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

Abstract

In this paper, a two parameters family Fβ1,β2 of maps of the plane having two different invariant subspaces is studied. We observe that, our model exhibits two chaotic attractors Ai, i=0,1, lying in these invariant subspaces and identify the parameters at which Ai 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 β1β2-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 undergoes a sequence of bifurcations: “a blowout bifurcation”, “a bifurcation to normal repulsion” and “a bifurcation to emergence a new chaotic attractor with an intermingled basin”. Numerical simulations are presented graphically to confirm the validity of our results.