Existence of multiple attractors and the nature of bifurcations in a discontinuous logistic map

We present the analytical investigations on a logistic map with a discontin uity at the centre. An explanation for the bifurcation phenomenon in discon tinuous systems is presented. We establish that whenever the elements of an n-cycle (n > 1) approach the discontinuities of the nth iterate of the map , a bifurcation other than the usual period-doubling one takes place. The p eriods of the cycles decrease in an arithmetic progression, as the control parameter is varied. The system also shows the presence of multiple attract ors. Our results are verified by numerical experiments as well.