Computing complex horseshoes by means of piecewise maps

TL;DR

This paper introduces a systematic numerical method for constructing complex horseshoe maps using piecewise functions and elementary transformations, enabling detailed visualization of nonlinear dynamical system properties.

Contribution

The paper presents a novel approach to compute horseshoe maps through piecewise maps and elementary transformations, enhancing the ability to illustrate complex dynamical behaviors.

Findings

Method effectively constructs complex horseshoe maps

Demonstrates the Wada property in nonlinear systems

Enables visualization of intricate folding structures

Abstract

A systematic procedure to numerically compute a horseshoe map is presented. This new method uses piecewise functions and expresses the required operations by means of elementary transformations, such as translations, scalings, projections and rotations. By repeatedly combining such transformations, arbitrarily complex folding structures can be created. We show the potential of these horseshoe piecewise maps to illustrate several central concepts of nonlinear dynamical systems, as for example the Wada property.