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Separatrix splitting at a Hamiltonian 02 bifurcation


We study the splitting of a separatrix in a generic unfolding of a degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We assume that the unperturbed fixed point has two purely imaginary eigenvalues and a non-semisimple double zero one. It is well known that a one-parameter unfolding of the corresponding Hamiltonian can be described by an integrable normal form. The normal form has a normally elliptic invariant manifold of dimension two. On this manifold, the truncated normal form has a separatrix loop. This loop shrinks to a point when the unfolding parameter vanishes. Unlike the normal form, in the original system the stable and unstable separatrices of the equilibrium do not coincide in general. The splitting of this loop is exponentially small compared to the small parameter. This phenomenon implies nonexistence of single-round homoclinic orbits and divergence of series in normal form theory. We derive an asymptotic expression for the separatrix splitting. We also discuss relations with the behavior of analytic continuation of the system in a complex neighborhood of the equilibrium.

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  1. Alfimov, G. L., Eleonsky, V.M., and Kulagin, N.E., Dynamical Systems in the Theory of Solitons in the Presence of Nonlocal Interactions, Chaos, 1992, vol. 2, no. 4, pp. 565–570.

    Article  MATH  MathSciNet  Google Scholar 

  2. Baldomá, I. and Seara, T.M., Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity, J. Nonlinear Sci., 2006, vol. 16, no. 6, pp. 543–582.

    Article  MATH  MathSciNet  Google Scholar 

  3. Baldomá, I., Fontich, E., Guàrdia, M., and Seara, T. M., Exponentially Small Splitting of Separatrices beyond Melnikov Analysis: Rigorous Results, J. Differential Equations, 2012, vol. 253, no. 12, pp. 3304–3439.

    Article  MATH  MathSciNet  Google Scholar 

  4. Baldomá, I., Castejón, O., and Seara, T.M., Exponentially Small Heteroclinic Breakdown in the Generic Hopf-Zero Singularity, J. Dynam. Differential Equations, 2013, vol. 25, no. 2, pp. 335–392.

    Article  MATH  MathSciNet  Google Scholar 

  5. Broer, H. W., Chow, S.-N., Kim, Y., and Vegter, G., A Normally Elliptic Hamiltonian Bifurcation, Z. Angew. Math. Phys., 1993, vol. 44, no. 3, pp. 389–432.

    Article  MATH  MathSciNet  Google Scholar 

  6. Champneys, A.R., Homoclinic Orbits in Reversible Systems and Their Applications in Mechanics, Fluids and Optics, Phys. D, 1998, vol. 112, nos. 1–2, pp. 158–86

    Article  MATH  MathSciNet  Google Scholar 

  7. Champneys, A.R., Codimension-One Persistence beyond All Orders of Homoclinic Orbits to Singular Saddle Centres in Reversible Systems, Nonlinearity, 2001, vol. 14, no. 1, pp. 87–112.

    Article  MATH  MathSciNet  Google Scholar 

  8. Gelfreich, V. G., Separatrix Splitting for a High-Frequency Perturbation of the Pendulum, Russ. J. Math. Phys., 2000, vol. 7, no. 1, pp. 48–71.

    MATH  MathSciNet  Google Scholar 

  9. Gelfreich, V. G., Splitting of a Small Separatrix Loop near the Saddle-Center Bifurcation in Area-Preserving Maps, Phys. D, 2000, vol. 136, nos. 3–4, pp. 266–279.

    Article  MATH  MathSciNet  Google Scholar 

  10. Gelfreich, V. and Lazutkin, V., Splitting of Separatrices: Perturbation Theory and Exponential Smallness, Russian Math. Surveys, 2001, vol. 56, no. 3, pp. 499–558; see also: Uspekhi Mat. Nauk, 2001, vol. 56, no. 3(339), pp. 79–142.

    Article  MathSciNet  Google Scholar 

  11. Gelfreich, V., Near Strongly Resonant Periodic Orbits in a Hamiltonian System, Proc. Natl. Acad. Sci. USA, 2002, vol. 99, no. 22, pp. 13975–13979.

    Article  MATH  MathSciNet  Google Scholar 

  12. Gelfreich, V. G. and Lerman, L. M., Almost Invariant Elliptic Manifold in a Singularly Perturbed Hamiltonian System, Nonlinearity, 2002, vol. 15, no. 2, pp. 447–457.

    Article  MATH  MathSciNet  Google Scholar 

  13. Gelfreich, V.G. and Lerman, L.M., Long-Periodic Orbits and Invariant Tori in a Singularly Perturbed Hamiltonian System, Phys. D, 2003, vol. 176, nos. 3–4, pp. 125–146.

    Article  MATH  MathSciNet  Google Scholar 

  14. Gelfreich, V. and Simó, C., High-Precision Computations of Divergent Asymptotic Series and Homoclinic Phenomena, Discrete Contin. Dyn. Syst. Ser. B, 2008, vol. 10, nos. 2–3, pp. 511–536.

    MATH  MathSciNet  Google Scholar 

  15. Giorgilli, A., Unstable Equilibria of Hamiltonian Systems, Discrete Contin. Dynam. Systems, 2001, vol. 7, no. 4, pp. 855–871.

    Article  MATH  MathSciNet  Google Scholar 

  16. Grotta Ragazzo, C., Irregular Dynamics and Homoclinic Orbits to Hamiltonian Saddle Centers, Comm. Pure Appl. Math., 1997, vol. 50, no. 2, pp. 105–147.

    Article  MATH  MathSciNet  Google Scholar 

  17. Haragus, M. and Iooss, G., Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, London: Springer, 2011.

    Book  MATH  Google Scholar 

  18. Iooss, G. and Adelmeyer, M., Topics in Bifurcation Theory and Applications, Adv. Ser. Nonlinear Dynam., vol. 3, River Edge, N.J.: World Sci., 1992.

    MATH  Google Scholar 

  19. Iooss, G. and Lombardi, E., Polynomial Normal Forms with Exponentially Small Remainder for Analytic Vector Fields, J. Differential Equations, 2005, vol. 212, no. 1, pp. 1–61.

    Article  MATH  MathSciNet  Google Scholar 

  20. Iooss, G. and Lombardi, E., Normal Forms with Exponentially Small Remainder: Application to Homoclinic Connections for the Reversible 02+ Resonance, C. R. Math. Acad. Sci. Paris, Ser. 1, 2004, vol. 339, no. 12, pp. 831–838.

    Article  MATH  MathSciNet  Google Scholar 

  21. Jézéquel, T., Bernard, P., and Lombardi, E., Homoclinic Connections with Many Loops near a 02 Resonant Fixed Point for Hamiltonian Systems, arXiv:1401.1509 (2014), 79 pp.

    Google Scholar 

  22. Koltsova, O., Families ofMulti-Round Homoclinic and Periodic Orbits near a Saddle-Center Equilibrium, Regul. Chaotic Dyn., 2003, vol. 8, no. 3, pp. 191–200.

    Article  MATH  MathSciNet  Google Scholar 

  23. Koltsova, O. and Lerman, L.M., Periodic and Homoclinic Orbits in a Two-Parameter Unfolding of a Hamiltonian System with a Homoclinic Orbit to a Saddle-Center, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1995, vol. 5, no. 2, pp. 397–408.

    Article  MATH  MathSciNet  Google Scholar 

  24. Koltsova, O.Yu. and Lerman, L. M., Families of Transverse Poincaré Homoclinic Orbits in 2NDimensional Hamiltonian Systems Close to the System with a Loop to a Saddle-Center, Int. J. Bifurcation & Chaos, 1996, vol. 6, no. 6, pp. 991–1006.

    Article  MATH  MathSciNet  Google Scholar 

  25. Lerman, L.M. and Gelfreich, V.G., Fast-Slow Hamiltonian Dynamics near a Ghost Separatrix Loop, J. Math. Sci. (N. Y.), 2005, vol. 126, no. 5, pp. 1445–1466; see also: Sovrem. Mat. Prilozh., 2003, no. 8, pp. 85–107.

    Article  MATH  MathSciNet  Google Scholar 

  26. Lerman, L.M., Hamiltonian Systems with a Separatrix Loop of a Saddle-Center, Selecta Math. (N. S.), 1991, vol. 10, pp. 297–306; see also: Methods of the Qualitative Theory of Differential Equations, E.A. Leontovich–Andronova (Ed.), Gorki: GGU, 1987, pp. 89–103.

    MathSciNet  Google Scholar 

  27. Llibre, J., Martínez, R., and Simó, C., Transversality of the Invariant Manifolds Associated to the Lyapunov Family of Periodic Orbits near L2 in the Restricted Three-body Problem, J. Differential Equations, 1985, vol. 58, no. 1, pp. 104–156.

    Article  MATH  MathSciNet  Google Scholar 

  28. Lombardi, E., Oscillatory Integrals and Phenomena beyond All Algebraic Orders: With Applications to Homoclinic Orbits in Reversible Systems, Lecture Notes in Math., vol. 1741, Berlin: Springer, 2000.

    Book  Google Scholar 

  29. Mielke, A., Holmes, P., and O’Reiley, O., Cascades of Homoclinic Orbits to, and Chaos near, a Hamiltonian Saddle-Center, J. Dynam. Differential Equations, 1992, vol. 4, no. 1, pp. 95–126.

    Article  MATH  MathSciNet  Google Scholar 

  30. Moser, J., On the Generalization of a Theorem of A. Liapounoff, Comm. Pure Appl. Math., 1958, vol. 11, pp. 257–271.

    Article  MATH  MathSciNet  Google Scholar 

  31. Treschev, D., Splitting of Separatrices for a Pendulum with Rapidly Oscillating Suspension Point, Russian J. Math. Phys., 1997, vol. 5, no. 1, pp. 63–98.

    MATH  MathSciNet  Google Scholar 

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Correspondence to Vassili Gelfreich.

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Dedicated to our colleagues Prof. Sergey Bolotin and Prof. Dmitry Treschev on the occasion of their anniversaries

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Gelfreich, V., Lerman, L. Separatrix splitting at a Hamiltonian 02 bifurcation. Regul. Chaot. Dyn. 19, 635–655 (2014).

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MSC2010 numbers

  • 37J20
  • 37J45
  • 70K50
  • 70K70


  • Hamiltonian bifurcation
  • homoclinic orbit
  • separatrix splitting
  • asymptotics beyond all orders