Bifurcation analysis of parametrically excited Rayleigh-Lienard oscillators

A parametrically excited Rayleigh-Lienard oscillator 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 and the stability of steady-state periodic solutions as well as parametric excitation-response and frequency-response curves are determined. Comparison with the parametrically excited Lienard oscillator is performed and analytic approximate solutions are checked using numerical integration. Dulac's criterion, the Poincare-Bendixson theorem, and energy considerations are used in order to study the existence and characteristics of limit cycles of the two coupled equations. A limit cycle corresponds to a modulated motion for the Rayleigh-Lienard oscillator. Modulated motion can be also obtained for very low values of the parametric excitation, and in this case, an approximate analytic solution is easily constructed. If the parametric excitation is increased, an infinite-period bifurcation is observed because the modulation period lengthens and becomes infinite, while the modulation amplitude remains finite and suddenly the attractor settles down into a periodic motion. Floquet's theory is used to evaluate the stability of the periodic solutions, and in certain cases, symmetry-breaking bifurcations are predicted. Numerical simulations confirm this scenario and detect chaos and unbounded motions in the instability regions of the periodic solutions.