Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 049, 17 pages      arXiv:1907.01161      https://doi.org/10.3842/SIGMA.2019.049
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

Heteroclinic Orbits and Nonintegrability in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers

Kazuyuki Yagasaki and Shogo Yamanaka
Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan

Received January 29, 2019, in final form June 21, 2019; Published online July 02, 2019

Abstract
We consider a class of two-degree-of-freedom Hamiltonian systems with saddle-centers connected by heteroclinic orbits and discuss some relationships between the existence of transverse heteroclinic orbits and nonintegrability. By the Lyapunov center theorem there is a family of periodic orbits near each of the saddle-centers, and the Hessian matrices of the Hamiltonian at the two saddle-centers are assumed to have the same number of positive eigenvalues. We show that if the associated Jacobian matrices have the same pair of purely imaginary eigenvalues, then the stable and unstable manifolds of the periodic orbits intersect transversely on the same Hamiltonian energy surface when sufficient conditions obtained in previous work for real-meromorphic nonintegrability of the Hamiltonian systems hold; if not, then these manifolds intersect transversely on the same energy surface, have quadratic tangencies or do not intersect whether the sufficient conditions hold or not. Our theory is illustrated for a system with quartic single-well potential and some numerical results are given to support the theoretical results.

Key words: nonintegrability; Hamiltonian system; heteroclinic orbits; saddle-center; Melnikov method; Morales-Ramis theory; differential Galois theory; monodromy.

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References

  1. Ayoul M., Zung N.T., Galoisian obstructions to non-Hamiltonian integrability, C. R. Math. Acad. Sci. Paris 348 (2010), 1323-1326, arXiv:0901.4586.
  2. Bogoyavlenskij O.I., Extended integrability and bi-Hamiltonian systems, Comm. Math. Phys. 196 (1998), 19-51.
  3. Champneys A.R., Lord G.J., Computation of homoclinic solutions to periodic orbits in a reduced water-wave problem, Phys. D 102 (1997), 101-124.
  4. Crespo T., Hajto Z., Algebraic groups and differential Galois theory, Graduate Studies in Mathematics, Vol. 122, Amer. Math. Soc., Providence, RI, 2011.
  5. Doedel E.J., Oldeman B.E., AUTO-07P: Continuation and bifurcation software for ordinary differential equations, 2012, available at http://indy.cs.concordia.ca/auto.
  6. Dovbysh S.A., The splitting of separatrices, the branching of solutions and non-integrability in the problem of the motion of a spherical pendulum with an oscillating suspension point, J. Appl. Math. Mech. 70 (2006), 42-55.
  7. Grotta Ragazzo C., Nonintegrability of some Hamiltonian systems, scattering and analytic continuation, Comm. Math. Phys. 166 (1994), 255-277.
  8. Guckenheimer J., Holmes P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, Vol. 42, Springer-Verlag, New York, 1983.
  9. Ilyashenko Y., Yakovenko S., Lectures on analytic differential equations, Graduate Studies in Mathematics, Vol. 86, Amer. Math. Soc., Providence, RI, 2008.
  10. Iwasaki K., Kimura H., Shimomura S., Yoshida M., From Gauss to Painlevé. A modern theory of special functions, Aspects of Mathematics, Vol. E16, Friedr. Vieweg & Sohn, Braunschweig, 1991.
  11. Kozlov V.V., Symmetries, topology and resonances in Hamiltonian mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Vol. 31, Springer-Verlag, Berlin, 1996.
  12. Lerman L.M., Hamiltonian systems with loops of a separatrix of a saddle-center, Selecta Math. Soviet. 10 (1991), 297-306.
  13. Maciejewski A.J., Przybylska M., Nonintegrability of the Suslov problem, J. Math. Phys. 45 (2004), 1065-1078.
  14. Maciejewski A.J., Przybylska M., Differential Galois approach to the non-integrability of the heavy top problem, Ann. Fac. Sci. Toulouse Math. 14 (2005), 123-160, arXiv:math.DS/0404367.
  15. Melnikov V.K., On the stability of a center for time-periodic perturbations, Trans. Moscow Math. Soc. 12 (1963), 3-52.
  16. Meyer K.R., Offin D.C., Introduction to Hamiltonian dynamical systems and the $N$-body problem, 3rd ed., Applied Mathematical Sciences, Vol. 90, Springer, Cham, 2017.
  17. Morales-Ruiz J.J., Differential Galois theory and non-integrability of Hamiltonian systems, Progress in Mathematics, Vol. 179, Birkhäuser Verlag, Basel, 1999.
  18. Morales-Ruiz J.J., Peris J.M., On a Galoisian approach to the splitting of separatrices, Ann. Fac. Sci. Toulouse Math. 8 (1999), 125-141.
  19. Morales-Ruiz J.J., Ramis J.P., Galoisian obstructions to integrability of Hamiltonian systems, Methods Appl. Anal. 8 (2001), 33-95.
  20. Moser J., Stable and random motions in dynamical systems, Annals of Mathematics Studies, Vol. 77, Princeton University Press, Princeton, N.J., 1973.
  21. Poincaré H., New methods of celestial mechanics, Vols. I-III, AIP Press, New York, 1982.
  22. Sakajo T., Yagasaki K., Chaotic motion of the $N$-vortex problem on a sphere. I. Saddle-centers in two-degree-of-freedom Hamiltonians, J. Nonlinear Sci. 18 (2008), 485-525.
  23. Simó C. (Editor), Hamiltonian systems with three or more degrees of freedom, Nato Science Series C, Vol. 533, Kluwer, Dordrech, 1999.
  24. van der Put M., Singer M.F., Galois theory of linear differential equations, Grundlehren der Mathematischen Wissenschaften, Vol. 328, Springer-Verlag, Berlin, 2003.
  25. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.
  26. Wiggins S., Introduction to applied nonlinear dynamical systems and chaos, 2nd ed., Texts in Applied Mathematics, Vol. 2, Springer-Verlag, New York, 2003.
  27. Yagasaki K., Horseshoes in two-degree-of-freedom Hamiltonian systems with saddle-centers, Arch. Ration. Mech. Anal. 154 (2000), 275-296.
  28. Yagasaki K., 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.
  29. Yagasaki K., Galoisian obstructions to integrability and Melnikov criteria for chaos in two-degree-of-freedom Hamiltonian systems with saddle centres, Nonlinearity 16 (2003), 2003-2012.
  30. Yagasaki K., Yamanaka S., Nonintegrability of dynamical systems with homo- and heteroclinic orbits, J. Differential Equations 263 (2017), 1009-1027.
  31. Ziglin S.L., Splitting of the separatrices and the nonexistence of first integrals in systems of differential equations of Hamiltonian type with two degrees of freedom, Math. USSR Izvestiya 31 (1988), 407-421.
  32. Ziglin S.L., The absence of an additional real-analytic first integral in some problems of dynamics, Funct. Anal. Appl. 31 (1997), 3-9.

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