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 f 1 and f 2 , with f 1 being dissipative. Depending on a control parameter d , f 2 can be conservative (for d = d c = 1 ) or dissipative (for d > d c ). 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 − d c ) − γ , η ∝ ( d − d c ) σ , and Δ ∝ ( d − d c ) α . Extending an argument by Jiang et al. (2004), we derive γ = σ + α , which agrees well with numerical simulation data.