Is one dimensional Poincaré map sufficient to describe the chaotic dynamics of a three dimensional system?

摘要

Study of continuous dynamical system through Poincaré map is one of the most popular topics in nonlinear analysis. This is done by taking intersections of the orbit of flow by a hyper-plane parallel to one of the coordinate hyper-planes of co-dimension one. Naturally for a 3D-attractor, the Poincaré map gives rise to 2D points, which can describe the dynamics of the attractor properly. In a very special case, sometimes these 2D points are considered as their 1D-projections to obtain a 1D map. However, this is an artificial way of reducing the 2D map by dropping one of the variables. Sometimes it is found that the two coordinates of the points on the Poincaré section are functionally related. This also reduces the 2D Poincaré map to a 1D map. This reduction is natural, and not artificial as mentioned above. In the present study, this issue is being highlighted. In fact, we find out some examples, which show that even this natural reduction of the 2D Poincaré map is not always justified, because the resultant 1D map may fail to generate the original dynamics properly. This proves that to describe the dynamics of the 3D chaotic attractor, the minimum dimension of the Poincaré map must be two, in general.