Eur. Phys. J. Plus (2012) 127: 59
https://doi.org/10.1140/epjp/i2012-12059-1

Regular Article

Dynamics of a biological system with time-delayed noise

¹ Laboratory of Modeling and Simulation in Engineering, Biomimetics and Prototypes, Faculty of Science, University of Yaoundé I, Box 812, Yaoundé, Cameroon
² Department of Physics, Faculty of Science, University of Douala, Box 24 157, Douala, Cameroon

^* e-mail: ryamapi@yahoo.fr

Received: 28 December 2011
Published online: 28 May 2012

Abstract

This work studies the dynamics of a biological system with a time-delayed random excitation. The model used is a multi-limit-cycle variation of the Van der Pol oscillator introduced to analyze enzymatic substrate reactions in brain waves. We found birhythmicity in the system and observed that the two frequencies are strongly influenced by the nonlinear coefficients. In the presence of a random excitation, such as a Gaussian white noise, the stability of the attractor is measured by calculating Kramer’s escape time. The activation energy is also estimated. Taking into account the time delay on the Gaussian white noise, we showed that the problem of the traditional Kramer escape time can be extended to analyze a bistable system under the influence of a noise made up of a superposition of a Gaussian white noise and its replicas delayed of time τ. The escape times for the intervals 0 < t < τ and τ < t < 2τ are calculated analytically.