| Abstract: |
| We study necessary conditions for existence of real-analytic first integrals and real-analytic integrability for perturbations of integrable systems in the sense of Bogoyavlenskij including non-Hamiltonian ones. Moreover, we compare our results with classical results of Poincar\`{e} and Kozlov for systems written in action and angle coordinates and discuss their relationships with Melnikov methods for periodic perturbations of single-degree-of-freedom Hamiltonian systems. The latter discussion reveals that the perturbed systems can be real-analytically nonintgrable even if there exists no transverse homoclinic orbit to a periodic orbit. This is a joint work with Kazuyuki Yagasaki (Kyoto University). |
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