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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

Kazuyuki Yagasaki 1,

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.
Keywords: Hénon-Heiles system., Melnikov method, Hamiltonian system, chaos, saddle-center, Smale horseshoe.
Mathematics Subject Classification: Primary: 37J45, 70H07, 70H09; Secondary: 34C37, 34E10, 37J3.
Citation: Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete and Continuous Dynamical Systems, 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, Redwood City, CA, 1978.

[2]

J. Cresson, Hyperbolicity, transversality and analytic first integrals, J. Differential Equations, 196 (2004), 289-300. doi: doi:10.1016/j.jde.2003.10.002.

[3]

R. Cushman, Examples of nonintegrable analytic Hamiltonian vector fields with no small divisor, Trans. Amer. Math. Soc., 238 (1978), 45-55.

[4]

H. Dankowicz, Looking for chaos: An extension and alternative to Melnikov's method, Int. J. Bifurcation Chaos, 6 (1996), 485-496. doi: doi:10.1142/S0218127496000205.

[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, Montreal, 1997.

[6]

J. R. Dormand and P. J. Prince, Practical Runge-Kutta processes, SIAM J. Sci. Stat. Comput., 10 (1989), 977-989. doi: doi:10.1137/0910057.

[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), 119-197.

[8]

N. Fenichel, Persistence and smoothness of invariant manifolds for flow, Ind. Univ. Math. J., 21 (1971), 193-225. doi: doi:10.1512/iumj.1971.21.21017.

[9]

A. Goriely, "Integrability and Nonintegrability of Dynamical Systems," World Scientific, New Jersey, 2001. doi: doi:10.1142/9789812811943.

[10]

C. Grotta-Ragazzo, Nonintegrability of some Hamiltonian systems, scattering and analytic continuation, Commun. Math. Phys., 166 (1994), 255-277. doi: doi:10.1007/BF02112316.

[11]

C. Grotta-Ragazzo, Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers, Commun. Pure Appl. Math., 50 (1997), 105-147. doi: doi:10.1002/(SICI)1097-0312(199702)50:2<105::AID-CPA1>3.0.CO;2-G.

[12]

J. Guckenheimer and P. J. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer-Verlag, New York, 1983.

[13]

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

[14]

G. Haller, "Chaos near Resonances," Springer-Verlag, New York, 1999.

[15]

M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments, Astron. J., 69 (1964), 73-79. doi: doi:10.1086/109234.

[16]

H. Ito, Non-integrability of Hénon-Heiles system and a theorem of Ziglin, Kodai Math. J., 8 (1985), 120-138. doi: doi:10.2996/kmj/1138037004.

[17]

H. Ito, A criterion for non-integrability of Hamiltonian systems with nonhomogeneous potentials, J. Appl. Math. Phys. (ZAMP), 38 (1987), 459-476. doi: doi:10.1007/BF00944963.

[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), 397-408. doi: doi:10.1142/S0218127495000338.

[19]

L. M. Lerman, Hamiltonian systems with loops of a separatrix of a saddle-center, Selecta Math. Sov., 10 (1991), 297-306.

[20]

V. K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-56.

[21]

K. R. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem," Springer-Verlag, New York, 1992.

[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), 95-126. doi: doi:10.1007/BF01048157.

[23]

J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems," Birkhäuser, Basel, 1999.

[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), 125-141.

[25]

J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems I, Methods Appl. Anal., 8 (2001), 33-96.

[26]

H. E. Nusse and J. A. Yorke, "Dynamics: Numerical Explorations," 2nd edition, Springer-Verlag, New York, 1997.

[27]

J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems," Springer-Verlag, New York, 1985.

[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), 91-102. doi: doi:10.1070/SM1970v010n01ABEH001588.

[29]

A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations, in: "Dynamics Reported," Vol. 2, (eds. U. Kirchgraber and H.O. Walther), John Wiley and Sons, Chichester, (1989), 89-169.

[30]

S. Wiggins, "Global Bifurcations and Chaos-Analytical Methods," Springer-Verlag, New York, 1988.

[31]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer-Verlag, New York, 1990.

[32]

S. Wiggins, "Normally Hyperbolic Invariant Manifolds in Dynamical Systems," Springer-Verlag, New York, 1994.

[33]

K. Yagasaki, Horseshoes in two-degree-of-freedom Hamiltonian systems with saddle-centers, Arch. Rational Mech. Anal., 154 (2000), 275-296. doi: doi:10.1007/s002050000094.

[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), 55-62. doi: doi:10.1016/S0375-9601(01)00324-3.

[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), 45-52. doi: doi:10.1016/S0375-9601(02)00936-2.

[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), 2003-2012. doi: doi:10.1088/0951-7715/16/6/307.

[37]

K. Yagasaki, Homoclinic and heteroclinic orbits to invariant tori in multi-degree-of-freedom Hamiltonian systems with saddle-centers, Nonlinearity, 18 (2005), 1331-1350. doi: doi:10.1088/0951-7715/18/3/020.

[38]

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

[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.

[40]

S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics, I, Funct. Anal. Appl., 16 (1982), 181-189. doi: doi:10.1007/BF01081586.

show all references

References:
[1]

R. Abraham and J. E. Marsden, "Foundations of Mechanics," 2nd edition, Addison-Wesley, Redwood City, CA, 1978.

[2]

J. Cresson, Hyperbolicity, transversality and analytic first integrals, J. Differential Equations, 196 (2004), 289-300. doi: doi:10.1016/j.jde.2003.10.002.

[3]

R. Cushman, Examples of nonintegrable analytic Hamiltonian vector fields with no small divisor, Trans. Amer. Math. Soc., 238 (1978), 45-55.

[4]

H. Dankowicz, Looking for chaos: An extension and alternative to Melnikov's method, Int. J. Bifurcation Chaos, 6 (1996), 485-496. doi: doi:10.1142/S0218127496000205.

[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, Montreal, 1997.

[6]

J. R. Dormand and P. J. Prince, Practical Runge-Kutta processes, SIAM J. Sci. Stat. Comput., 10 (1989), 977-989. doi: doi:10.1137/0910057.

[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), 119-197.

[8]

N. Fenichel, Persistence and smoothness of invariant manifolds for flow, Ind. Univ. Math. J., 21 (1971), 193-225. doi: doi:10.1512/iumj.1971.21.21017.

[9]

A. Goriely, "Integrability and Nonintegrability of Dynamical Systems," World Scientific, New Jersey, 2001. doi: doi:10.1142/9789812811943.

[10]

C. Grotta-Ragazzo, Nonintegrability of some Hamiltonian systems, scattering and analytic continuation, Commun. Math. Phys., 166 (1994), 255-277. doi: doi:10.1007/BF02112316.

[11]

C. Grotta-Ragazzo, Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers, Commun. Pure Appl. Math., 50 (1997), 105-147. doi: doi:10.1002/(SICI)1097-0312(199702)50:2<105::AID-CPA1>3.0.CO;2-G.

[12]

J. Guckenheimer and P. J. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Springer-Verlag, New York, 1983.

[13]

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

[14]

G. Haller, "Chaos near Resonances," Springer-Verlag, New York, 1999.

[15]

M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments, Astron. J., 69 (1964), 73-79. doi: doi:10.1086/109234.

[16]

H. Ito, Non-integrability of Hénon-Heiles system and a theorem of Ziglin, Kodai Math. J., 8 (1985), 120-138. doi: doi:10.2996/kmj/1138037004.

[17]

H. Ito, A criterion for non-integrability of Hamiltonian systems with nonhomogeneous potentials, J. Appl. Math. Phys. (ZAMP), 38 (1987), 459-476. doi: doi:10.1007/BF00944963.

[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), 397-408. doi: doi:10.1142/S0218127495000338.

[19]

L. M. Lerman, Hamiltonian systems with loops of a separatrix of a saddle-center, Selecta Math. Sov., 10 (1991), 297-306.

[20]

V. K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-56.

[21]

K. R. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem," Springer-Verlag, New York, 1992.

[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), 95-126. doi: doi:10.1007/BF01048157.

[23]

J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems," Birkhäuser, Basel, 1999.

[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), 125-141.

[25]

J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems I, Methods Appl. Anal., 8 (2001), 33-96.

[26]

H. E. Nusse and J. A. Yorke, "Dynamics: Numerical Explorations," 2nd edition, Springer-Verlag, New York, 1997.

[27]

J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems," Springer-Verlag, New York, 1985.

[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), 91-102. doi: doi:10.1070/SM1970v010n01ABEH001588.

[29]

A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations, in: "Dynamics Reported," Vol. 2, (eds. U. Kirchgraber and H.O. Walther), John Wiley and Sons, Chichester, (1989), 89-169.

[30]

S. Wiggins, "Global Bifurcations and Chaos-Analytical Methods," Springer-Verlag, New York, 1988.

[31]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer-Verlag, New York, 1990.

[32]

S. Wiggins, "Normally Hyperbolic Invariant Manifolds in Dynamical Systems," Springer-Verlag, New York, 1994.

[33]

K. Yagasaki, Horseshoes in two-degree-of-freedom Hamiltonian systems with saddle-centers, Arch. Rational Mech. Anal., 154 (2000), 275-296. doi: doi:10.1007/s002050000094.

[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), 55-62. doi: doi:10.1016/S0375-9601(01)00324-3.

[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), 45-52. doi: doi:10.1016/S0375-9601(02)00936-2.

[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), 2003-2012. doi: doi:10.1088/0951-7715/16/6/307.

[37]

K. Yagasaki, Homoclinic and heteroclinic orbits to invariant tori in multi-degree-of-freedom Hamiltonian systems with saddle-centers, Nonlinearity, 18 (2005), 1331-1350. doi: doi:10.1088/0951-7715/18/3/020.

[38]

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

[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.

[40]

S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics, I, Funct. Anal. Appl., 16 (1982), 181-189. doi: doi:10.1007/BF01081586.

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