Modulated motion and infinite-period homoclinic bifurcation for parametrically excited Liénard systems

Abstract

A parametrically excited Liénard system is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Two coupled equations for the amplitude and the phase of solutions are derived. Their fixed points correspond to limit cycles for the Liénard system and we determine stability of steady-state response as well as response-parametric excitation and response-frequency curves. We use the Poincaré–Bendixson theorem, the Dulac's criterion and energy considerations to study existence and characteristics of limit cycles of the two coupled equations. A limit cycle corresponds to a modulated motion in the Liénard system. We show that modulated motion can also be obtained for very low values of the parametric excitation and construct an approximate analytic solution. Moreover, we observe an unusual infinite-period homoclinic bifurcation, because in certain cases due to the symmetry of the two coupled equations two stable limit cycles approach a saddle point and merge to form a greater stable limit cycle. Subsequently, this limit cycle and another unstable limit cycle coalesce and annihilate through a saddle-node bifurcation. Comparison with the solution obtained by the numerical integration confirms the validity of our analysis.