Proliferation of stability in phase and parameter spaces of nonlinear systems

TL;DR

This paper demonstrates how composing maps in discrete nonlinear systems can generate multiple identical stable structures in phase and parameter spaces, enhancing understanding of stability proliferation.

Contribution

It introduces a method to multiply and relocate stable domains in nonlinear systems using map composition, applicable to various two-dimensional systems.

Findings

Generated multiple identical ISSs in parameter space

Proliferation of riddled basins of attraction

Method applicable to any 2D nonlinear system

Abstract

In this work we show how the composition of maps allows us to multiply, enlarge and move stable domains in phase and parameter spaces of discrete nonlinear systems. Using H\'enon maps with distinct parameters we generate many identical copies of isoperiodic stable structures (ISSs) in the parameter space and attractors in phase space. The equivalence of the identical ISSs is checked by the largest Lyapunov exponent analysis and the multiplied basins of attraction become riddled. Our proliferation procedure should be applicable to any two-dimensional nonlinear system.