Subharmonic Melnikov functions and nonintegrability for autonomous and non-autonomous perturbations of single-degree-of-freedom Hamiltonian systems near periodic orbits. (English) Zbl 1543.37057
Summary: We study autonomous and non-autonomous perturbations of single-degree-of-freedom Hamiltonian systems and give sufficient conditions for their real-analytic non-integrability near periodic orbits of the unperturbed systems such that the first integrals and commutative vector fields depend analytically on the small parameter by using the subharmonic Melnikov functions. Moreover, we show that autonomous dissipative perturbations prevent real-analytic integrability of these systems. Our results reveal that the perturbed systems can be real-analytically non-integrable even if there is no homoclinic/heteroclinic orbit in the unperturbed systems. We illustrate our theory with a periodically forced duffing equation and a damped Morse oscillator.
MSC:
| 37J30 | Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria) |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37J46 | Periodic, homoclinic and heteroclinic orbits of finite-dimensional Hamiltonian systems |
| 34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |
| 37C27 | Periodic orbits of vector fields and flows |
| 34C25 | Periodic solutions to ordinary differential equations |
| 70H08 | Nearly integrable Hamiltonian systems, KAM theory |
| 70K43 | Quasi-periodic motions and invariant tori for nonlinear problems in mechanics |
Keywords:
integrability; subharmonic Melnikov function; dissipative systems; planar systems; nearly integrable systemsReferences:
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