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.

Partial and complete periodic synchronization in coupled discontinuous map lattices

“…[15][16][17][18] In fact, there are many practical systems that show discontinuous dynamical properties, such as nerve cells, [19] electronic circuits, [20] relaxation oscillators, and impact oscillators. [21][22][23][24][25] Here we study the synchronized patterns on the coupled map lattices (CMLs) whose individual dynamics is both discontinuous and non-invertible. One of the authors and his cooperators have conducted intensive investigation on those kinds of individual dynamical systems, and found new types of characteristic dynamics compared with those in systems that are smooth everywhere, which include type-V intermittencies, [23] hole-induced crises, [25] multiple Devil's staircases, [26] coexistence of attractors, [27,28] and so on.…”

Discontinuous bifurcation and coexistence of attractors in a piecewise linear map with a gap

“…The interesting phenomena observed in these systems include type V intermittencies, [11,21] crises in-duced by collision of attractors with discontinuities or mapping holes, [23−25] multiple Devil's staircases, [26,27] coexistence of attractors, [12,22] and many others. [28,29] In this paper, we report our observation on a series of characteristic coexistence of a period-5 orbit with different types of attractors (periodic or chaotic attractors). Section 2 shows the model and system employed in the work.…”