In this paper, a one parameter family (F β) 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 A β, 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

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