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July  2018, 38(7): 3439-3457. doi: 10.3934/dcds.2018147

Non-floquet invariant tori in reversible systems

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

Received  July 2017 Revised  February 2018 Published  April 2018

Fund Project: This work is supported by National Natural Science Foundation of China (11501234).

In this paper we obtain a theorem about the persistence of non-floquet invariant tori of analytic reversible systems by an improved KAM iteration. This theorem can be applied to solve the persistence problem of completely hyperbolic-type degenerate invariant tori for a class of reversible system.

Citation: Xiaocai Wang. Non-floquet invariant tori in reversible systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3439-3457. doi: 10.3934/dcds.2018147
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. W. BroerJ. 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.  Google Scholar

[5]

H. W. BroerM. C. CiocciH. 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.  Google Scholar

[6]

H. W. BroerM. 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.  Google Scholar

[7]

L. H. Eliasson, Perturbations of stable invariant tori, Ann. Scuola Norm. Sup. Pisa, 15 (1988), 119-147.   Google Scholar

[8]

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.  Google Scholar

[9]

Y. HanY. Li and Y. Yi, Degenerate lower-dimensional tori in Hamiltonian systems, J. Differ. Equations, 227 (2006), 670-691.  doi: 10.1016/j.jde.2006.02.006.  Google Scholar

[10]

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

[11]

S. B. Kuksin, Nearly Integrable Infinte Dimensional Hamiltonian Systems, Lecture Notes in Math., Vol. 1556, Springer-Verlag, New York/Berlin, 1993. doi: 10.1007/BFb0092243.  Google Scholar

[12]

Y. Li and Y. Yi, Persistence of lower-dimensional tori of general types in Hamiltonian systems, Trans. Amer. Math. Soc., 357 (2005), 1565-1600.  doi: 10.1090/S0002-9947-04-03564-0.  Google Scholar

[13]

(1022821) J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608.  doi: 10.1007/BF01221590.  Google Scholar

[14]

J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: A survey, Physics D, 112 (1998), 1-39.  doi: 10.1016/S0167-2789(97)00199-1.  Google Scholar

[15]

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

[16]

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

[17]

J. Moser, Stable and Random Motions in Dynamical Systems, with Special Emphasis on Celestial Mechanics, Princeton University Press, Princeton, NJ, 2001.  Google Scholar

[18]

I. O. Parasyuk, Conservation of quasiperiodic motions in reversible multifrequency systems, Dokl. Akad. Nauk Ukrain. SSR. Ser.A, 9 (1982), 19-22, 85.   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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.  Google Scholar

[23]

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, Birkhäuser, Basel, 12 (1994), 184-199. doi: 10.1007/978-3-0348-7515-8_14.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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.  Google Scholar

[28]

X. WangJ. 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.  Google Scholar

[29]

X. WangD. 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.  Google Scholar

[30]

X. WangJ. 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.  Google Scholar

[31]

X. WangJ. Xu and D. Zhang, On the persistence of degenerate lower-dimensional tori in reversible systems, Ergodic Theory Dynam. Systems, 35 (2015), 2311-2333.  doi: 10.1017/etds.2014.34.  Google Scholar

[32]

X. WangD. Zhang and J. Xu, On the persistence of lower-dimensional elliptic tori with prescribed frequency in reversible systems, Discrete and Continuous Dynamical System, 36 (2016), 1677-1692.  doi: 10.3934/dcds.2016.36.1677.  Google Scholar

[33]

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.  Google Scholar

[34]

H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89.  doi: 10.1090/S0002-9947-1934-1501735-3.  Google Scholar

[35]

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.  Google Scholar

[36]

J. Xu and J. You, Persistence of the non-twist invariant tori for nearly integrable Hamiltonian systems, Proc. Amer. Math. Soc., 138 (2010), 2385-2395.  doi: 10.1090/S0002-9939-10-10151-8.  Google Scholar

[37]

J. Xu, Persistence of elliptic lower-dimensional invariant tori for small perturbation of degenerate integrable Hamiltonian systems, J. Math. Anal. Appl., 208 (1997), 372-387.  doi: 10.1006/jmaa.1997.5313.  Google Scholar

[38]

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

[39]

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

[40]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. Ⅰ, Comm. Pure Appl. Math., 28 (1975), 1-140.  doi: 10.1002/cpa.3160280104.  Google Scholar

[41]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. Ⅱ, Comm. Pure Appl. Math., 29 (1976), 49-111.  doi: 10.1002/cpa.3160290104.  Google Scholar

[42]

D. ZhangJ. Xu and X. Wang, A New KAM iteration with nearly infinitely small steps in reversible systems of polynomial character, Qual. Theory Dyn. Syst, 17 (2018), 271-289.  doi: 10.1007/s12346-017-0229-0.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. W. BroerJ. 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.  Google Scholar

[5]

H. W. BroerM. C. CiocciH. 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.  Google Scholar

[6]

H. W. BroerM. 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.  Google Scholar

[7]

L. H. Eliasson, Perturbations of stable invariant tori, Ann. Scuola Norm. Sup. Pisa, 15 (1988), 119-147.   Google Scholar

[8]

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.  Google Scholar

[9]

Y. HanY. Li and Y. Yi, Degenerate lower-dimensional tori in Hamiltonian systems, J. Differ. Equations, 227 (2006), 670-691.  doi: 10.1016/j.jde.2006.02.006.  Google Scholar

[10]

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

[11]

S. B. Kuksin, Nearly Integrable Infinte Dimensional Hamiltonian Systems, Lecture Notes in Math., Vol. 1556, Springer-Verlag, New York/Berlin, 1993. doi: 10.1007/BFb0092243.  Google Scholar

[12]

Y. Li and Y. Yi, Persistence of lower-dimensional tori of general types in Hamiltonian systems, Trans. Amer. Math. Soc., 357 (2005), 1565-1600.  doi: 10.1090/S0002-9947-04-03564-0.  Google Scholar

[13]

(1022821) J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608.  doi: 10.1007/BF01221590.  Google Scholar

[14]

J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: A survey, Physics D, 112 (1998), 1-39.  doi: 10.1016/S0167-2789(97)00199-1.  Google Scholar

[15]

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

[16]

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

[17]

J. Moser, Stable and Random Motions in Dynamical Systems, with Special Emphasis on Celestial Mechanics, Princeton University Press, Princeton, NJ, 2001.  Google Scholar

[18]

I. O. Parasyuk, Conservation of quasiperiodic motions in reversible multifrequency systems, Dokl. Akad. Nauk Ukrain. SSR. Ser.A, 9 (1982), 19-22, 85.   Google Scholar

[19]

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

[20]

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

[21]

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

[22]

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.  Google Scholar

[23]

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, Birkhäuser, Basel, 12 (1994), 184-199. doi: 10.1007/978-3-0348-7515-8_14.  Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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.  Google Scholar

[28]

X. WangJ. 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.  Google Scholar

[29]

X. WangD. 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.  Google Scholar

[30]

X. WangJ. 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.  Google Scholar

[31]

X. WangJ. Xu and D. Zhang, On the persistence of degenerate lower-dimensional tori in reversible systems, Ergodic Theory Dynam. Systems, 35 (2015), 2311-2333.  doi: 10.1017/etds.2014.34.  Google Scholar

[32]

X. WangD. Zhang and J. Xu, On the persistence of lower-dimensional elliptic tori with prescribed frequency in reversible systems, Discrete and Continuous Dynamical System, 36 (2016), 1677-1692.  doi: 10.3934/dcds.2016.36.1677.  Google Scholar

[33]

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.  Google Scholar

[34]

H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89.  doi: 10.1090/S0002-9947-1934-1501735-3.  Google Scholar

[35]

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.  Google Scholar

[36]

J. Xu and J. You, Persistence of the non-twist invariant tori for nearly integrable Hamiltonian systems, Proc. Amer. Math. Soc., 138 (2010), 2385-2395.  doi: 10.1090/S0002-9939-10-10151-8.  Google Scholar

[37]

J. Xu, Persistence of elliptic lower-dimensional invariant tori for small perturbation of degenerate integrable Hamiltonian systems, J. Math. Anal. Appl., 208 (1997), 372-387.  doi: 10.1006/jmaa.1997.5313.  Google Scholar

[38]

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

[39]

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

[40]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. Ⅰ, Comm. Pure Appl. Math., 28 (1975), 1-140.  doi: 10.1002/cpa.3160280104.  Google Scholar

[41]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. Ⅱ, Comm. Pure Appl. Math., 29 (1976), 49-111.  doi: 10.1002/cpa.3160290104.  Google Scholar

[42]

D. ZhangJ. Xu and X. Wang, A New KAM iteration with nearly infinitely small steps in reversible systems of polynomial character, Qual. Theory Dyn. Syst, 17 (2018), 271-289.  doi: 10.1007/s12346-017-0229-0.  Google Scholar

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