Homoclinic Orbits, Melnikov’s Method, and Chaos

In the last chapter, we discussed the near resonance continuation theory for periodic orbits of periodically perturbed oscillators. For the case where the unperturbed oscillator has a regular period annulus, we found that there is generally an infinite number of resonances at which a first-order perturbation theory can be used to prove the existence of perturbed periodic orbits. But, as mentioned previously, we cannot conclude from the results of our analysis that the perturbed oscillator has infinitely many periodic orbits. To do so would seem to require a condition that might be impossible to satisfy. Indeed, the nonzero amplitude of the perturbation would have to be made sufficiently small for each of an infinite sequence of continuations corresponding to an infinite sequence of resonant unperturbed periodic orbits that approaches the boundary of a period annulus.