Development of the Poincare cross-section method: Visualization the three-dimensional sections of four-dimensional flows

TL;DR

This paper introduces a novel method using three-dimensional Poincare sections to visualize and classify four-dimensional flows, enhancing understanding of complex attractor topologies.

Contribution

It develops an algorithm for three-dimensional Poincare sections to classify four-dimensional attractors, providing a universal approach for analyzing evolving systems.

Findings

Obtained 3D Poincare section traces of 4D attractors

Enabled primary classification based on topology

Demonstrated universality for systems with iterative equations

Abstract

The theme of the article is the application of the Poincare section method for visual classification of attractors in the four-dimensional phase space; the purpose of the study is to introduce consideration of three-dimensional Poincare sections, and develop an algorithm for their use for classification of four-dimensional flows by the type of attractors. The article contains the obtained in three-dimensional Poincare sections traces of 4D attractors, allowing you to get an idea of their topology, as well as to conduct their primary classification. It is important that the model discussed in the article is formulated as universal, which allows one to study the evolution of systems (if they can be described iterative equations) consisting of subsystems interacting along arbitrary laws.