Scientific visualization of Poincare maps

Poincare maps have proved to be a valuable tool in the analysis of several dynamical systems modeled by differential equations. These maps are generated by reducing the continuous flow to a two dimensional discrete dynamics. From a map it is possible to identify the chaos phenomenon in a system under the influence of an external parameter. If this external parameter is variable, one can study the behavior of the system by interpolating the set of corresponding Poincare maps. Despite its usefulness, the computer graphics work carried out so far has been limited to the display and plot of Poincare maps. In this paper a prototype for the computer analysis of Poincare maps is described. We show that, from the point-of-view of computer graphics, we can process Poincare maps as noisy images. This approach not only facilitates the partition of Poincare maps into regular and chaotic regions but also offers possibilities of visualizing the continuous evolution of a system by varying the external parameters. Some results are given to illustrate the functionalities of the prototype. (C) 1998 Elsevier Science Ltd. All rights reserved.