Explicit solutions of normal form of driven oscillatory systems in entrainment bands

Abstract

As in a prior article (Ref. 1), we consider an oscillatory dissipative system driven by external sinusoidal perturbations of given amplitude Q and frequency ω. The kinetic equations are transformed to normal form and solved for small Q near a Hopf bifurcation to oscillations in the autonomous system. Whereas before we chose irrational ratios of the frequency of the autonomous system ωn to ω, with quasiperiodic response of the system to the perturbation, we now choose rational coprime ratios, with periodic response (entrainment). The dissipative system has either two variables or is adequately described by two variables near the bifurcation. We obtain explicit solutions and develop these in detail for ωn/ω=1; 1:2; 2:1; 1:3; 3:1. We choose a specific dissipative model (Brusselator) and test the theory by comparison with full numerical solutions. The analytic solutions of the theory give an excellent approximation for the autonomous system near the bifurcation. The theoretically predicted and calculated entrainment bands agree very well for small Q in the vicinity of the bifurcation (small μ); deviations increase with increasing Q and μ. The theory is applicable to one or two external periodic perturbations.