over;
- The trace and determinant of the Jacobian matrix do not allow deductions about the stability of equilibrium points in the same way as for planar systems, it is necessary to examine the real part of the eigenvalues.

Basically the method of analysis models that of systems in the plane:

- Determine the equilibrium points;
- Linearise the system about each equilibrium point in turn;
- Examine the real part of the eigenvalues of the Jacobian matrix and apply the results of the Hartman-Grobman theorem.