The occurrence of locally riddled basins and on-off intermittency in a parametric nonlinear system

•Rigorous analysis for the occurrence of locally riddled basin and on-off intermittency is given.•Using the monotonicity and concavity of the fiber maps in our model, we give a classification, with respect to the number of attracting invariant graphs.•We examine the bifurcation effects that appear in our model on varying parameter; in particular, a non regular temporal bursting and the appearance of an on-off intermittency is investigated.•The nonhysteretic blowout bifurcation of the chaotic attractor is also illustrated. In this paper, a one parameter family (Fβ)β∈(0,1) of maps of the unit square I×I is studied. We observe that, for some values of parameter, our model exhibits one chaotic attractor Aβ,0, lying in the invariant subspace I×{0}, with a locally riddled basin of attraction. Rigorous analysis for the occurrence of locally riddled basin is given. Using the monotonicity and concavity of the fiber maps in our model, we give a classification, with respect to the number of attracting invariant graphs. We examine the bifurcation effects that appear in our model on varying parameter; in particular, a nonregular temporal bursting and the appearance of an on-off intermittency is investigated. Finally, we show that the model undergoes a nonhysteretic blowout bifurcation.