Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-28T10:44:52.521Z Has data issue: false hasContentIssue false
  • English
  • Français

On the persistence of degenerate lower-dimensional tori in reversible systems

Published online by Cambridge University Press:  07 August 2014

XIAOCAI WANG ,
JUNXIANG XU  and
DONGFENG ZHANG
XIAOCAI WANG
Affiliation:
Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, PR China email [email protected], [email protected]
JUNXIANG XU
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, PR China email [email protected], [email protected]
DONGFENG ZHANG
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, PR China email [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This work focuses on the persistence of lower-dimensional tori with prescribed frequencies and singular normal matrices in reversible systems. By the Kolmogorov–Arnold–Moser theory and the special structure of unperturbed nonlinear terms in the differential equation, we prove that the invariant torus with given frequency persists under small perturbations. Our result is a generalization of X. Wang et al [Degenerate lower dimensional tori in reversible systems. J. Math. Anal. Appl.387 (2012), 776–790].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 7 , October 2015 , pp. 2311 - 2333
Copyright
© Cambridge University Press, 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, V. I.. Reversible systems. Nonlinear and Turbulent Processes in Physics, Vol. 3 (Kiev, 1983). Harwood Academic Publishers, Chur, 1984, pp. 1161–1174.Google Scholar
Broer, H. W., Huitema, G. B. and Sevryuk, M. B.. Quasi-Periodic Motions in Families of Dynamical Systems, Order Amidst Chaos (Lecture Notes in Mathematics, 1645). Springer-Verlag, Berlin, 1996.Google Scholar
Broer, H. W. and Huitema, G. B.. Unfoldings of quasi-periodic tori in reversible systems. J. Dynam. Differential Equations 7 (1995), 191–212.CrossRefGoogle Scholar
Broer, H. W., Hoo, J. and Naudot, V.. Normal linear stability of quasi-periodic tori. J. Differential Equations 232 (2007), 355–418.CrossRefGoogle Scholar
Broer, H. W., Ciocci, M. C., Hanßmann, H. and Vanderbauwhede, A.. Quasi-periodic stability of normally resonant tori. Phys. D 238 (2009), 309–318.CrossRefGoogle Scholar
Broer, H. W., Ciocci, M. C. and Hanßmann, H.. The quasi-periodic reversible Hopf bifurcation. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 17 (2007), 2605–2623.CrossRefGoogle Scholar
Han, Y., Li, Y. and Yi, Y.. Degenerate lower-dimensional tori in Hamiltonian systems. J. Differential Equations 227 (2006), 670–691.CrossRefGoogle Scholar
Hanßmann, H.. Quasi-periodic bifurcations in reversible systems. Regul. Chaotic Dyn. 16 (2011), 51–60.CrossRefGoogle Scholar
Liu, B.. On lower dimensional invariant tori in reversible systems. J. Differential Equations 176 (2001), 158–194.CrossRefGoogle Scholar
Li, Y. and Yi, Y.. Persistence of lower-dimensional tori of general types in Hamiltonian systems. Trans. Amer. Math. Soc. 357 (2005), 1565–1600.CrossRefGoogle Scholar
Moser, J.. Convergent series expansions for quasi-periodic motions. Math. Ann. 169 (1967), 136–176.CrossRefGoogle Scholar
Moser, J.. Stable and Random Motions in Dynamical Systems, with Special Emphasis on Celestial Mechanics (Annals of Mathematics Studies, 77). Princeton University Press, Princeton, NJ, 1973.Google Scholar
Parasyuk, I. O.. Conservation of quasiperiodic motions in reversible multifrequency systems. Dokl. Akad. Nauk Ukrain. SSR. Ser. A 9 (1982), 19–22.Google Scholar
Pöschel, J.. On elliptic lower-dimensional tori in Hamiltonian systems. Math. Z. 202 (1989), 559–608.CrossRefGoogle Scholar
Pöschel, J.. A lecture on the classical KAM theorem. Smooth Ergodic Theory and Its Applications, AMS Summer Research Institute (Seattle, WA, 1999) (Proceedings of Symposia in Pure Mathematics, 69). Eds. Katok, A., de la Llave, R., Pesin, Ya. and Weiss, H.. American Mathematical Society, Providence, RI, 2001, pp. 707–732.CrossRefGoogle Scholar
Pöschel, J.. Integrability of Hamiltonian systems on Cantor sets. Commun. Pure Appl. Math. 35 (1982), 653–696.CrossRefGoogle Scholar
Sevryuk, M. B.. Reversible Systems (Lecture Notes in Mathematics, 1211). Springer-Verlag, Berlin, 1986.CrossRefGoogle Scholar
Sevryuk, M. B.. Invariant m-dimensional tori of reversible systems with phase space of dimension greater than 2m. J. Soviet Math. 51 (1990), 2374–2386.CrossRefGoogle Scholar
Sevryuk, M. B.. New results in the reversible KAM theory. Seminar on Dynamical Systems. Eds. Kuksin, S. B., Lazutkin, V. F. and Pöschel, J.. Birkhäuser, Basel, 1994, pp. 184–199.CrossRefGoogle Scholar
Sevryuk, M. B.. The iteration–approximation decoupling in the reversible KAM theory. Chaos 5 (1995), 552–565.CrossRefGoogle ScholarPubMed
Sevryuk, M. B.. Partial preservation of frequencies in KAM theory. Nonlinearity 19 (2006), 1099–1140.CrossRefGoogle Scholar
Tkhai, V. N.. On the stability of mechanical systems under the action of position forces. J. Appl. Math. Mech. 44 (1981), 40–48.Google Scholar
Tkhai, V. N.. Reversibility of mechanical systems. J. Appl. Math. Mech. 55 (1991), 461–468.CrossRefGoogle Scholar
Wei, B.. Perturbations of lower dimensional tori in the resonant zone for reversible systems. J. Math. Anal. Appl. 253 (2001), 558–577.Google Scholar
Wang, X. and Xu, J.. Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann’s non-degeneracy condition. Discrete Contin. Dynam. Syst. Ser. A 25 (2009), 701–718.CrossRefGoogle Scholar
Wang, X., Xu, J. and Zhang, D.. Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems. Discrete Contin. Dynam. Syst. Ser. B 14 (2010), 1237–1249.Google Scholar
Wang, X., Zhang, D. and Xu, J.. Persistence of lower dimensional tori for a class of nearly integrable reversible systems. Acta Appl. Math. 115 (2011), 193–207.CrossRefGoogle Scholar
Wang, X., Xu, J. and Zhang, D.. Degenerate lower dimensional tori in reversible systems. J. Math. Anal. Appl. 387 (2012), 776–790.CrossRefGoogle Scholar
Wei, B.. Perturbations of lower dimensional tori in the resonant zone for reversible systems. J. Math. Anal. Appl. 253 (2001), 558–557.Google Scholar
Whitney, H.. Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36 (1934), 63–89.CrossRefGoogle Scholar
Xu, J.. Normal form of reversible systems and persistence of lower dimensional tori under weaker nonresonance conditions. SIAM J. Math. Anal. 36 (2004), 233–255.CrossRefGoogle Scholar
Xu, J. and Jiang, S.. Reducibility for a class of nonlinear quasi-periodic differential equations with degenerate equilibrium point under small perturbation. Ergod. Th. & Dynam. Sys. 31 (2010), 599–611.CrossRefGoogle Scholar
Xu, J.. On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point. J. Differential Equations 250 (2011), 551–571.CrossRefGoogle Scholar
Xu, J.. On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems. Discrete Contin. Dyn. Syst. 33 (2013), 2593–2619.CrossRefGoogle Scholar
You, J.. A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems. Commun. Math. Phys. 192 (1998), 145–168.CrossRefGoogle Scholar