Harmonic and subharmonic bifurcation in the Brussel model with periodic force

Abstract

A famous model, the chemical reaction-Brussel model with periodic force, is investigated. We study the regular Hopf bifurcation and singular Hopf bifurcation from a basic equilibrium, and show the existence of the subharmonic solutions by using the averaging method and perturbed methods and bifurcation equations. By our analysis it can be shown that the homoclinic orbits do not occur, so we can conjecture that the harmonic oscillation can make successive subharmonic bifurcations, until a chaotic state ultimately develops. The results and methods in this paper are our first step in theoretically treating the transition to a chaotic state in the Brussel model and are appropriate to investigating the general nonlinear oscillation with periodic force.