C. Chicone, LYAPUNOV-SCHMIDT REDUCTION AND MELNIKOV INTEGRALS FOR BIFURCATION OF PERIODIC-SOLUTIONS IN COUPLED OSCILLATORS, Journal of differential equations, 112(2), 1994, pp. 407-447
We consider a system of autonomous ordinary differential equations dep
ending on a small parameter such that the unperturbed system has an in
variant manifold of periodic solutions. The problem addressed here is
the determination of sufficient geometric conditions for some of the p
eriodic solutions on this invariant manifold to survive after perturba
tion. The main idea is to use a Lyapunov-Schmidt reduction for an appr
opriate displacement function in order to obtain the bifurcation funct
ion for the problem in a form which can be recognized as a generalizat
ion of the subharmonic Melnikov function. Thus, the multidimensional b
ifurcation problem can be cast in a form where the geometry of the pro
blem is clearly incorporated. An important application can be made in
case the uncoupled system of differential equations is a system of osc
illators in resonance. In this case the invariant manifold of periodic
solutions is just the product of the uncoupled oscillations. When eac
h of the oscillators has one degree of freedom, the bifurcation functi
on is computed by quadrature along the unperturbed oscillations. Addit
ional applications include the computation of entrainment domains for
a sinusoidally forced van der Pol oscillator and the computation of mu
tual synchronization domains for a system of inductively coupled van d
er Pol oscillators. (c) 1994 Academic Press, Inc.