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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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[12]

G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory Dynam. Systems, 27 (2007), 427-457.  doi: 10.1017/S0143385706000757.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

M. W. Hirsch, Differential Topology, 2nd edition, Springer-Verlag, New York, 1994.  Google Scholar

[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.  Google Scholar

[17]

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

[18] J. W. Milnor, Topology from the Differentiable Viewpoint, 2 edition, Princeton, NJ, Princeton Univ. Press, 1997.   Google Scholar
[19]

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

[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.   Google Scholar
[21]

M. Nagumo, A theory of degree of mapping based on infinitesimal analysis, Amer. J. Math., 73 (1951), 485-496.  doi: 10.2307/2372303.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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]

V. I. Arnold, Ordinary Differential Equations, Translated from the Russian by Roger Cooke, Second printing of the 1992 edition, Universitext, Springer-Verlag, Berlin, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[12]

G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory Dynam. Systems, 27 (2007), 427-457.  doi: 10.1017/S0143385706000757.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

M. W. Hirsch, Differential Topology, 2nd edition, Springer-Verlag, New York, 1994.  Google Scholar

[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.  Google Scholar

[17]

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

[18] J. W. Milnor, Topology from the Differentiable Viewpoint, 2 edition, Princeton, NJ, Princeton Univ. Press, 1997.   Google Scholar
[19]

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

[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.   Google Scholar
[21]

M. Nagumo, A theory of degree of mapping based on infinitesimal analysis, Amer. J. Math., 73 (1951), 485-496.  doi: 10.2307/2372303.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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