Persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems

We study persistence of periodic and homoclinic orbits, first integrals commutative vector fields in dynamical systems depending on a small parameter $\varepsilon>0$ give several necessary conditions for their persistence. Here we treat orbits not only to equilibria but also orbits. discuss some relationships these results with the standard subharmonic Melnikov methods time-periodic perturbations single-degree-of-freedom Hamiltonian systems, another version method autonomous multi-degree-of-freedom systems. In particular, show that integral which converges or as perturbation tends zero does exist near unperturbed perturbed if functions are identically connected open sets. illustrate our theory four examples: The periodically forced Duffing oscillator, two identical pendula coupled harmonic rigid body three-mode truncation buckled beam.