Simple scenarios of onset of chaos in three-dimensional maps

TL;DR

This paper qualitatively describes two primary routes to chaos in three-dimensional maps, supported by numerical analysis of Henon-like maps and models in nonholonomic mechanics.

Contribution

It introduces a detailed qualitative framework for understanding chaos onset in 3D maps, focusing on Shilnikov and Lorenz-like scenarios, with numerical illustrations.

Findings

Identification of Shilnikov scenario in spiral chaos transition

Description of Lorenz-like and figure-eight attractor transitions

Numerical validation using Henon-like and nonholonomic mechanics models

Abstract

We give a qualitative description of two main routes to chaos in three-dimensional maps. We discuss Shilnikov scenario of transition to spiral chaos and a scenario of transition to discrete Lorenz-like and figure-eight strange attractors. The theory is illustrated by numerical analysis of three-dimensional Henon-like maps and Poincare maps in models of nonholonomic mechanics.