The Melnikov Theory for Subharmonics and Their Bifurcations in Forced Oscillations

Source abstract

The subharmonic Melnikov theory for periodic perturbations of planar Hamiltonian systems is improved. An approximation to the associated Poincaré map in action-angle coordinates is explicitly constructed, and existence, stability, and bifurcation theorems for subharmonics are obtained. In particular, simple formulas for determining the stability of subharmonics and invariant circles bifurcating from them at Hopf bifurcations are obtained, and a degenerate resonance case, which was not appropriately treated in previous references, is discussed. Furthermore, the weak nonlinearity case, in which the unperturbed system is linear, is studied. The results are also useful to describe dynamics near the unperturbed centers in strongly nonlinear systems. Several examples are given to illustrate our theory.