This study provides a method for calculating first-order approximations of the Lyapunov exponents of systems with discontinuity in the presence of noise in order to characterize the stability of motions in those systems. This paper in particular illustrates the results of the ways in which noise influences period-doubling bifurcation in the parameter space of an integrate-and-fire model with a periodically modulated reset level. For evaluating a stochastic version of period-doubling bifurcation, the first-passage-time problem of the Ornstein-Uhlenbeck process is used to define a Markov operator governing the transition of a reset-level phase density. The results on the stochastic bifurcation are thus compared with numerical computations of angles and moduli of eigenvalues of the Markov operator that describes the firing properties of the model.

• Received 31 August 2000

DOI:https://doi.org/10.1103/PhysRevE.65.021901