Discrete-time Poincare Maps

A convenient method for visualizing continuous trajectories is to construct an equivalent discrete-time mapping by a periodic stroboscopic sampling of points along a trajectory. One way of accomplishing this is by the so-called Poincare map (or surface-of-section) method (see figure 4.1). In general, an N — l)-dimensional surface-of-section 5 C F is chosen, and we consider the sequence of successive in- 

Poincare maps of this form have the obvious advantage of being much simpler to study than their differential-equation counterparts, without sacrificing any of the essential behavioral properties. They may also be studied as generic systems to help abstract behaviors of more complicated systems. 

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A Poincare map is established by cutting across the trajectories in a certain region in the phase space, say with dimension n, with a surface that is one dimension less than the dimension of the phase space, n — 1. One such cut is also shown in Fig. 1. The equation that produces the return to the crossing the next time is a discrete evolution equation and is called the Poincare map. The dynamics of the continuous system that creates the Poincare map can be analyzed by the discrete equation. Therefore, chaotic behavior of the Poincare map can be used to identify chaos in the continuous system. For example, for certain parameters, the Henon discrete evolution equation is the Poincare map for the Lorenz systems. 

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