A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems

Xiaocai Wang; Junxiang Xu; Dongfeng Zhang

doi:10.3934/dcds.2017092

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April 2017, 37(4): 2141-2160. doi: 10.3934/dcds.2017092

A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems

Xiaocai Wang ^1, , Junxiang Xu ^2, and Dongfeng Zhang ^2,

1.	Faculty of mathematics and physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, China
2.	Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China

Received April 2016 Revised November 2016 Published December 2016

Fund Project: The first author is supported by National Natural Science Foundation of China (11501234) and Qing Lan Project. The second author is supported by National Natural Science Foundation of China (11371090). The third author is supported by National Natural Science Foundation of China (11001048).

In this paper we consider the persistence of elliptic lower dimensional invariant tori with one normal frequency in reversible systems, andprove that if the frequency mapping

$ω(y) ∈ \mathbb{R}^n$

and normal frequency mapping

$λ(y) ∈ \mathbb{R}$

satisfy that

$\text{deg} (ω/λ ,\mathcal{O},ω_0/λ_0)≠ 0,$

where

$ω_0 =ω(y_0)$

and

$λ_0 = λ(y_0)$

satisfy Melnikov's non-resonance conditions for some

$y_0∈\mathcal{O}$

, then the direction of this frequency for the invariant torus persists under small perturbations. Our result is a generalization of X. Wang et al[Persistence of lower dimensional elliptic invariant tori for a class of nearly integrablereversible systems, Discrete and Continuous Dynamical Systems series B, 14 (2010), 1237-1249].

Keywords: KAM iteration, lower-dimensional tori, topological degree.

Mathematics Subject Classification: Primary:37J40;Secondary:37E99, 47A55, 37F50.

Citation: Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2141-2160. doi: 10.3934/dcds.2017092

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]	V. I. Arnold, Ordinary Differential Equations, Translated from the Russian by Roger Cooke, Second printing of the 1992 edition, Universitext, Springer-Verlag, Berlin, 2006.
[3]	L. Biasco, L. Chierchia and E. Valdinoci, Elliptic two-dimensional invariant tori for the planetary three-body problem, Arch. Rational Mech. Anal., 170 (2003), 91-135. doi: 10.1007/s00205-003-0269-2.
[4]	L. Biasco, L. Chierchia and E. Valdinoci, N-dimensional elliptic invariant tori for the planar (N+1)-body problem, SIAM J. Math. Anal., 37 (2006), 1560-1588. doi: 10.1137/S0036141004443646.
[5]	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.
[6]	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.
[7]	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.
[8]	H. W. Broer, M. C. Ciocci, H. Hanßss mann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori, Physica D, 238 (2009), 309-318. doi: 10.1016/j.physd.2008.10.004.
[9]	H. W. Broer, M. C. Ciocci and H. Hanßss mann, The quasi-periodic reversible Hopf bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2605-2623. doi: 10.1142/S021812740701866X.
[10]	L. Corsi, R. Feola and G. Gentile, Lower-dimensional invariant tori for perturbations of a class of non-convex Hamiltonian functions, J. Stat. Phys., 150 (2013), 156-180. doi: 10.1007/s10955-012-0682-8.
[11]	L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., Ⅳ Serie, 15 (1988), 115-147.
[12]	G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory Dynam. Systems, 27 (2007), 427-457. doi: 10.1017/S0143385706000757.
[13]	A. Giorgilli, U. Locatelli and M. Sansottera, On the convergence of an algorithm constructing the normal form for elliptic lower dimensional tori in planetary systems, Celestial Mech. Dynam. Astronom., 119 (2014), 397-424. doi: 10.1007/s10569-014-9562-7.
[14]	H. Hanßssmann, Quasi-periodic bifurcations in reversible systems, Regular and Chaotic Dynamics, 16 (2011), 51-60. doi: 10.1134/S1560354710520059.
[15]	M. W. Hirsch, Differential Topology, 2^nd edition, Springer-Verlag, New York, 1994.
[16]	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.
[17]	B. Liu, On lower dimensional invariant tori in reversible systems, J. Differ. Equations, 176 (2001), 158-194. doi: 10.1006/jdeq.2000.3960.
[18]	J. W. Milnor, Topology from the Differentiable Viewpoint, 2 edition, Princeton, NJ, Princeton Univ. Press, 1997.
[19]	J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176. doi: 10.1007/BF01399536.
[20]	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.
[21]	M. Nagumo, A theory of degree of mapping based on infinitesimal analysis, Amer. J. Math., 73 (1951), 485-496. doi: 10.2307/2372303.
[22]	I. O. Parasyuk, Conservation of quasiperiodic motions in reversible multifrequency systems, Dokl. Akad. Nauk Ukrain. SSR. Ser.A, 9 (1982), 19-22.
[23]	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 69, Amer. Math. Soc. , Providence, RI (2001), 707–732. doi: 10.1090/pspum/069/1858551.
[24]	M. B. Sevryuk, Reversible Systems, Lecture Notes in Math. 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.
[25]	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.
[26]	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.
[27]	M. B. Sevryuk, The iteration-approximation decoupling in the reversible KAM theory, Chaos, 5 (1995), 552-565. doi: 10.1063/1.166125.
[28]	M. B. Sevryuk, Partial preservation of frequency in KAM theory, Nonlinearity, 19 (2006), 1099-1140. doi: 10.1088/0951-7715/19/5/005.
[29]	V. N. Tkhai, Reversibility of mechanical systems, J. Appl.Math. Mech, 55 (1991), 461-468. doi: 10.1016/0021-8928(91)90007-H.
[30]	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.
[31]	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.
[32]	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.
[33]	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.
[34]	X. Wang, J. 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.
[35]	X. Wang, D. Zhang and J. Xu, On the persistence of lower-dimensional elliptic tori with prescribed frequency in reversible systems, Acta Appl. Math., 115 (2011), 193-207. doi: 10.3934/dcds.2016.36.1677.
[36]	B. Wei, Perturbations of lower dimensional tori in the resonant zone for reversible systems, J. Math. Anal. Appl, 253 (2001), 558-557. doi: 10.1006/jmaa.2000.7165.
[37]	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.
[38]	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.
[39]	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.

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]	V. I. Arnold, Ordinary Differential Equations, Translated from the Russian by Roger Cooke, Second printing of the 1992 edition, Universitext, Springer-Verlag, Berlin, 2006.
[3]	L. Biasco, L. Chierchia and E. Valdinoci, Elliptic two-dimensional invariant tori for the planetary three-body problem, Arch. Rational Mech. Anal., 170 (2003), 91-135. doi: 10.1007/s00205-003-0269-2.
[4]	L. Biasco, L. Chierchia and E. Valdinoci, N-dimensional elliptic invariant tori for the planar (N+1)-body problem, SIAM J. Math. Anal., 37 (2006), 1560-1588. doi: 10.1137/S0036141004443646.
[5]	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.
[6]	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.
[7]	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.
[8]	H. W. Broer, M. C. Ciocci, H. Hanßss mann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori, Physica D, 238 (2009), 309-318. doi: 10.1016/j.physd.2008.10.004.
[9]	H. W. Broer, M. C. Ciocci and H. Hanßss mann, The quasi-periodic reversible Hopf bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 2605-2623. doi: 10.1142/S021812740701866X.
[10]	L. Corsi, R. Feola and G. Gentile, Lower-dimensional invariant tori for perturbations of a class of non-convex Hamiltonian functions, J. Stat. Phys., 150 (2013), 156-180. doi: 10.1007/s10955-012-0682-8.
[11]	L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., Ⅳ Serie, 15 (1988), 115-147.
[12]	G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory Dynam. Systems, 27 (2007), 427-457. doi: 10.1017/S0143385706000757.
[13]	A. Giorgilli, U. Locatelli and M. Sansottera, On the convergence of an algorithm constructing the normal form for elliptic lower dimensional tori in planetary systems, Celestial Mech. Dynam. Astronom., 119 (2014), 397-424. doi: 10.1007/s10569-014-9562-7.
[14]	H. Hanßssmann, Quasi-periodic bifurcations in reversible systems, Regular and Chaotic Dynamics, 16 (2011), 51-60. doi: 10.1134/S1560354710520059.
[15]	M. W. Hirsch, Differential Topology, 2^nd edition, Springer-Verlag, New York, 1994.
[16]	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.
[17]	B. Liu, On lower dimensional invariant tori in reversible systems, J. Differ. Equations, 176 (2001), 158-194. doi: 10.1006/jdeq.2000.3960.
[18]	J. W. Milnor, Topology from the Differentiable Viewpoint, 2 edition, Princeton, NJ, Princeton Univ. Press, 1997.
[19]	J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176. doi: 10.1007/BF01399536.
[20]	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.
[21]	M. Nagumo, A theory of degree of mapping based on infinitesimal analysis, Amer. J. Math., 73 (1951), 485-496. doi: 10.2307/2372303.
[22]	I. O. Parasyuk, Conservation of quasiperiodic motions in reversible multifrequency systems, Dokl. Akad. Nauk Ukrain. SSR. Ser.A, 9 (1982), 19-22.
[23]	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 69, Amer. Math. Soc. , Providence, RI (2001), 707–732. doi: 10.1090/pspum/069/1858551.
[24]	M. B. Sevryuk, Reversible Systems, Lecture Notes in Math. 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.
[25]	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.
[26]	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.
[27]	M. B. Sevryuk, The iteration-approximation decoupling in the reversible KAM theory, Chaos, 5 (1995), 552-565. doi: 10.1063/1.166125.
[28]	M. B. Sevryuk, Partial preservation of frequency in KAM theory, Nonlinearity, 19 (2006), 1099-1140. doi: 10.1088/0951-7715/19/5/005.
[29]	V. N. Tkhai, Reversibility of mechanical systems, J. Appl.Math. Mech, 55 (1991), 461-468. doi: 10.1016/0021-8928(91)90007-H.
[30]	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.
[31]	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.
[32]	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.
[33]	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.
[34]	X. Wang, J. 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.
[35]	X. Wang, D. Zhang and J. Xu, On the persistence of lower-dimensional elliptic tori with prescribed frequency in reversible systems, Acta Appl. Math., 115 (2011), 193-207. doi: 10.3934/dcds.2016.36.1677.
[36]	B. Wei, Perturbations of lower dimensional tori in the resonant zone for reversible systems, J. Math. Anal. Appl, 253 (2001), 558-557. doi: 10.1006/jmaa.2000.7165.
[37]	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.
[38]	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.
[39]	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.

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American Institute of Mathematical Sciences

A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems

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A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems

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