Analysis on recurrence behavior in oscillating networks of biologically relevant organic reactions

Abstract

In this paper, we present a new method based on dynamical system theory to study certain type of slow-fast motions in dynamical systems, for which geometric singular perturbation theory may not be applicable. The method is then applied to consider recurrence behavior in an oscillating network model which is biologically related to organic reactions. We analyze the stability and bifurcation of the equilibrium of the system, and find the conditions for the existence of recurrence, i.e., there exists a "window" in bifurcation diagram between a saddle-node bifurcation point and a Hopf bifurcation point, where the equilibrium is unstable. Simulations are given to show a very good agreement with analytical predictions.

Keywords

• slow-fast motion
• oscillating network
• organic reaction
• sustained oscillation
• recurrence
• stability and bifurcation
• saddle-node bifurcation
• hopf bifurcation
• limit cycle