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A characterization of isochronous centres in terms of symmetries.

Emilio Freire, Gasull, Armengol, Guillamon, Antoni 2 (2004)

Revista Matemática Iberoamericana

We present a description of isochronous centres of planar vector fields X by means of their groups of symmetries. More precisely, given a normalizer U of X (i.e., [X,U]= µ X, where µ is a scalar function), we provide a necessary and sufficient isochronicity condition based on µ. This criterion extends the result of Sabatini and Villarini that establishes the equivalence between isochronicity and the existence of commutators ([X,U]= 0). We put also special emphasis on the mechanical aspects of isochronicity;...

Approximative sequences and almost homoclinic solutions for a class of second order perturbed Hamiltonian systems

Marek Izydorek, Joanna Janczewska (2014)

Banach Center Publications

In this work we will consider a class of second order perturbed Hamiltonian systems of the form q ̈ + V q ( t , q ) = f ( t ) , where t ∈ ℝ, q ∈ ℝⁿ, with a superquadratic growth condition on a time periodic potential V: ℝ × ℝⁿ → ℝ and a small aperiodic forcing term f: ℝ → ℝⁿ. To get an almost homoclinic solution we approximate the original system by time periodic ones with larger and larger time periods. These approximative systems admit periodic solutions, and an almost homoclinic solution for the original system is obtained...

Conley type index and Hamiltonian inclusions

Zdzisław Dzedzej (2010)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

This paper is based mainly on the joint paper with W. Kryszewski [Dzedzej, Z., Kryszewski, W.: Conley type index applied to Hamiltonian inclusions. J. Math. Anal. Appl. 347 (2008), 96–112.], where cohomological Conley type index for multivalued flows has been applied to prove the existence of nontrivial periodic solutions for asymptotically linear Hamiltonian inclusions. Some proofs and additional remarks concerning definition of the index and special cases are given.

Connecting orbits of time dependent Lagrangian systems

Patrick Bernard (2002)

Annales de l’institut Fourier

We generalize to higher dimension results of Birkhoff and Mather on the existence of orbits wandering in regions of instability of twist maps. This generalization is strongly inspired by the one proposed by Mather. However, its advantage is that it contains most of the results of Birkhoff and Mather on twist maps.

Density of chaotic dynamics in periodically forced pendulum-type equations

Elena Bosetto, Enrico Serra, Susanna Terracini (2001)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We announce that a class of problems containing the classical periodically forced pendulum equation displays the main features of chaotic dynamics for a dense set of forcing terms in a space of periodic functions with zero mean value. The approach is based on global variational methods.

Diffusion time and splitting of separatrices for nearly integrable isochronous Hamiltonian systems

Massimiliano Berti, Philippe Bolle (2000)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider the problem of Arnold’s diffusion for nearly integrable isochronous Hamiltonian systems. We prove a shadowing theorem which improves the known estimates for the diffusion time. We also justify for three time scales systems that the splitting of the separatrices is correctly predicted by the Poincaré-Melnikov function.

Existence of solutions and of multiple solutions for nonlinear nonsmooth periodic systems

Evgenia H. Papageorgiou, Nikolaos S. Papageorgiou (2004)

Czechoslovak Mathematical Journal

In this paper we examine nonlinear periodic systems driven by the vectorial p -Laplacian and with a nondifferentiable, locally Lipschitz nonlinearity. Our approach is based on the nonsmooth critical point theory and uses the subdifferential theory for locally Lipschitz functions. We prove existence and multiplicity results for the “sublinear” problem. For the semilinear problem (i.e. p = 2 ) using a nonsmooth multidimensional version of the Ambrosetti-Rabinowitz condition, we prove an existence theorem...

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