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Coupling linearity and twist: an extension of the Poincaré–Birkhoff theorem for Hamiltonian systems

Abstract

We provide an extension of the Poincaré–Birkhoff Theorem for systems coupling linear components with twisting components. Applications are given both to weakly coupled Hamiltonian systems where, e.g., a superlinear or sublinear behaviour is assumed in the nonlinear part of the coupling in order to recover the needed twist conditions, and to local perturbations of superintegrable systems, showing the survival of a number of periodic solutions from a lower-dimensional torus.

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Acknowledgements

We are grateful to two anonymous referees whose comments helped to improve and clarify this manuscript. The authors have been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The paper has been written while P.G. was a postdoctoral fellow of the Istituto Nazionale di Alta Matematica, funded by the project Mathtech–CNR–INdAM.

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Correspondence to Paolo Gidoni.

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Fonda, A., Gidoni, P. Coupling linearity and twist: an extension of the Poincaré–Birkhoff theorem for Hamiltonian systems. Nonlinear Differ. Equ. Appl. 27, 55 (2020). https://doi.org/10.1007/s00030-020-00653-9

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Mathematics Subject Classification

  • 34C25
  • 70H12
  • 37J45

Keywords

  • Periodic solutions
  • Hamiltonian systems
  • Poincaré–Birkhoff theorem