\(\Psi\)-series and obstructions to integrability of periodically perturbed one degree of freedom Hamiltonians. (English) Zbl 1037.70509
Summary: A connection between the \(\Psi\)-series local expansion of the solution of a perturbed system of ODEs and the evaluation of the Mel’nikov vector with the method of residues has recently been found by Goriely and Tabor. By following an analogous procedure, we find a straightforward relation between the failure of the compatibility condition of the Painlevé test and the absence of an analytic integral for periodically perturbed Hamiltonians whose unperturbed part does not necessarily possess a homoclinic loop. We apply these results to a periodically perturbed anharmonic oscillator.
MSC:
| 70H05 | Hamilton’s equations |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 37J05 | Relations of dynamical systems with symplectic geometry and topology (MSC2010) |
| 70K40 | Forced motions for nonlinear problems in mechanics |
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