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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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. Google Scholar

[6]

A. Giorgilli, Persistence of invariant tori.,, , ().   Google Scholar

[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.  Google Scholar

[8]

A. Giorgilli and E. Zehnder, Exponential stability for time dependent potentials, Z. Angew. Math. Phys., 43 (1992), 827-855. doi: 10.1007/BF00913410.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

L. Chierchia and G. Gallavotti, Drift and diffusion in phase space, Ann. Inst. H. Poincaré Phys. Théor., 60 (1994), 144pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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. Google Scholar

[6]

A. Giorgilli, Persistence of invariant tori.,, , ().   Google Scholar

[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.  Google Scholar

[8]

A. Giorgilli and E. Zehnder, Exponential stability for time dependent potentials, Z. Angew. Math. Phys., 43 (1992), 827-855. doi: 10.1007/BF00913410.  Google Scholar

[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.  Google Scholar

[1]

Ghobad Barmalzan, Ali Akbar Hosseinzadeh, Narayanaswamy Balakrishnan. Stochastic comparisons of series-parallel and parallel-series systems with dependence between components and also of subsystems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021101

[2]

Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623

[3]

Jaume Llibre, Yuzhou Tian. Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021228

[4]

David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353

[5]

Juan Carlos Marrero. Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes. Journal of Geometric Mechanics, 2010, 2 (3) : 243-263. doi: 10.3934/jgm.2010.2.243

[6]

Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684

[7]

Rui L. Fernandes, Yuxuan Zhang. Local and global integrability of Lie brackets. Journal of Geometric Mechanics, 2021, 13 (3) : 355-384. doi: 10.3934/jgm.2021024

[8]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[9]

Guillaume Duval, Andrzej J. Maciejewski. Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4589-4615. doi: 10.3934/dcds.2014.34.4589

[10]

Mitsuru Shibayama. Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3707-3719. doi: 10.3934/dcds.2015.35.3707

[11]

A. Ghose Choudhury, Partha Guha. Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2465-2478. doi: 10.3934/dcdsb.2017126

[12]

Kaizhi Wang, Lin Wang, Jun Yan. Aubry-Mather theory for contact Hamiltonian systems II. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021128

[13]

Nathan Glatt-Holtz, Mohammed Ziane. Singular perturbation systems with stochastic forcing and the renormalization group method. Discrete & Continuous Dynamical Systems, 2010, 26 (4) : 1241-1268. doi: 10.3934/dcds.2010.26.1241

[14]

Dumitru Motreanu, Calogero Vetro, Francesca Vetro. Systems of quasilinear elliptic equations with dependence on the gradient via subsolution-supersolution method. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 309-321. doi: 10.3934/dcdss.2018017

[15]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[16]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196

[17]

Alexander Gomilko, Mariusz Lemańczyk, Thierry de la Rue. On Furstenberg systems of aperiodic multiplicative functions of Matomäki, Radziwiłł, and Tao. Journal of Modern Dynamics, 2021, 17: 529-555. doi: 10.3934/jmd.2021018

[18]

Vladimir Răsvan. On the central stability zone for linear discrete-time Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 734-741. doi: 10.3934/proc.2003.2003.734

[19]

Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas. Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5183-5208. doi: 10.3934/dcds.2021073

[20]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, Stock price fluctuation prediction method based on time series analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 915-915. doi: 10.3934/dcdss.2019061

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (109)
  • HTML views (1)
  • Cited by (2)

Other articles
by authors

[Back to Top]