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Degenerate with respect to parameters fold-Hopf bifurcations
A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems
1. | Faculty of mathematics and physics, Huaiyin Institute of Technology, Huaian, Jiangsu 223003, China |
2. | Department of Mathematics, Southeast University, Nanjing, Jiangsu 210096, China |
$ω(y) ∈ \mathbb{R}^n$ |
$λ(y) ∈ \mathbb{R}$ |
$\text{deg} (ω/λ ,\mathcal{O},ω_0/λ_0)≠ 0,$ |
$ω_0 =ω(y_0)$ |
$λ_0 = λ(y_0)$ |
$y_0∈\mathcal{O}$ |
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, 2nd 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, 2nd 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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