The occurrence of riddled basins and blowout bifurcations in a parametric nonlinear system

TL;DR

This paper investigates riddled basins and blowout bifurcations in a two-parameter nonlinear map, revealing how chaotic attractors and their basins evolve, including bifurcations and fractal basin boundaries, supported by numerical simulations.

Contribution

It introduces a detailed analysis of riddled basins and bifurcations in a parametric nonlinear system, including a semi-conjugation to a random walk model and fractal boundary characterization.

Findings

Identification of parameters with riddled basins

Occurrence of blowout bifurcations and chaotic saddles

Fractal boundary separating basins of attraction

Abstract

In this paper, a two parameters family $F_{\beta_1,\beta_2}$ of maps of the plane living two different subspaces invariant is studied. We observe that, our model exhibits two chaotic attractors $A_i$, $i=0,1$, lying in these invariant subspaces and identify the parameters at which $A_i$ has a locally riddled basin of attraction or becomes a chaotic saddle. Then, the occurrence of riddled basin in the global sense is investigated in an open region of $\beta_1\beta_2$-plane. We semi-conjugate our system to a random walk model and define a fractal boundary which separates the basins of attraction of the two chaotic attractors, then we describe riddled basin in detail. We show that the model undergos a sequence of bifurcations: "a blowout bifurcation", "a bifurcation to normal repulsion" and "a bifurcation by creating a new chaotic attractor with an intermingled basin". Numerical simulations are presented graphically to confirm the validity of our results.