January  2011, 29(1): 387-402. doi: 10.3934/dcds.2011.29.387

Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers

1. 

Department of Mechanical and Systems Engineering, Gifu University, Gifu 501-1193, Japan

Received  September 2007 Revised  September 2009 Published  September 2010

We consider two-degree-of-freedom Hamiltonian systems with saddle-centers, and develop a Melnikov-type technique for detecting creation of transverse homoclinic orbits by higher-order terms. We apply the technique to the generalized Hénon-Heiles system and give a positive answer to a remaining question of whether chaotic dynamics occurs for some parameter values although it is known to be nonintegrable in a complex analytical meaning.
Citation: Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387
References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition,, Addison-Wesley, (1978).   Google Scholar

[2]

J. Cresson, Hyperbolicity, transversality and analytic first integrals,, J. Differential Equations, 196 (2004), 289.  doi: doi:10.1016/j.jde.2003.10.002.  Google Scholar

[3]

R. Cushman, Examples of nonintegrable analytic Hamiltonian vector fields with no small divisor,, Trans. Amer. Math. Soc., 238 (1978), 45.   Google Scholar

[4]

H. Dankowicz, Looking for chaos: An extension and alternative to Melnikov's method,, Int. J. Bifurcation Chaos, 6 (1996), 485.  doi: doi:10.1142/S0218127496000205.  Google Scholar

[5]

E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede and X. Wang, "AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont),", Concordia University, (1997).   Google Scholar

[6]

J. R. Dormand and P. J. Prince, Practical Runge-Kutta processes,, SIAM J. Sci. Stat. Comput., 10 (1989), 977.  doi: doi:10.1137/0910057.  Google Scholar

[7]

S. A. Dovbysh, Transversal intersection of separatrices and branching of solutions as obstructions to the existence of an analytic integral in many-dimensional systems. I. Basic result: Separatrices of hyperbolic periodic points,, Collect. Math., 50 (1999), 119.   Google Scholar

[8]

N. Fenichel, Persistence and smoothness of invariant manifolds for flow,, Ind. Univ. Math. J., 21 (1971), 193.  doi: doi:10.1512/iumj.1971.21.21017.  Google Scholar

[9]

A. Goriely, "Integrability and Nonintegrability of Dynamical Systems,", World Scientific, (2001).  doi: doi:10.1142/9789812811943.  Google Scholar

[10]

C. Grotta-Ragazzo, Nonintegrability of some Hamiltonian systems, scattering and analytic continuation,, Commun. Math. Phys., 166 (1994), 255.  doi: doi:10.1007/BF02112316.  Google Scholar

[11]

C. Grotta-Ragazzo, Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers,, Commun. Pure Appl. Math., 50 (1997), 105.  doi: doi:10.1002/(SICI)1097-0312(199702)50:2<105::AID-CPA1>3.0.CO;2-G.  Google Scholar

[12]

J. Guckenheimer and P. J. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer-Verlag, (1983).   Google Scholar

[13]

E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations I," 2nd edition,, Springer-Verlag, (1993).   Google Scholar

[14]

G. Haller, "Chaos near Resonances,", Springer-Verlag, (1999).   Google Scholar

[15]

M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments,, Astron. J., 69 (1964), 73.  doi: doi:10.1086/109234.  Google Scholar

[16]

H. Ito, Non-integrability of Hénon-Heiles system and a theorem of Ziglin,, Kodai Math. J., 8 (1985), 120.  doi: doi:10.2996/kmj/1138037004.  Google Scholar

[17]

H. Ito, A criterion for non-integrability of Hamiltonian systems with nonhomogeneous potentials,, J. Appl. Math. Phys. (ZAMP), 38 (1987), 459.  doi: doi:10.1007/BF00944963.  Google Scholar

[18]

O. Y. Koltsova and L. M. Lerman, Periodic and homoclinic orbits in a two-parameter unfolding of a Hamiltonian system with a homoclinic orbit to a saddle-center,, Int. J. Bifurcation Chaos, 5 (1995), 397.  doi: doi:10.1142/S0218127495000338.  Google Scholar

[19]

L. M. Lerman, Hamiltonian systems with loops of a separatrix of a saddle-center,, Selecta Math. Sov., 10 (1991), 297.   Google Scholar

[20]

V. K. Melnikov, On the stability of the center for time periodic perturbations,, Trans. Moscow Math. Soc., 12 (1963), 1.   Google Scholar

[21]

K. R. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,", Springer-Verlag, (1992).   Google Scholar

[22]

A. Mielke, P. J. Holmes and O. O'Reilly, Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center,, J. Dyn. Diff. Eqn., 4 (1992), 95.  doi: doi:10.1007/BF01048157.  Google Scholar

[23]

J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,", Birkhäuser, (1999).   Google Scholar

[24]

J. J. Morales-Ruiz and J. M. Peris, On a Galoisian approach to the splitting of separatrices,, Ann. Fac. Sci. Toulouse Math., 8 (1999), 125.   Google Scholar

[25]

J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems I,, Methods Appl. Anal., 8 (2001), 33.   Google Scholar

[26]

H. E. Nusse and J. A. Yorke, "Dynamics: Numerical Explorations," 2nd edition,, Springer-Verlag, (1997).   Google Scholar

[27]

J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems,", Springer-Verlag, (1985).   Google Scholar

[28]

L. P. Shil'nikov, A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type,, Math. USSR Sbornik, 10 (1970), 91.  doi: doi:10.1070/SM1970v010n01ABEH001588.  Google Scholar

[29]

A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations,, in:, 2 (1989), 89.   Google Scholar

[30]

S. Wiggins, "Global Bifurcations and Chaos-Analytical Methods,", Springer-Verlag, (1988).   Google Scholar

[31]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,", Springer-Verlag, (1990).   Google Scholar

[32]

S. Wiggins, "Normally Hyperbolic Invariant Manifolds in Dynamical Systems,", Springer-Verlag, (1994).   Google Scholar

[33]

K. Yagasaki, Horseshoes in two-degree-of-freedom Hamiltonian systems with saddle-centers,, Arch. Rational Mech. Anal., 154 (2000), 275.  doi: doi:10.1007/s002050000094.  Google Scholar

[34]

K. Yagasaki, Homoclinic and heteroclinic behavior in an infinite-degree-of-freedom Hamiltonian system: Chaotic free vibrations of an undamped, buckled beam,, Phys. Lett. A, 285 (2001), 55.  doi: doi:10.1016/S0375-9601(01)00324-3.  Google Scholar

[35]

K. Yagasaki, Numerical evidence of fast diffusion in a three-degree-of-freedom Hamiltonian system with a saddle-center,, Phys. Lett. A, 301 (2002), 45.  doi: doi:10.1016/S0375-9601(02)00936-2.  Google Scholar

[36]

K. Yagasaki, Galoisian obstructions to integrability and Melnikov criteria for chaos in two-degree-of-freedom Hamiltonian systems with saddle-centers,, Nonlinearity, 16 (2003), 2003.  doi: doi:10.1088/0951-7715/16/6/307.  Google Scholar

[37]

K. Yagasaki, Homoclinic and heteroclinic orbits to invariant tori in multi-degree-of-freedom Hamiltonian systems with saddle-centers,, Nonlinearity, 18 (2005), 1331.  doi: doi:10.1088/0951-7715/18/3/020.  Google Scholar

[38]

K. Yagasaki, Numerical analysis for global bifurcations of periodic orbits in autonomous differential equations,, in preparation., ().   Google Scholar

[39]

K. Yagasaki, "HomMap: A Package of AUTO and Dynamics Drivers for Homoclinic Bifurcation Analysis for Periodic Orbits of Maps and ODEs, Version $2.0$,", in preparation., ().   Google Scholar

[40]

S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics, I,, Funct. Anal. Appl., 16 (1982), 181.  doi: doi:10.1007/BF01081586.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition,, Addison-Wesley, (1978).   Google Scholar

[2]

J. Cresson, Hyperbolicity, transversality and analytic first integrals,, J. Differential Equations, 196 (2004), 289.  doi: doi:10.1016/j.jde.2003.10.002.  Google Scholar

[3]

R. Cushman, Examples of nonintegrable analytic Hamiltonian vector fields with no small divisor,, Trans. Amer. Math. Soc., 238 (1978), 45.   Google Scholar

[4]

H. Dankowicz, Looking for chaos: An extension and alternative to Melnikov's method,, Int. J. Bifurcation Chaos, 6 (1996), 485.  doi: doi:10.1142/S0218127496000205.  Google Scholar

[5]

E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede and X. Wang, "AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont),", Concordia University, (1997).   Google Scholar

[6]

J. R. Dormand and P. J. Prince, Practical Runge-Kutta processes,, SIAM J. Sci. Stat. Comput., 10 (1989), 977.  doi: doi:10.1137/0910057.  Google Scholar

[7]

S. A. Dovbysh, Transversal intersection of separatrices and branching of solutions as obstructions to the existence of an analytic integral in many-dimensional systems. I. Basic result: Separatrices of hyperbolic periodic points,, Collect. Math., 50 (1999), 119.   Google Scholar

[8]

N. Fenichel, Persistence and smoothness of invariant manifolds for flow,, Ind. Univ. Math. J., 21 (1971), 193.  doi: doi:10.1512/iumj.1971.21.21017.  Google Scholar

[9]

A. Goriely, "Integrability and Nonintegrability of Dynamical Systems,", World Scientific, (2001).  doi: doi:10.1142/9789812811943.  Google Scholar

[10]

C. Grotta-Ragazzo, Nonintegrability of some Hamiltonian systems, scattering and analytic continuation,, Commun. Math. Phys., 166 (1994), 255.  doi: doi:10.1007/BF02112316.  Google Scholar

[11]

C. Grotta-Ragazzo, Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers,, Commun. Pure Appl. Math., 50 (1997), 105.  doi: doi:10.1002/(SICI)1097-0312(199702)50:2<105::AID-CPA1>3.0.CO;2-G.  Google Scholar

[12]

J. Guckenheimer and P. J. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields,", Springer-Verlag, (1983).   Google Scholar

[13]

E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations I," 2nd edition,, Springer-Verlag, (1993).   Google Scholar

[14]

G. Haller, "Chaos near Resonances,", Springer-Verlag, (1999).   Google Scholar

[15]

M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments,, Astron. J., 69 (1964), 73.  doi: doi:10.1086/109234.  Google Scholar

[16]

H. Ito, Non-integrability of Hénon-Heiles system and a theorem of Ziglin,, Kodai Math. J., 8 (1985), 120.  doi: doi:10.2996/kmj/1138037004.  Google Scholar

[17]

H. Ito, A criterion for non-integrability of Hamiltonian systems with nonhomogeneous potentials,, J. Appl. Math. Phys. (ZAMP), 38 (1987), 459.  doi: doi:10.1007/BF00944963.  Google Scholar

[18]

O. Y. Koltsova and L. M. Lerman, Periodic and homoclinic orbits in a two-parameter unfolding of a Hamiltonian system with a homoclinic orbit to a saddle-center,, Int. J. Bifurcation Chaos, 5 (1995), 397.  doi: doi:10.1142/S0218127495000338.  Google Scholar

[19]

L. M. Lerman, Hamiltonian systems with loops of a separatrix of a saddle-center,, Selecta Math. Sov., 10 (1991), 297.   Google Scholar

[20]

V. K. Melnikov, On the stability of the center for time periodic perturbations,, Trans. Moscow Math. Soc., 12 (1963), 1.   Google Scholar

[21]

K. R. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,", Springer-Verlag, (1992).   Google Scholar

[22]

A. Mielke, P. J. Holmes and O. O'Reilly, Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center,, J. Dyn. Diff. Eqn., 4 (1992), 95.  doi: doi:10.1007/BF01048157.  Google Scholar

[23]

J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems,", Birkhäuser, (1999).   Google Scholar

[24]

J. J. Morales-Ruiz and J. M. Peris, On a Galoisian approach to the splitting of separatrices,, Ann. Fac. Sci. Toulouse Math., 8 (1999), 125.   Google Scholar

[25]

J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems I,, Methods Appl. Anal., 8 (2001), 33.   Google Scholar

[26]

H. E. Nusse and J. A. Yorke, "Dynamics: Numerical Explorations," 2nd edition,, Springer-Verlag, (1997).   Google Scholar

[27]

J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems,", Springer-Verlag, (1985).   Google Scholar

[28]

L. P. Shil'nikov, A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type,, Math. USSR Sbornik, 10 (1970), 91.  doi: doi:10.1070/SM1970v010n01ABEH001588.  Google Scholar

[29]

A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations,, in:, 2 (1989), 89.   Google Scholar

[30]

S. Wiggins, "Global Bifurcations and Chaos-Analytical Methods,", Springer-Verlag, (1988).   Google Scholar

[31]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos,", Springer-Verlag, (1990).   Google Scholar

[32]

S. Wiggins, "Normally Hyperbolic Invariant Manifolds in Dynamical Systems,", Springer-Verlag, (1994).   Google Scholar

[33]

K. Yagasaki, Horseshoes in two-degree-of-freedom Hamiltonian systems with saddle-centers,, Arch. Rational Mech. Anal., 154 (2000), 275.  doi: doi:10.1007/s002050000094.  Google Scholar

[34]

K. Yagasaki, Homoclinic and heteroclinic behavior in an infinite-degree-of-freedom Hamiltonian system: Chaotic free vibrations of an undamped, buckled beam,, Phys. Lett. A, 285 (2001), 55.  doi: doi:10.1016/S0375-9601(01)00324-3.  Google Scholar

[35]

K. Yagasaki, Numerical evidence of fast diffusion in a three-degree-of-freedom Hamiltonian system with a saddle-center,, Phys. Lett. A, 301 (2002), 45.  doi: doi:10.1016/S0375-9601(02)00936-2.  Google Scholar

[36]

K. Yagasaki, Galoisian obstructions to integrability and Melnikov criteria for chaos in two-degree-of-freedom Hamiltonian systems with saddle-centers,, Nonlinearity, 16 (2003), 2003.  doi: doi:10.1088/0951-7715/16/6/307.  Google Scholar

[37]

K. Yagasaki, Homoclinic and heteroclinic orbits to invariant tori in multi-degree-of-freedom Hamiltonian systems with saddle-centers,, Nonlinearity, 18 (2005), 1331.  doi: doi:10.1088/0951-7715/18/3/020.  Google Scholar

[38]

K. Yagasaki, Numerical analysis for global bifurcations of periodic orbits in autonomous differential equations,, in preparation., ().   Google Scholar

[39]

K. Yagasaki, "HomMap: A Package of AUTO and Dynamics Drivers for Homoclinic Bifurcation Analysis for Periodic Orbits of Maps and ODEs, Version $2.0$,", in preparation., ().   Google Scholar

[40]

S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics, I,, Funct. Anal. Appl., 16 (1982), 181.  doi: doi:10.1007/BF01081586.  Google Scholar

[1]

Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079

[2]

Iliya D. Iliev, Chengzhi Li, Jiang Yu. Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes. Communications on Pure & Applied Analysis, 2010, 9 (3) : 583-610. doi: 10.3934/cpaa.2010.9.583

[3]

Jingbo Dou, Huaiyu Zhou. Liouville theorems for fractional Hénon equation and system on $\mathbb{R}^n$. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1915-1927. doi: 10.3934/cpaa.2015.14.1915

[4]

Shaoyun Shi, Wenlei Li. Non-integrability of generalized Yang-Mills Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1645-1655. doi: 10.3934/dcds.2013.33.1645

[5]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839

[6]

Xianwei Chen, Zhujun Jing, Xiangling Fu. Chaos control in a pendulum system with excitations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 373-383. doi: 10.3934/dcdsb.2015.20.373

[7]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83

[8]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations & Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59

[9]

Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443

[10]

Lingling Liu, Bo Gao, Dongmei Xiao, Weinian Zhang. Identification of focus and center in a 3-dimensional system. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 485-522. doi: 10.3934/dcdsb.2014.19.485

[11]

E. Fossas, J. M. Olm. Galerkin method and approximate tracking in a non-minimum phase bilinear system. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 53-76. doi: 10.3934/dcdsb.2007.7.53

[12]

Zhujun Jing, K.Y. Chan, Dashun Xu, Hongjun Cao. Bifurcations of periodic solutions and chaos in Josephson system. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 573-592. doi: 10.3934/dcds.2001.7.573

[13]

Ting Yang. Homoclinic orbits and chaos in the generalized Lorenz system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1097-1108. doi: 10.3934/dcdsb.2019210

[14]

V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277

[15]

Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789

[16]

Yulin Zhao, Siming Zhu. Higher order Melnikov function for a quartic hamiltonian with cuspidal loop. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 995-1018. doi: 10.3934/dcds.2002.8.995

[17]

Min Hu, Tao Li, Xingwu Chen. Bi-center problem and Hopf cyclicity of a Cubic Liénard system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 401-414. doi: 10.3934/dcdsb.2019187

[18]

Jian Zhang, Wen Zhang, Xiaoliang Xie. Existence and concentration of semiclassical solutions for Hamiltonian elliptic system. Communications on Pure & Applied Analysis, 2016, 15 (2) : 599-622. doi: 10.3934/cpaa.2016.15.599

[19]

Xing Huang, Wujun Lv. Stochastic functional Hamiltonian system with singular coefficients. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1257-1273. doi: 10.3934/cpaa.2020060

[20]

Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]