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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 and Continuous Dynamical Systems, 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.

[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. 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.

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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.

[21]

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

[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.

[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.

[24]

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

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[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.

[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.

[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.

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. 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.

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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.

[21]

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

[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.

[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.

[24]

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

[25]

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

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[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.

[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.

[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.

[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.

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