March  2016, 36(3): 1677-1692. doi: 10.3934/dcds.2016.36.1677

On the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems

1. 

Faculty of mathematics and physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, China

2. 

Department of Mathematics, Southeast University, Nanjing 210096

Received  October 2014 Revised  June 2015 Published  August 2015

This work focuses on the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems. By KAM method and the special structure of unperturbed nonlinear terms, we prove that the invariant torus with given frequency persists under small perturbations. Our result is a generalization of [22].
Citation: Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. On the persistence of lower-dimensional elliptic tori with prescribed frequencies in reversible systems. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1677-1692. doi: 10.3934/dcds.2016.36.1677
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 Math., 1645, Springer-Verlag, Berlin, 1996.

[3]

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

[4]

H. W. Broer, J. Hoo and V. Naudot, Normal linear stability of quasi-periodic tori, J. Differ. Equations, 232 (2007), 355-418. doi: 10.1016/j.jde.2006.08.022.

[5]

H. W. Broer, M. C. Ciocci, H. Hanßmann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori, Physica D, 238 (2009), 309-318. doi: 10.1016/j.physd.2008.10.004.

[6]

H. W. Broer, M. C. Ciocci and H. Hanßmann, The quasi-periodic reversible Hopf bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2605-2623. doi: 10.1142/S021812740701866X.

[7]

H. Hanßmann, Quasi-periodic bifurcations in reversible systems, Regular and Chaotic Dynamics, 16 (2011), 51-60. doi: 10.1134/S1560354710520059.

[8]

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

[9]

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

[10]

J. Moser, Stable and Random Motions in Dynamical Systems, with Special Emphasis on Celestial Mechanics, Annals Mathematics Studies, Vol. 77, Princeton University Press, Princeton, 1973.

[11]

I. O. Parasyuk, Conservation of quasiperiodic motions in reversible multifrequency systems, Dokl. Akad. Nauk Ukrain. SSR. Ser. A, 9 (1982), 19-22.

[12]

J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608. doi: 10.1007/BF01221590.

[13]

J. Pöschel, A lecture on the classical KAM theorem, in Smooth Ergodic Theory and Its Applications, AMS Summer Research Institute (Seattle, 1999) (eds. A. Katok, R. de la Llave, Ya. Pesin and H. Weiss), Proc. Symposia in Pure Mathematics, 69, Amer. Math. Soc., Providence, RI, 2001, 707-732. doi: 10.1090/pspum/069/1858551.

[14]

M. B. Sevryuk, Reversible Systems, Lecture Notes in Math., 1211, Springer-Verlag, Berlin, 1986.

[15]

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

[16]

M. B. Sevryuk, New results in the reversible KAM theory, in Seminar on Dynamical Systems (eds. S. B. Kuksin, V. F. Lazutkin and J. Pöschel), Progr. Nonlinear Differential Equations Appl., 12, Birkhäuser, Basel, 1994, 184-199. doi: 10.1007/978-3-0348-7515-8_14.

[17]

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

[18]

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

[19]

V. N. Tkhai, Reversibility of mechanical systems, J. Appl. Math. Mech., 55 (1991), 461-468. doi: 10.1016/0021-8928(91)90007-H.

[20]

X. Wang and J. Xu, Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition, Discrete and Continuous Dynamical Systems series A, 25 (2009), 701-718. doi: 10.3934/dcds.2009.25.701.

[21]

X. Wang, J. Xu and D. Zhang, Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems, Discrete and Continuous Dynamical Systems Series B, 14 (2010), 1237-1249. doi: 10.3934/dcdsb.2010.14.1237.

[22]

X. Wang, D. Zhang and J. Xu, Persistence of lower dimensional tori for a class of nearly integrable reversible systems, Acta Applicanda Mathematicae, 115 (2011), 193-207. doi: 10.1007/s10440-011-9615-9.

[23]

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

[24]

X. Wang, J. Xu and D. Zhang, On the persistence of degenerate lower-dimensional tori in reversible systems, Ergodic Theory and Dynamical Systems, available on CJO2014.

[25]

B. Wei, Perturbations of lower dimensional tori in the resonant zone for reversible systems, J. Math. Anal. Appl., 253 (2001), 558-577. doi: 10.1006/jmaa.2000.7165.

[26]

J. Xu, Normal form of reversible systems and persistence of lower dimensional tori under weaker nonresonance conditions, SIAM J. Math. Anal., 36 (2004), 233-255. doi: 10.1137/S0036141003421923.

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 Math., 1645, Springer-Verlag, Berlin, 1996.

[3]

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

[4]

H. W. Broer, J. Hoo and V. Naudot, Normal linear stability of quasi-periodic tori, J. Differ. Equations, 232 (2007), 355-418. doi: 10.1016/j.jde.2006.08.022.

[5]

H. W. Broer, M. C. Ciocci, H. Hanßmann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori, Physica D, 238 (2009), 309-318. doi: 10.1016/j.physd.2008.10.004.

[6]

H. W. Broer, M. C. Ciocci and H. Hanßmann, The quasi-periodic reversible Hopf bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2605-2623. doi: 10.1142/S021812740701866X.

[7]

H. Hanßmann, Quasi-periodic bifurcations in reversible systems, Regular and Chaotic Dynamics, 16 (2011), 51-60. doi: 10.1134/S1560354710520059.

[8]

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

[9]

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

[10]

J. Moser, Stable and Random Motions in Dynamical Systems, with Special Emphasis on Celestial Mechanics, Annals Mathematics Studies, Vol. 77, Princeton University Press, Princeton, 1973.

[11]

I. O. Parasyuk, Conservation of quasiperiodic motions in reversible multifrequency systems, Dokl. Akad. Nauk Ukrain. SSR. Ser. A, 9 (1982), 19-22.

[12]

J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608. doi: 10.1007/BF01221590.

[13]

J. Pöschel, A lecture on the classical KAM theorem, in Smooth Ergodic Theory and Its Applications, AMS Summer Research Institute (Seattle, 1999) (eds. A. Katok, R. de la Llave, Ya. Pesin and H. Weiss), Proc. Symposia in Pure Mathematics, 69, Amer. Math. Soc., Providence, RI, 2001, 707-732. doi: 10.1090/pspum/069/1858551.

[14]

M. B. Sevryuk, Reversible Systems, Lecture Notes in Math., 1211, Springer-Verlag, Berlin, 1986.

[15]

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

[16]

M. B. Sevryuk, New results in the reversible KAM theory, in Seminar on Dynamical Systems (eds. S. B. Kuksin, V. F. Lazutkin and J. Pöschel), Progr. Nonlinear Differential Equations Appl., 12, Birkhäuser, Basel, 1994, 184-199. doi: 10.1007/978-3-0348-7515-8_14.

[17]

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

[18]

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

[19]

V. N. Tkhai, Reversibility of mechanical systems, J. Appl. Math. Mech., 55 (1991), 461-468. doi: 10.1016/0021-8928(91)90007-H.

[20]

X. Wang and J. Xu, Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition, Discrete and Continuous Dynamical Systems series A, 25 (2009), 701-718. doi: 10.3934/dcds.2009.25.701.

[21]

X. Wang, J. Xu and D. Zhang, Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems, Discrete and Continuous Dynamical Systems Series B, 14 (2010), 1237-1249. doi: 10.3934/dcdsb.2010.14.1237.

[22]

X. Wang, D. Zhang and J. Xu, Persistence of lower dimensional tori for a class of nearly integrable reversible systems, Acta Applicanda Mathematicae, 115 (2011), 193-207. doi: 10.1007/s10440-011-9615-9.

[23]

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

[24]

X. Wang, J. Xu and D. Zhang, On the persistence of degenerate lower-dimensional tori in reversible systems, Ergodic Theory and Dynamical Systems, available on CJO2014.

[25]

B. Wei, Perturbations of lower dimensional tori in the resonant zone for reversible systems, J. Math. Anal. Appl., 253 (2001), 558-577. doi: 10.1006/jmaa.2000.7165.

[26]

J. Xu, Normal form of reversible systems and persistence of lower dimensional tori under weaker nonresonance conditions, SIAM J. Math. Anal., 36 (2004), 233-255. doi: 10.1137/S0036141003421923.

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