July  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.   Google Scholar

[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.  Google Scholar

[3]

H. Broer, G. Huitema and M. Sevryuk, Quasi-periodic motions in families of dynamical systems, in Lecture Notes in Mathematics,, Springer, 1645 (1996).   Google Scholar

[4]

A.D. Bruno, Analytic form of differential equations,, \emph{Trans. Moscow Math. Soc.}, 25 (1971), 131.   Google Scholar

[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.  Google Scholar

[6]

L.H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems,, \emph{Ann. Scuola Norm. Sup. Pisa.}, 15 (1988), 115.   Google Scholar

[7]

S.M. Graff, On the conservation of hyperbolic invariant tori for Hamiltonian systems,, \emph{J. Differ. Eqs.}, 15 (1974), 1.   Google Scholar

[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.   Google Scholar

[9]

S.B. Kuksin, Nearly integrable infinite dimensional Hamiltonian systems, in Lecture Notes in Mathematics,, Springer-Verlag, 1556 (1993).   Google Scholar

[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.   Google Scholar

[11]

J. Moser, Convergent series expansions for quasi-periodic motions,, \emph{Math. Ann.}, 169 (1967), 136.   Google Scholar

[12]

J. Pöschel, On elliptic lower dimensional tori in Hamiltonian systems,, \emph{Math. Z.}, 202 (1989), 559.  doi: 10.1007/BF01221590.  Google Scholar

[13]

J. Pöschel, A lecture on the classical KAM theorem,, \emph{School on Dynamical Systems}, (1992).   Google Scholar

[14]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations,, \emph{Ann. Scuola Norm. Sup. Pisa.}, 23 (1996), 119.   Google Scholar

[15]

H. Rüssmann, On twist Hamiltonians. Talk on the Colloque International: Mécanique céleste et systèmes hamiltonians,, \emph{Marseille}, (1990).   Google Scholar

[16]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems,, \emph{Regular and Chaotic Dynamics}, 6 (2001), 119.  doi: 10.1070/RD2001v006n02ABEH000169.  Google Scholar

[17]

M.B. Sevryuk, KAM-stable Hamiltonians,, \emph{J. Dynamics Control Systems}, 1 (1995), 351.  doi: 10.1007/BF02269374.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

show all references

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.   Google Scholar

[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.  Google Scholar

[3]

H. Broer, G. Huitema and M. Sevryuk, Quasi-periodic motions in families of dynamical systems, in Lecture Notes in Mathematics,, Springer, 1645 (1996).   Google Scholar

[4]

A.D. Bruno, Analytic form of differential equations,, \emph{Trans. Moscow Math. Soc.}, 25 (1971), 131.   Google Scholar

[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.  Google Scholar

[6]

L.H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems,, \emph{Ann. Scuola Norm. Sup. Pisa.}, 15 (1988), 115.   Google Scholar

[7]

S.M. Graff, On the conservation of hyperbolic invariant tori for Hamiltonian systems,, \emph{J. Differ. Eqs.}, 15 (1974), 1.   Google Scholar

[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.   Google Scholar

[9]

S.B. Kuksin, Nearly integrable infinite dimensional Hamiltonian systems, in Lecture Notes in Mathematics,, Springer-Verlag, 1556 (1993).   Google Scholar

[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.   Google Scholar

[11]

J. Moser, Convergent series expansions for quasi-periodic motions,, \emph{Math. Ann.}, 169 (1967), 136.   Google Scholar

[12]

J. Pöschel, On elliptic lower dimensional tori in Hamiltonian systems,, \emph{Math. Z.}, 202 (1989), 559.  doi: 10.1007/BF01221590.  Google Scholar

[13]

J. Pöschel, A lecture on the classical KAM theorem,, \emph{School on Dynamical Systems}, (1992).   Google Scholar

[14]

J. Pöschel, A KAM-theorem for some nonlinear partial differential equations,, \emph{Ann. Scuola Norm. Sup. Pisa.}, 23 (1996), 119.   Google Scholar

[15]

H. Rüssmann, On twist Hamiltonians. Talk on the Colloque International: Mécanique céleste et systèmes hamiltonians,, \emph{Marseille}, (1990).   Google Scholar

[16]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems,, \emph{Regular and Chaotic Dynamics}, 6 (2001), 119.  doi: 10.1070/RD2001v006n02ABEH000169.  Google Scholar

[17]

M.B. Sevryuk, KAM-stable Hamiltonians,, \emph{J. Dynamics Control Systems}, 1 (1995), 351.  doi: 10.1007/BF02269374.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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