Dynamical properties of discontinuous and noninvertible two-dimensional map

Abstract

This paper reports five different features in piecewise continuous and noninvertible system described by two conservative maps or one conservative map and another dissipative map. The features are as follows: the stochastic web bounded by images of discontinuous borderline is the only chaotic trajectory; phase collapse is caused by the irreversibility which makes some points to have two pre-images; fat fractal forbidden web induced by irreversibility; riddled-like attraction basin in stochastic web and forbidden web, when there are different attractors coexisting; and the impossibility to tell to which attractor an initial condition will approach. In an integrate-and-fire circuit, we find that the features still exist in the circuit system described by two dissipative maps, so they are common features in discontinuous and noninvertible maps.