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.

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.

[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.

[1]

Xiaocai Wang, Junxiang Xu. Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 701-718. doi: 10.3934/dcds.2009.25.701

[2]

Fuzhong Cong, Yong Li. Invariant hyperbolic tori for Hamiltonian systems with degeneracy. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 371-382. doi: 10.3934/dcds.1997.3.371

[3]

Dongfeng Zhang, Junxiang Xu. On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 635-655. doi: 10.3934/dcds.2006.16.635

[4]

Xiaocai Wang. Non-floquet invariant tori in reversible systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3439-3457. doi: 10.3934/dcds.2018147

[5]

C. Chandre. Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 457-465. doi: 10.3934/dcdsb.2002.2.457

[6]

Hans Koch. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 633-646. doi: 10.3934/dcds.2002.8.633

[7]

Pedro J. Torres, Zhibo Cheng, Jingli Ren. Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2155-2168. doi: 10.3934/dcds.2013.33.2155

[8]

Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure & Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587

[9]

Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569

[10]

Denis G. Gaidashev. Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 63-102. doi: 10.3934/dcds.2005.13.63

[11]

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

[12]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017

[13]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116

[14]

Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure & Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941

[15]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785

[16]

Roman Šimon Hilscher. On general Sturmian theory for abnormal linear Hamiltonian systems. Conference Publications, 2011, 2011 (Special) : 684-691. doi: 10.3934/proc.2011.2011.684

[17]

Asaf Kislev. Compactly supported Hamiltonian loops with a non-zero Calabi invariant. Electronic Research Announcements, 2014, 21: 80-88. doi: 10.3934/era.2014.21.80

[18]

Sandra Lucente, Eugenio Montefusco. Non-hamiltonian Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 761-770. doi: 10.3934/dcdss.2013.6.761

[19]

Amadeu Delshams, Pere Gutiérrez, Tere M. Seara. Exponentially small splitting for whiskered tori in Hamiltonian systems: flow-box coordinates and upper bounds. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 785-826. doi: 10.3934/dcds.2004.11.785

[20]

Amadeu Delshams, Pere Gutiérrez. Exponentially small splitting for whiskered tori in Hamiltonian systems: continuation of transverse homoclinic orbits. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 757-783. doi: 10.3934/dcds.2004.11.757

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]