August  2018, 38(8): 3735-3763. doi: 10.3934/dcds.2018162

Degenerate lower dimensional invariant tori in reversible system

School of Mathematical Sciences, Peking University, Beijing 100871, China

* Corresponding author: Shengqing Hu

Received  June 2017 Revised  January 2018 Published  May 2018

Fund Project: The second author is supported by NNSF grant 11231001.

In this paper, we are concerned with the existence of lower dimensional invariant tori in nearly integrable reversible systems. By KAM method, we prove that under some reasonable assumptions, there are many so-called degenerate lower dimensional invariant tori, that is one of normal frequencies is zero.

Citation: Shengqing Hu, Bin Liu. Degenerate lower dimensional invariant tori in reversible system. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3735-3763. doi: 10.3934/dcds.2018162
References:
[1]

V. I. Arnold, Reversible systems, in Nonlinear and Turbulent Processes in Physics, Vol. 3 (Kiev, 1983), Harwood Academic Publ., Chur, 1984, 1161–1174.

[2]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems, Order Amidst Chaos (Lecture notes in Mathematics, 1645). Springer-Verlag, Berlin, 1996.

[3]

H. W. Broer and G. B. Huitema, Unfolding of quasi-periodic tori in reversible systems, J. Dynam. Differential Equations, 7 (1995), 191-212.  doi: 10.1007/BF02218818.

[4]

L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm, Suo. Pisa, 15 (1988), 115-147. 

[5]

S. M. Graff, On the continuation of hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations, 15 (1974), 1-69.  doi: 10.1016/0022-0396(74)90086-2.

[6]

B. Liu, On lower dimensional invariant tori in reversible system, J. Differential Equations, 176 (2001), 158-194.  doi: 10.1006/jdeq.2000.3960.

[7]

J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176.  doi: 10.1007/BF01399536.

[8]

M. B. Sevryuk, Reversible Systems, Lecture Notes in Math., Vol. 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.

[9]

M. B. Sevryuk, Invariant $m$-dimensional tori of reversible systems with a phase space of dimension greater than $2m$, J. Soviet Math., 51 (1990), 2374-2386.  doi: 10.1007/BF01094996.

[10]

M. B. Sevryuk, New results in the reversible KAM theory, in Seminar on Dynamical Systems, (Eds. S. B. Kuksin, V. F. Lazutkin and J. Poschel. ) Birkhauser, Basel, 12 (1994), 184–199. doi: 10.1007/978-3-0348-7515-8_14.

[11]

M. B. Sevryuk, The iteration-approximation decoupling in the reversible KAM theory, Chaos, 5 (1995), 552-565.  doi: 10.1063/1.166125.

[12]

M. B. Sevryuk, Partial preservation frequencies in KAM theory, Nonlinearity, 19 (2006), 1099-1140.  doi: 10.1088/0951-7715/19/5/005.

[13]

X. C. WangJ. X. Xu and D. F. Zhang, Degenerate lower dimensional tori in reversible system, J. Math. Anal. Appl., 387 (2012), 776-790.  doi: 10.1016/j.jmaa.2011.09.030.

[14]

X. C. WangJ. X. Xu and D. F. Zhang, On the persistence of lower-dimensional tori in reversible system, Ergodic Theory and Dynamical Systems, 35 (2015), 2311-2333.  doi: 10.1017/etds.2014.34.

[15]

J. X. Xu, On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point, J. Differential Equations., 250 (2011), 551-571.  doi: 10.1016/j.jde.2010.09.030.

[16]

J. G. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian system, Commun. Math. Phys., 192 (1998), 145-168.  doi: 10.1007/s002200050294.

[17]

J. G. You, Perturbations of lower dimensional tori for Hamiltonian systems, J. Differential Equations., 152 (1999), 1-29.  doi: 10.1006/jdeq.1998.3515.

show all references

References:
[1]

V. I. Arnold, Reversible systems, in Nonlinear and Turbulent Processes in Physics, Vol. 3 (Kiev, 1983), Harwood Academic Publ., Chur, 1984, 1161–1174.

[2]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems, Order Amidst Chaos (Lecture notes in Mathematics, 1645). Springer-Verlag, Berlin, 1996.

[3]

H. W. Broer and G. B. Huitema, Unfolding of quasi-periodic tori in reversible systems, J. Dynam. Differential Equations, 7 (1995), 191-212.  doi: 10.1007/BF02218818.

[4]

L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm, Suo. Pisa, 15 (1988), 115-147. 

[5]

S. M. Graff, On the continuation of hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations, 15 (1974), 1-69.  doi: 10.1016/0022-0396(74)90086-2.

[6]

B. Liu, On lower dimensional invariant tori in reversible system, J. Differential Equations, 176 (2001), 158-194.  doi: 10.1006/jdeq.2000.3960.

[7]

J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176.  doi: 10.1007/BF01399536.

[8]

M. B. Sevryuk, Reversible Systems, Lecture Notes in Math., Vol. 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.

[9]

M. B. Sevryuk, Invariant $m$-dimensional tori of reversible systems with a phase space of dimension greater than $2m$, J. Soviet Math., 51 (1990), 2374-2386.  doi: 10.1007/BF01094996.

[10]

M. B. Sevryuk, New results in the reversible KAM theory, in Seminar on Dynamical Systems, (Eds. S. B. Kuksin, V. F. Lazutkin and J. Poschel. ) Birkhauser, Basel, 12 (1994), 184–199. doi: 10.1007/978-3-0348-7515-8_14.

[11]

M. B. Sevryuk, The iteration-approximation decoupling in the reversible KAM theory, Chaos, 5 (1995), 552-565.  doi: 10.1063/1.166125.

[12]

M. B. Sevryuk, Partial preservation frequencies in KAM theory, Nonlinearity, 19 (2006), 1099-1140.  doi: 10.1088/0951-7715/19/5/005.

[13]

X. C. WangJ. X. Xu and D. F. Zhang, Degenerate lower dimensional tori in reversible system, J. Math. Anal. Appl., 387 (2012), 776-790.  doi: 10.1016/j.jmaa.2011.09.030.

[14]

X. C. WangJ. X. Xu and D. F. Zhang, On the persistence of lower-dimensional tori in reversible system, Ergodic Theory and Dynamical Systems, 35 (2015), 2311-2333.  doi: 10.1017/etds.2014.34.

[15]

J. X. Xu, On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point, J. Differential Equations., 250 (2011), 551-571.  doi: 10.1016/j.jde.2010.09.030.

[16]

J. G. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian system, Commun. Math. Phys., 192 (1998), 145-168.  doi: 10.1007/s002200050294.

[17]

J. G. You, Perturbations of lower dimensional tori for Hamiltonian systems, J. Differential Equations., 152 (1999), 1-29.  doi: 10.1006/jdeq.1998.3515.

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