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

TL;DR

This paper investigates whether a one-dimensional return map can adequately represent the chaotic dynamics of a three-dimensional system, concluding that a two-dimensional map is generally necessary for accurate description.

Contribution

It demonstrates through examples that reducing a 2D Poincare map to 1D can fail to capture the original dynamics, emphasizing the importance of a 2D map for 3D chaotic systems.

Findings

1D maps may not reproduce the original dynamics accurately.

Natural reduction to 1D is often unjustified.

2D Poincare maps are generally essential for 3D chaos analysis.

Abstract

Study of continuous dynamical system through Poincare 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 Poincare 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 Poincare section are functionally related. This also reduces the 2D Poincare 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 Poincare map is not always justified, because the resultant 1D map may fail to generate the original dynamics. This proves that to describe the dynamics of the 3D chaotic attractor, the minimum dimension of the Poincare map must be two, in general.