# American Institue of Mathematical Sciences

2016, 15(4): 1233-1250. doi: 10.3934/cpaa.2016.15.1233

## Persistence of the hyperbolic lower dimensional non-twist invariant torus in a class of Hamiltonian systems

 1 Department of Mathematics and Physics, Hefei University, Hefei, MO 230601, China 2 School of Machinery and Electronic Information, China University of Geosciences, Wuhan, MO 430074, China 3 School of Mathematical Physics, Xuzhou Institute of Technology, Xuzhou, MO 221000, China

Received  August 2015 Revised  February 2016 Published  April 2016

We consider a class of nearly integrable Hamiltonian systems with Hamiltonian being $H(\theta,I,u,v)=h(I)+\frac{1}{2}\sum_{j=1}^{m}\Omega_j(u_j^2-v_j^2)+f(\theta,I,u,v)$. By introducing external parameter and KAM methods, we prove that, if the frequency mapping has nonzero Brouwer topological degree at some Diophantine frequency, the hyperbolic invariant torus with this frequency persists under small perturbations.
Citation: Lei Wang, Quan Yuan, Jia Li. Persistence of the hyperbolic lower dimensional non-twist invariant torus in a class of Hamiltonian systems. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1233-1250. doi: 10.3934/cpaa.2016.15.1233
##### References:
 [1] V.I. Arnold, Proof of a theorem by A. N. Kolmogorov on the persistence of quasi-periodic motions under small perturbations of the Hamiltonian,, \emph{Uspehi Mat. Nauk}, 18 (1963), 13. [2] Q.Y. Bi and J.X. Xu, Persistence of lower dimensional hyperbolic invariant tori for nearly integrable symplectic mappings,, \emph{Qual. Theory Dyn. Syst.}, 13 (2014), 269. doi: 10.1007/s12346-014-0117-9. [3] H. Broer, G. Huitema and M. Sevryuk, Quasi-periodic motions in families of dynamical systems, in Lecture Notes in Mathematics,, Springer, 1645 (1996). [4] A.D. Bruno, Analytic form of differential equations,, \emph{Trans. Moscow Math. Soc.}, 25 (1971), 131. [5] C. Cheng and Y. Sun, Existence of KAM tori in degenerate Hamiltonian systems,, \emph{J. Differential Equations}, 114 (1994), 288. doi: 10.1006/jdeq.1994.1152. [6] L.H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems,, \emph{Ann. Scuola Norm. Sup. Pisa.}, 15 (1988), 115. [7] S.M. Graff, On the conservation of hyperbolic invariant tori for Hamiltonian systems,, \emph{J. Differ. Eqs.}, 15 (1974), 1. [8] A.N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function,, \emph{Dokl. Akad. Nauk. SSSR}, 98 (1954), 527. [9] S.B. Kuksin, Nearly integrable infinite dimensional Hamiltonian systems, in Lecture Notes in Mathematics,, Springer-Verlag, 1556 (1993). [10] V.K. Melnikov, On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function,, \emph{Soviet Math. Dokl.}, 6 (1965), 1592. [11] J. Moser, Convergent series expansions for quasi-periodic motions,, \emph{Math. Ann.}, 169 (1967), 136. [12] J. Pöschel, On elliptic lower dimensional tori in Hamiltonian systems,, \emph{Math. Z.}, 202 (1989), 559. doi: 10.1007/BF01221590. [13] J. Pöschel, A lecture on the classical KAM theorem,, \emph{School on Dynamical Systems}, (1992). [14] J. Pöschel, A KAM-theorem for some nonlinear partial differential equations,, \emph{Ann. Scuola Norm. Sup. Pisa.}, 23 (1996), 119. [15] H. Rüssmann, On twist Hamiltonians. Talk on the Colloque International: Mécanique céleste et systèmes hamiltonians,, \emph{Marseille}, (1990). [16] H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems,, \emph{Regular and Chaotic Dynamics}, 6 (2001), 119. doi: 10.1070/RD2001v006n02ABEH000169. [17] M.B. Sevryuk, KAM-stable Hamiltonians,, \emph{J. Dynamics Control Systems}, 1 (1995), 351. doi: 10.1007/BF02269374. [18] X. Wang, J. Xu and D. Zhang, Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems,, \emph{Discrete Contin. Dyn. Syst., 14 (2010), 1237. [19] J. Xu, Persistence of elliptic lower dimensional invariant Tori for small perturbation of degenerate integrable Hamiltonian systems,, \emph{Journal of Mathematical Analysis and Applications}, 208 (1997), 372. doi: 10.1006/jmaa.1997.5313. [20] J.X. Xu, J.G. You and Q.J. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy,, \emph{Mathematische Zeitschrift}, 226 (1997), 375. doi: 10.1007/PL00004344. [21] J.X. Xu and J.G. You, Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy condition,, \emph{Journal of Differential Equations}, 235 (2007), 609. doi: 10.1016/j.jde.2006.12.001. [22] J.X. Xu and J.G. You, Persistence of the non-twist torus in nearly integrable Hamiltonian systems,, \emph{Pro Math Amer Soc.}, 138 (2010), 2385. doi: 10.1090/S0002-9939-10-10151-8. [23] J.G. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems,, \emph{Commun. Math. Phys.}, 192 (1998), 145. doi: 10.1007/s002200050294. [24] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problem. I and II,, \emph{Commun. Pure Appl. Math.}, 28 (1975), 91. [25] D. Zhang and J. Xu, On invariant tori of vector field under weaker non-degeneracy condition, \emph{Nonlinear Differ. Equ. Appl.}, 22 (2015), 1381.

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##### References:
 [1] V.I. Arnold, Proof of a theorem by A. N. Kolmogorov on the persistence of quasi-periodic motions under small perturbations of the Hamiltonian,, \emph{Uspehi Mat. Nauk}, 18 (1963), 13. [2] Q.Y. Bi and J.X. Xu, Persistence of lower dimensional hyperbolic invariant tori for nearly integrable symplectic mappings,, \emph{Qual. Theory Dyn. Syst.}, 13 (2014), 269. doi: 10.1007/s12346-014-0117-9. [3] H. Broer, G. Huitema and M. Sevryuk, Quasi-periodic motions in families of dynamical systems, in Lecture Notes in Mathematics,, Springer, 1645 (1996). [4] A.D. Bruno, Analytic form of differential equations,, \emph{Trans. Moscow Math. Soc.}, 25 (1971), 131. [5] C. Cheng and Y. Sun, Existence of KAM tori in degenerate Hamiltonian systems,, \emph{J. Differential Equations}, 114 (1994), 288. doi: 10.1006/jdeq.1994.1152. [6] L.H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems,, \emph{Ann. Scuola Norm. Sup. Pisa.}, 15 (1988), 115. [7] S.M. Graff, On the conservation of hyperbolic invariant tori for Hamiltonian systems,, \emph{J. Differ. Eqs.}, 15 (1974), 1. [8] A.N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function,, \emph{Dokl. Akad. Nauk. SSSR}, 98 (1954), 527. [9] S.B. Kuksin, Nearly integrable infinite dimensional Hamiltonian systems, in Lecture Notes in Mathematics,, Springer-Verlag, 1556 (1993). [10] V.K. Melnikov, On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function,, \emph{Soviet Math. Dokl.}, 6 (1965), 1592. [11] J. Moser, Convergent series expansions for quasi-periodic motions,, \emph{Math. Ann.}, 169 (1967), 136. [12] J. Pöschel, On elliptic lower dimensional tori in Hamiltonian systems,, \emph{Math. Z.}, 202 (1989), 559. doi: 10.1007/BF01221590. [13] J. Pöschel, A lecture on the classical KAM theorem,, \emph{School on Dynamical Systems}, (1992). [14] J. Pöschel, A KAM-theorem for some nonlinear partial differential equations,, \emph{Ann. Scuola Norm. Sup. Pisa.}, 23 (1996), 119. [15] H. Rüssmann, On twist Hamiltonians. Talk on the Colloque International: Mécanique céleste et systèmes hamiltonians,, \emph{Marseille}, (1990). [16] H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems,, \emph{Regular and Chaotic Dynamics}, 6 (2001), 119. doi: 10.1070/RD2001v006n02ABEH000169. [17] M.B. Sevryuk, KAM-stable Hamiltonians,, \emph{J. Dynamics Control Systems}, 1 (1995), 351. doi: 10.1007/BF02269374. [18] X. Wang, J. Xu and D. Zhang, Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems,, \emph{Discrete Contin. Dyn. Syst., 14 (2010), 1237. [19] J. Xu, Persistence of elliptic lower dimensional invariant Tori for small perturbation of degenerate integrable Hamiltonian systems,, \emph{Journal of Mathematical Analysis and Applications}, 208 (1997), 372. doi: 10.1006/jmaa.1997.5313. [20] J.X. Xu, J.G. You and Q.J. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy,, \emph{Mathematische Zeitschrift}, 226 (1997), 375. doi: 10.1007/PL00004344. [21] J.X. Xu and J.G. You, Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy condition,, \emph{Journal of Differential Equations}, 235 (2007), 609. doi: 10.1016/j.jde.2006.12.001. [22] J.X. Xu and J.G. You, Persistence of the non-twist torus in nearly integrable Hamiltonian systems,, \emph{Pro Math Amer Soc.}, 138 (2010), 2385. doi: 10.1090/S0002-9939-10-10151-8. [23] J.G. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems,, \emph{Commun. Math. Phys.}, 192 (1998), 145. doi: 10.1007/s002200050294. [24] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problem. I and II,, \emph{Commun. Pure Appl. Math.}, 28 (1975), 91. [25] D. Zhang and J. Xu, On invariant tori of vector field under weaker non-degeneracy condition, \emph{Nonlinear Differ. Equ. Appl.}, 22 (2015), 1381.
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