Dynamical Properties of Two Discontinuous and Noninvertible 2-d Dissipative Maps

摘要

We report special crisis features in an integrate-and-fire model of an electronic relaxation oscillator, which can be described by composition of two mapping functions, including a dissipative map f1on [1,χ0)(0<χ01, f2 is dissipative. The composed mapping function on [0,1] is dissipative, discontinuous and noninvertible. When d=l, the phase plane is composed of a stochastic web formed by forward images of the borderline, period-1 and period-6 elliptical islands, and a fat fractal forbidden network. When d>1, the stochastic web suddenly changes to a transient web and the elliptical islands reduces to period-1 or period-6 point attractors. This can be defined as a special crisis. We find that the averaged transient lifetime <τ> shows very good scaling behaviors: <τ>∝(d-dc)-γ with γ=1.18±0.02. This behavior almost does not change when system parameters are greatly changed. In another dissipative, discontinuous and noninvertible mapping system all the characteristics of the phase space remain unchanged, however the similar crisis was not observed. We believe it can be found with some revisions of the mapping functions.