American Institute of Mathematical Sciences

August  2016, 9(4): 1109-1118. doi: 10.3934/dcdss.2016044

Normal forms à la Moser for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium

 1 School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom, United Kingdom

Received  March 2015 Revised  August 2015 Published  August 2016

The classical theorem of Moser, on the existence of a normal form in the neighbourhood of a hyperbolic equilibrium, is extended to a class of real-analytic Hamiltonians with aperiodically time-dependent perturbations. A stronger result is obtained in the case in which the perturbing function exhibits a time decay.
Citation: Alessandro Fortunati, Stephen Wiggins. Normal forms à la Moser for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1109-1118. doi: 10.3934/dcdss.2016044
References:
 [1] L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 144pp. [2] A. Fortunati and S. Wiggins, Persistence of Diophantine flows for quadratic nearly integrable Hamiltonians under slowly decaying aperiodic time dependence, Regul. Chaotic Dyn., 19 (2014), 586-600. doi: 10.1134/S1560354714050062. [3] A. Fortunati and S. Wiggins, A Kolmogorov theorem for nearly integrable Poisson systems with asymptotically decaying time-dependent perturbation, Regul. Chaotic Dyn., 20 (2015), 476-485. doi: 10.1134/S1560354715040061. [4] G. Gallavotti, Hamilton-Jacobi's equation and Arnold's diffusion near invariant tori in a priori unstable isochronous systems, Rend. Sem. Mat. Univ. Politec. Torino, 55 (1997), 291-302 (1999), Jacobian conjecture and dynamical systems (Torino, 1997). [5] A. Giorgilli, On a Theorem of Lyapounov, Rendiconti dell'Istituto Lombardo Accademia di Scienze e Lettere, Classe di Scienze Matematiche e Naturali, 146 (2012), 133-160. [6] A. Giorgilli, Persistence of invariant tori., http://www.mat.unimi.it/users/antonio/hamsys/hamsys.html. [7] A. Giorgilli, Exponential stability of Hamiltonian systems, in Dynamical systems. Part I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003, 87-198. [8] A. Giorgilli and E. Zehnder, Exponential stability for time dependent potentials, Z. Angew. Math. Phys., 43 (1992), 827-855. doi: 10.1007/BF00913410. [9] J. Moser, The analytic invariants of an area-preserving mapping near a hyperbolic fixed point, Comm. Pure Appl. Math., 9 (1956), 673-692. doi: 10.1002/cpa.3160090404.

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References:
 [1] L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 144pp. [2] A. Fortunati and S. Wiggins, Persistence of Diophantine flows for quadratic nearly integrable Hamiltonians under slowly decaying aperiodic time dependence, Regul. Chaotic Dyn., 19 (2014), 586-600. doi: 10.1134/S1560354714050062. [3] A. Fortunati and S. Wiggins, A Kolmogorov theorem for nearly integrable Poisson systems with asymptotically decaying time-dependent perturbation, Regul. Chaotic Dyn., 20 (2015), 476-485. doi: 10.1134/S1560354715040061. [4] G. Gallavotti, Hamilton-Jacobi's equation and Arnold's diffusion near invariant tori in a priori unstable isochronous systems, Rend. Sem. Mat. Univ. Politec. Torino, 55 (1997), 291-302 (1999), Jacobian conjecture and dynamical systems (Torino, 1997). [5] A. Giorgilli, On a Theorem of Lyapounov, Rendiconti dell'Istituto Lombardo Accademia di Scienze e Lettere, Classe di Scienze Matematiche e Naturali, 146 (2012), 133-160. [6] A. Giorgilli, Persistence of invariant tori., http://www.mat.unimi.it/users/antonio/hamsys/hamsys.html. [7] A. Giorgilli, Exponential stability of Hamiltonian systems, in Dynamical systems. Part I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003, 87-198. [8] A. Giorgilli and E. Zehnder, Exponential stability for time dependent potentials, Z. Angew. Math. Phys., 43 (1992), 827-855. doi: 10.1007/BF00913410. [9] J. Moser, The analytic invariants of an area-preserving mapping near a hyperbolic fixed point, Comm. Pure Appl. Math., 9 (1956), 673-692. doi: 10.1002/cpa.3160090404.
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