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

Kazuyuki Yagasaki

doi:10.3934/dcds.2011.29.387

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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

Abstract
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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 & 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. Google Scholar
[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. Google Scholar
[3]	R. Cushman, Examples of nonintegrable analytic Hamiltonian vector fields with no small divisor, Trans. Amer. Math. Soc., 238 (1978), 45-55. Google Scholar
[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. 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, Montreal, 1997. Google Scholar
[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. 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), 119-197. Google Scholar
[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. Google Scholar
[9]	A. Goriely, "Integrability and Nonintegrability of Dynamical Systems," World Scientific, New Jersey, 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), 255-277. 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), 105-147. 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, New York, 1983. Google Scholar
[13]	E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations I," 2nd edition, Springer-Verlag, Berlin, 1993. Google Scholar
[14]	G. Haller, "Chaos near Resonances," Springer-Verlag, New York, 1999. Google Scholar
[15]	M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments, Astron. J., 69 (1964), 73-79. 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), 120-138. 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), 459-476. 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), 397-408. 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), 297-306. Google Scholar
[20]	V. K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-56. Google Scholar
[21]	K. R. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem," Springer-Verlag, New York, 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), 95-126. doi: doi:10.1007/BF01048157. Google Scholar
[23]	J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems," Birkhäuser, Basel, 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), 125-141. Google Scholar
[25]	J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems I, Methods Appl. Anal., 8 (2001), 33-96. Google Scholar
[26]	H. E. Nusse and J. A. Yorke, "Dynamics: Numerical Explorations," 2nd edition, Springer-Verlag, New York, 1997. Google Scholar
[27]	J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems," Springer-Verlag, New York, 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), 91-102. doi: doi:10.1070/SM1970v010n01ABEH001588. Google Scholar
[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. Google Scholar
[30]	S. Wiggins, "Global Bifurcations and Chaos-Analytical Methods," Springer-Verlag, New York, 1988. Google Scholar
[31]	S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer-Verlag, New York, 1990. Google Scholar
[32]	S. Wiggins, "Normally Hyperbolic Invariant Manifolds in Dynamical Systems," Springer-Verlag, New York, 1994. Google Scholar
[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. 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), 55-62. 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), 45-52. 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), 2003-2012. 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), 1331-1350. 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), 181-189. 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, Redwood City, CA, 1978. Google Scholar
[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. Google Scholar
[3]	R. Cushman, Examples of nonintegrable analytic Hamiltonian vector fields with no small divisor, Trans. Amer. Math. Soc., 238 (1978), 45-55. Google Scholar
[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. 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, Montreal, 1997. Google Scholar
[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. 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), 119-197. Google Scholar
[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. Google Scholar
[9]	A. Goriely, "Integrability and Nonintegrability of Dynamical Systems," World Scientific, New Jersey, 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), 255-277. 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), 105-147. 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, New York, 1983. Google Scholar
[13]	E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations I," 2nd edition, Springer-Verlag, Berlin, 1993. Google Scholar
[14]	G. Haller, "Chaos near Resonances," Springer-Verlag, New York, 1999. Google Scholar
[15]	M. Hénon and C. Heiles, The applicability of the third integral of motion: Some numerical experiments, Astron. J., 69 (1964), 73-79. 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), 120-138. 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), 459-476. 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), 397-408. 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), 297-306. Google Scholar
[20]	V. K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-56. Google Scholar
[21]	K. R. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem," Springer-Verlag, New York, 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), 95-126. doi: doi:10.1007/BF01048157. Google Scholar
[23]	J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems," Birkhäuser, Basel, 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), 125-141. Google Scholar
[25]	J. J. Morales-Ruiz and J. P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems I, Methods Appl. Anal., 8 (2001), 33-96. Google Scholar
[26]	H. E. Nusse and J. A. Yorke, "Dynamics: Numerical Explorations," 2nd edition, Springer-Verlag, New York, 1997. Google Scholar
[27]	J. A. Sanders and F. Verhulst, "Averaging Methods in Nonlinear Dynamical Systems," Springer-Verlag, New York, 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), 91-102. doi: doi:10.1070/SM1970v010n01ABEH001588. Google Scholar
[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. Google Scholar
[30]	S. Wiggins, "Global Bifurcations and Chaos-Analytical Methods," Springer-Verlag, New York, 1988. Google Scholar
[31]	S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," Springer-Verlag, New York, 1990. Google Scholar
[32]	S. Wiggins, "Normally Hyperbolic Invariant Manifolds in Dynamical Systems," Springer-Verlag, New York, 1994. Google Scholar
[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. 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), 55-62. 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), 45-52. 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), 2003-2012. 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), 1331-1350. 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), 181-189. doi: doi:10.1007/BF01081586. Google Scholar

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

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