March  2003, 9(2): 233-262. doi: 10.3934/dcds.2003.9.233

On the stability of some properly--degenerate Hamiltonian systems with two degrees of freedom

1. 

Settore di Analisi Funzionale, SISSA/ISAS, Via Beirut 2–4, 34013 Trieste, Italy

2. 

Dipartimento di Matematica, Università "Roma Tre", Largo S. L. Murialdo 1, 00146 Roma, Italy

Received  October 2001 Revised  April 2002 Published  December 2002

Properly degenerate nearly--integrable Hamiltonian systems with two degrees of freedom such that the "intermediate system" depend explicitly upon the angle--variable conjugated to the non--degenerate action--variable are considered and, in particular, model problems motivated by classical examples of Celestial Mechanics, are investigated. Under suitable "convexity" assumptions on the intermediate Hamiltonian, it is proved that, in every energy surface, the action variables stay forever close to their initial values. In "non convex" cases, stability holds up to a small set where, in principle, the degenerate action--variable might (in exponentially long times) drift away from its initial value by a quantity independent of the perturbation. Proofs are based on a "blow up" (complex) analysis near separatrices, KAM techniques and energy conservation arguments.
Citation: Luca Biasco, Luigi Chierchia. On the stability of some properly--degenerate Hamiltonian systems with two degrees of freedom. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 233-262. doi: 10.3934/dcds.2003.9.233
[1]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069

[2]

Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413

[3]

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

[4]

Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257

[5]

Shanshan Liu, Maoan Han. Bifurcation of limit cycles in a family of piecewise smooth systems via averaging theory. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 3115-3124. doi: 10.3934/dcdss.2020133

[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]

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

[8]

Yuri Kifer. Another proof of the averaging principle for fully coupled dynamical systems with hyperbolic fast motions. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1187-1201. doi: 10.3934/dcds.2005.13.1187

[9]

Marcel Freitag. The fast signal diffusion limit in nonlinear chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1109-1128. doi: 10.3934/dcdsb.2019211

[10]

Atul Narang, Sergei S. Pilyugin. Toward an Integrated Physiological Theory of Microbial Growth: From Subcellular Variables to Population Dynamics. Mathematical Biosciences & Engineering, 2005, 2 (1) : 169-206. doi: 10.3934/mbe.2005.2.169

[11]

Andrey V. Kremnev, Alexander S. Kuleshov. Nonlinear dynamics and stability of the skateboard. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 85-103. doi: 10.3934/dcdss.2010.3.85

[12]

Evgeny L. Korotyaev. Estimates for solutions of KDV on the phase space of periodic distributions in terms of action variables. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 219-225. doi: 10.3934/dcds.2011.30.219

[13]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3295-3317. doi: 10.3934/dcds.2020406

[14]

Qihuai Liu, Pedro J. Torres. Orbital dynamics on invariant sets of contact Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021297

[15]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57

[16]

Kaizhi Wang, Lin Wang, Jun Yan. Aubry-Mather theory for contact Hamiltonian systems II. Discrete & Continuous Dynamical Systems, 2022, 42 (2) : 555-595. doi: 10.3934/dcds.2021128

[17]

Jiying Ma, Dongmei Xiao. Nonlinear dynamics of a mathematical model on action potential duration and calcium transient in paced cardiac cells. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2377-2396. doi: 10.3934/dcdsb.2013.18.2377

[18]

Jaume Llibre, Clàudia Valls. Hopf bifurcation for some analytic differential systems in $\R^3$ via averaging theory. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 779-790. doi: 10.3934/dcds.2011.30.779

[19]

Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112

[20]

Xiao-Ping Wang, Xianmin Xu. A dynamic theory for contact angle hysteresis on chemically rough boundary. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 1061-1073. doi: 10.3934/dcds.2017044

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (46)

Other articles
by authors

[Back to Top]