Stochastic period-doubling bifurcation analysis of a Rössler system with a bounded random parameter

Abstract

This paper aims to study the stochastic period-doubling bifurcation of the three-dimensional Rössler system with an arch-like bounded random parameter. First, we transform the stochastic Rössler system into its equivalent deterministic one in the sense of minimal residual error by the Chebyshev polynomial approximation method. Then, we explore the dynamical behaviour of the stochastic Rössler system through its equivalent deterministic system by numerical simulations. The numerical results show that some stochastic period-doubling bifurcation, akin to the conventional one in the deterministic case, may also appear in the stochastic Rössler system. In addition, we also examine the influence of the random parameter intensity on bifurcation phenomena in the stochastic Rössler system.

Interpreting a period-adding bifurcation scenario in neural bursting patterns using border-collision bifurcation in a discontinuous map of a slow control variable

“…Employing this simple discontinuous map controlled by a slow variable, we interpret not only the structure of the first return map formed by ISI series of the periodic bursting but also the mechanism for the transition from period k to period k + 1 bursting. The effect of noise on behaviours near the bifurcation point, [35,55,56] such as spiking or bursting near the parameter region from spiking to bursting, was studied in a previous study. [35] Type I bursting and type II bursting were induced when noise was introduced into the first and third equations of the theoretical models, respectively.…”