Lyapunov-Schmidt Reduction and Melnikov Integrals for Bifurcation of Periodic Solutions in Coupled Oscillators

We consider a system of autonomous ordinary differential equations depending on a small parameter such that the unperturbed system has an invariant manifold of periodic solutions. The problem addressed here is the determination of sufficient geometric conditions for some of the periodic solutions on this invariant manifold to survive after perturbation. The main idea is to use a Lyapunov-Schmidt reduction for an appropriate displacement function in order to obtain the bifurcation function for the problem in a form which can be recognized as a generalization of the subharmonic Melnikov function. Thus, the multidimensional bifurcation problem can be cast in a form where the geometry of the problem is clearly incorporated. An important application can be made in case the uncoupled system of differential equations is a system of oscillators in resonance. In this case the invariant manifold of periodic solutions is just the product of the uncoupled oscillations. When each of the oscillators has one degree of freedom, the bifurcation function is computed by quadrature along the unperturbed oscillations. Additional applications include the computation of entrainment domains for a sinusoidally forced van der Pol oscillator and the computation of mutual synchronization domains for a system of inductively coupled van der Pol oscillators.