Explicit solutions of normal form of driven oscillatory systems

摘要

We consider an oscillatory dissipative system driven by external sinusoidal perturbations of given amplitudeQand frequency ohgr;. The kinetic equations are transformed to normal form and solved for smallQ, near a Hopf bifurcation to oscillations in the autonomous system, for ratios ohgr;nto the autonomous frequency of irrational so that the response of the system is quasiperiodic. The system is assumed to have either two variables or is adequately described by two variables near the bifurcation, and we obtain explicit solutions for this general case. The equations show interesting effects of external perturbations on limit cycles, both stable and unstable. Next we treat a specific model (Brusselator) and show by comparison with results of numerical integration that the theory predicts well the shape of the perturbed limit cycle, its variation with changes in constraints and parameters, and the point of transition from quasiperiodic to periodic response.