# American Institute of Mathematical Sciences

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

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##### 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
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