A Coupled Oscillatory-Reaction System Exhibiting Bifurcation Scheme Leading to a Chaos

Abstract

An oscillatory reaction system is numerically studied, which exhibits a bifurcation scheme leading to a chaos. The model system consists of coupled limit-cycle oscillators and posseses the symmetry for the permutation of the oscillator numbers. In this work the cases of N = 2 and 3 are studied, where N is the number of the oscillators. There arise three distinct states: The phase locked state, the periodic state and the chaos. The main bifurcation scheme leading to the chaos is, (the phase locked state)→(the periodic state)→(the chaos). The analysis of the numerical data is carried out by calculating the power spectrum, the divergence rate of nearby trajectories, the distribution function of the concentration variable and the locus of the concentration variable in the phase space. The chaos is characterized by the broad peaks in the power spectrum and a finite positive value of the divergence rate. The stability analysis of the periodic state shows that a new period of oscillation is not necessarily created at the onset of the chaos. This may be due to the fact that the chaos is of a Lorenz type. The phase diagram including various states is drawn for N = 2, which indicates that several types of the bifurcation schemes leading to the chaos can possibly exist.