A riddled basin escaping crisis and the universality in an integrate-and-fire circuit

Abstract

We investigate an integrate-and-fire model of an electronic relaxation oscillator, which can be described by the discontinuous and non-invertible composition of two mapping functions f1 and f2, with f1 being dissipative. Depending on a control parameter d, f2 can be conservative (for d=dc=1) or dissipative (for d>dc). We find a kind of crisis, which is induced by the escape from a riddled-like attraction basin sea in the phase space. The averaged crisis transient lifetime (〈τ〉), the relative measure of the fat fractal forbidden network (η), and the measure of the escaping hole (Δ) show clear scaling behaviors: 〈τ〉∝(d−dc)−γ, η∝(d−dc)σ, and Δ∝(d−dc)α. Extending an argument by Jiang et al. (2004), we derive γ=σ+α, which agrees well with numerical simulation data.