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

Abstract

In this paper, a two parameters family F β 1 ,β 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 β 1 β 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.