A filtered H\'enon map

TL;DR

This paper investigates how inserting a linear filter into the feedback loop of the Hénon map affects its chaotic behavior, revealing complex dynamics relevant for chaos-based communication systems.

Contribution

It introduces a method to analyze the impact of linear filters on the Hénon map's dynamics using Lyapunov exponents, highlighting new behaviors like bifurcations and attractor coexistence.

Findings

Filter coefficients can induce or suppress chaos in the Hénon map.

Complex bifurcation cascades and attractor coexistence are observed.

Results are applicable to bandlimited chaos communication systems.

Abstract

In this paper, we use Lyapunov exponents to analyze how the dynamical properties of the H\'enon map change as a function of the coefficients of a linear filter inserted in its feedback loop. We show that the generated orbits can be chaotic or not, depending on the filter coefficients. The dynamics of the system presents complex behavior, including cascades of bifurcations, coexistence of attractors, crises, and "shrimps". The obtained results are relevant in the context of bandlimited chaos-based communication systems, that have recently been proposed in the literature.