# 2 1 Poincare return mappings

The solution of (2.1) corresponds to a trajectory (or flow) in some n-dimensional phase space. There is an old and very ambitious programme of Poincard which seeks to determine the general behaviour of (2.1) 8s /-**>, but only when n - 2 is the problem completely solved. Because trajectories cannot intersect in phase space a two-dimensional topology limits the asymptotic behaviour to two generic states: Either trajectories approach a stable attractor (stationary solution) or a limit cycle (periodic solution) alter an infinite time. In the first case the attractor has dimension 0 (a point) and in the second, dimension I (a closed curve). When n>2 trajectories can behave in a "ar more exotic fashion. Trajectories can now cross and develop complicated knotted configurations wit hout actually intersecting. The detailed behaviour of (2.1) when nss3 is not yet known. A theorem of Ruelle and Takens  shows that in these higher dimensional cases the fate of generic trajectories is to approach a non-empty, finite region of the phase space, containing neither attracting points nor limit cycles and in which neighbouring trajectories rapidly diverge from each other when evolved backwards or forwards in time. Trajectories will enter this attracting set and then wander around it in a chaotic fashion. Ruelle and Takens termed this set a 'strange attractor1. A strange attractor is defined to be a set which attracts all nearby solution trajectories and which has the structure of M x C where M is a smooth manifold and C a Cantor set, or product of Cantor sets.

Clearly, the degree of complexity in the description of (2.1) can be reduced by lowering the dimension of the problem. A useful technique for effecting this reduction which simultaneously converts the system under study from a differential to a difference equation was suggested by Poincare . We place an (« -1) dimensional cross-section, X, through the n-dimensional phase space so that ~ is intersected infinitely often by the phase flow. The discrete sequence of intersections that the flow makes with £ gives a difference equation describing an induced mapping of X into itself. This mapping is called the Poincare return mapping of the dynamical system (2.1). Tf (2.1) exhibits chaotic behaviour it will be mirrored by the erratic behaviour of the associated Poincare map. If the solution of (2 H U periodic then the phase flow will be a closed orbit and the Poincare map simply a fixed point In general, the cross-section X will be intersected infinitely often by the phase orbit but onh a fitvu- number of times in any finite open interval of time.