Breakdown of a 2D Heteroclinic Connection in the Hopf-Zero Singularity (I)

Article

Abstract

In this paper we study a beyond all orders phenomenon which appears in the analytic unfoldings of the Hopf-zero singularity. It consists in the breakdown of a two-dimensional heteroclinic surface which exists in the truncated normal form of this singularity at any order. The results in this paper are twofold: on the one hand, we give results for generic unfoldings which lead to sharp exponentially small upper bounds of the difference between these manifolds. On the other hand, we provide asymptotic formulas for this difference by means of the Melnikov function for some non-generic unfoldings.

Keywords

Exponentially small splitting Hopf-zero bifurcation Melnikov function Borel transform 

Mathematics Subject Classification

34E10 E4E15 37C29 37G99 

Notes

Acknowledgements

The three authors are really grateful to the two anonymous referees who carefully reviewed the manuscript. Specially the comments and suggestions about the introduction have greatly improved the quality of the final version. The authors are in debt with Jordi Villanueva for his help in the proof of Theorem 2.9. The authors have been partially supported by the Spanish MINECO-FEDER Grant MTM2015-65715-P and the Catalan Grant 2014SGR504. Tere M-Seara is also supported by the Russian Scientific Foundation Grant 14-41-00044 and European Marie Curie Action FP7-PEOPLE-2012-IRSES: BREUDS.

References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Volume 55 of National Bureau of Standards Applied Mathematics Series. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC (1964)Google Scholar
  2. Baldomá, I.: The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems. Nonlinearity 19(6), 1415–1446 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. Baldomá, I., Fontich, E.: Exponentially small splitting of invariant manifolds of parabolic points. Mem. Am. Math. Soc. 167(792):x–83 (2004)Google Scholar
  4. Baldomá, I., Fontich, E.: Exponentially small splitting of separatrices in a weakly hyperbolic case. J. Differ. Equ. 210(1), 106–134 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. Baldomá, I., Martín, P.: The inner equation for generalized standard maps. SIAM J. Appl. Dyn. Syst. 11(3), 1062–1097 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. Baldomá, I., Seara, T.M.: Breakdown of heteroclinic orbits for some analytic unfoldings of the Hopf-zero singularity. J. Nonlinear Sci. 16(6), 543–582 (2006)MathSciNetCrossRefMATHGoogle Scholar
  7. Baldomá, I., Seara, T.M.: The inner equation for generic analytic unfoldings of the Hopf-zero singularity. Discrete Contin. Dyn. Syst. Ser. B 10(2–3), 323–347 (2008)MathSciNetMATHGoogle Scholar
  8. Baldomá, I., Fontich, E., Guardia, M., Seara, T.M.: Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results. J. Differ. Equ. 253(12), 3304–3439 (2012)MathSciNetCrossRefMATHGoogle Scholar
  9. Baldomá, I., Castejón, O., Seara, T.M.: Exponentially small heteroclinic breakdown in the generic Hopf-Zero singularity. J. Dyn. Differ. Equ. 25(2), 335–392 (2013)MathSciNetCrossRefMATHGoogle Scholar
  10. Baldomá, I., Castejón, O., Seara, T.M.: Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (II). The generic case. Preprint arXiv:1608.01116 (2016)
  11. Benseny, A., Olivé, C.: High precision angles between invariant manifolds for rapidly forced Hamiltonian systems. In: Proceedings Equadiff91, pp. 315–319 (1993)Google Scholar
  12. Broer, H.W., Roussarie, R.: Exponential confinement of chaos in the bifurcation sets of real analytic diffeomorphisms. In: Global Analysis of Dynamical Systems, pp. 167–210. Inst. Phys., Bristol (2001)Google Scholar
  13. Broer, H.W., Takens, F.: Formally symmetric normal forms and genericity. In: Dynamics Reported, Volume 2 of Dynamics Report Series Dynamical Systems and Applications, pp. 39–59. Wiley, Chichester (1989)Google Scholar
  14. Broer, H.W., Tangerman, F.M.: From a differentiable to a real analytic perturbation theory, applications to the Kupka Smale theorems. Ergodic Theory Dyn. Syst. 6(3), 345–362 (1986)MathSciNetCrossRefMATHGoogle Scholar
  15. Broer, H.W., Vegter, G.: Subordinate Šil’nikov bifurcations near some singularities of vector fields having low codimension. Ergodic Theory Dyn. Syst. 4, 509–525 (1984)CrossRefMATHGoogle Scholar
  16. Castejón, O.: Study of invariant manifolds in two different problems: the Hopf-zero singularity and neural synchrony. Ph.D. thesis, UPC (2015)Google Scholar
  17. Champneys, A.R., Kirk, V.: The entwined wiggling of homoclinic curves emerging from saddle-node/Hopf instabilities. Physica D 195(1–2), 77–105 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. Delshams, A., Ramírez-Ros, R.: Singular separatrix splitting and the Melnikov method: an experimental study. Exp. Math. 8(1), 29–48 (1999)MathSciNetCrossRefMATHGoogle Scholar
  19. Delshams, A., Seara, T.M.: An asymptotic expression for the splitting of separatrices of the rapidly forced pendulum. Commun. Math. Phys. 150(3), 433–463 (1992)MathSciNetCrossRefMATHGoogle Scholar
  20. Delshams, A., Seara, T.M.: Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom. Math. Phys. Electron. J. 3, Paper 4 (1997)Google Scholar
  21. Dumortier, F., Ibáñez, S.: Singularities of vector fields on \(text{R}^3\). Nonlinearity 11(4), 1037–1047 (1998)MathSciNetCrossRefMATHGoogle Scholar
  22. Dumortier, F., Ibáñez, S., Kokubu, H., Simó, C.: About the unfolding of a Hopf-zero singularity. Discrete Contin. Dyn. Syst. 33(10), 4435–4471 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. Fontich, E.: Rapidly forced planar vector fields and splitting of separatrices. J. Differ. Equ. 119(2), 310–335 (1995)MathSciNetCrossRefMATHGoogle Scholar
  24. Fontich, E., Simó, C.: Invariant manifolds for near identity differentiable maps and splitting of separatrices. Ergodic Theory Dyn. Syst. 10(2), 319–346 (1990a)MathSciNetMATHGoogle Scholar
  25. Fontich, E., Simó, C.: The splitting of separatrices for analytic diffeomorphisms. Ergodic Theory Dyn. Syst. 10(2), 295–318 (1990b)MathSciNetMATHGoogle Scholar
  26. Freire, E., Gamero, E., Rodríguez-Luis, A.J., Algaba, A.: A note on the triple-zero linear degeneracy: normal forms, dynamical and bifurcation behaviors of an unfolding. Int. J. Bifur. Chaos Appl. Sci. Eng. 12(12), 2799–2820 (2002)MathSciNetCrossRefMATHGoogle Scholar
  27. Gaivão, J.P., Gelfreich, V.G.: Splitting of separatrices for the Hamiltonian–Hopf bifurcation with the Swift–Hohenberg equation as an example. Nonlinearity 24(3), 677–698 (2011)MathSciNetCrossRefMATHGoogle Scholar
  28. Gavrilov, N.K.: On some bifurcations of an equilibrium with one zero and a pair of pure imaginary eigenvalues. In: Leontovich-Andronova, E.A. (ed.) Methods of the Qualitative Theory of Differential Equations (Russian), pp. 33–40. Gorky State University, Gorky (1978)Google Scholar
  29. Gavrilov, N.K.: On bifurcations of codimension two equilibria of divergence-free vector fields. In: Leontovich-Andronova, E.A. (ed.) Methods of the Qualitative Theory of Differential Equations (Russian), pp. 46–54. Gorky State University, Gorky (1985)Google Scholar
  30. Gavrilov, N.K., Roshchin, N.V.: On stability of an equilibrium with one zero and a pair of pure imaginary eigenvalues. In: Leontovich-Andronova, E.A. (ed.) Methods of the Qualitative Theory of Differential Equations, pp. 41–49. Gorky State University, Gorky (1983)Google Scholar
  31. Gelfreich, V.G.: Separatrices splitting for the rapidly forced pendulum. In Seminar on Dynamical Systems (St. Petersburg, 1991), Volume 12 of Programs in Nonlinear Differential Equations Applications, pp. 47–67. Birkhäuser, Basel (1994)Google Scholar
  32. Gelfreich, V.G.: Melnikov method and exponentially small splitting of separatrices. Physica D 101(3–4), 227–248 (1997a)MathSciNetCrossRefMATHGoogle Scholar
  33. Gelfreich, V.G.: Reference systems for splittings of separatrices. Nonlinearity 10(1), 175–193 (1997b)MathSciNetCrossRefMATHGoogle Scholar
  34. Gelfreich, V.G.: A proof of the exponentially small transversality of the separatrices for the standard map. Commun. Math. Phys. 201(1), 155–216 (1999)MathSciNetCrossRefMATHGoogle Scholar
  35. Gelfreich, V.G.: Separatrix splitting for a high-frequency perturbation of the pendulum. Russ. J. Math. Phys. 7(1), 48–71 (2000)MathSciNetMATHGoogle Scholar
  36. Gelfreich, V.: Chaotic zone in the Bogdanov–Takens bifurcation for diffeomorphisms. In: Analysis and Applications—ISAAC 2001 (Berlin), volume 10 of International Society of Analytics Applied Computing, pp. 187–197. Kluwer, Dordrecht (2003)Google Scholar
  37. Gelfreich, V.G., Brännstrom, N.: Asymptotic series for the splitting of separatrices near a Hamiltonian bifurcation (2008). arXiv preprint arXiv:0806.2403
  38. Gelfreich, V., Naudot, V.: Analytic invariants associated with a parabolic fixed point in \(\mathbb{C}^2\). Ergodic Theory Dyn. Syst. 28(6), 1815–1848 (2008)CrossRefMATHGoogle Scholar
  39. Gelfreich, V., Naudot, V.: Width of the homoclinic zone in the parameter space for quadratic maps. Exp. Math. 18(4), 409–427 (2009)MathSciNetCrossRefMATHGoogle Scholar
  40. Gelfreich, V.G., Sauzin, D.: Borel summation and splitting of separatrices for the Hénon map. Ann. Inst. Fourier (Grenoble) 51(2), 513–567 (2001)MathSciNetCrossRefMATHGoogle Scholar
  41. Gelfreich, V.G., Simó, C.: High-precision computations of divergent asymptotic series and homoclinic phenomena. Discrete Contin. Dyn. Syst. Ser. B 10(2–3), 511–536 (2008)MathSciNetMATHGoogle Scholar
  42. Gelfreich, V., Simó, C., Vieiro, A.: Dynamics of symplectic maps near a double resonance. Physica D 243(1), 92–110 (2013)MathSciNetCrossRefMATHGoogle Scholar
  43. Guardia, M.: Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discrete Contin. Dyn. Syst. 33(7), 2829–2859 (2013)MathSciNetCrossRefMATHGoogle Scholar
  44. Guardia, M., Olivé, C., Seara, T.M.: Exponentially small splitting for the pendulum: a classical problem revisited. J. Nonlinear Sci. 20(5), 595–685 (2010)MathSciNetCrossRefMATHGoogle Scholar
  45. Guckenheimer, J.: On a codimension two bifurcation. Dyn. Syst. Turbul. Warwick 1980, 99–142 (1981)MathSciNetMATHGoogle Scholar
  46. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields Volume 42 of Applied Mathematical Sciences. Springer, New York (1983)CrossRefMATHGoogle Scholar
  47. Holmes, P., Marsden, J., Scheurle, J.: Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations. In: Hamiltonian Dynamical Systems, volume 81 of Contemporary Mathematics. American Mathematical Society, Providence, RI (1988)Google Scholar
  48. Jézéquel, T., Bernard, P., Lombardi, E.: Homoclinic orbits with many loops near a \(0^2i\omega \) resonant fixed point of Hamiltonian systems. Discrete Contin. Dyn. Syst. 36(6), 3153–3225 (2016)MathSciNetMATHGoogle Scholar
  49. Kruskal, M.D., Segur, H.: Asymptotics beyond all orders in a model of crystal growth. Stud. Appl. Math. 85(2), 129–181 (1991)MathSciNetCrossRefMATHGoogle Scholar
  50. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, Volume 112 of Applied Mathematical Sciences, third edn. Springer, New York (2004)CrossRefGoogle Scholar
  51. Lamb, J.W., Teixeira, M.A., Webster, K.: Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in \({\bf R}^3\). Preprint http://www.ime.unicamp.br/~teixeira/LTW04.pdf (2004)
  52. Lazaro Ochoa, J.T.: On normal forms and splitting of separatrices in reversible systems. Ph.D. thesis, Universitat Politècnica de Catalunya (2003)Google Scholar
  53. Lazutkin, V.F.: Splitting of separatrices for the Chirikov standard map. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 300 (Teor. Predst. Din. Sist. Spets. Vyp. 8):25–55, 285, (2003)Google Scholar
  54. Lochak, P., Marco, J.-P., Sauzin, D.: On the splitting of invariant manifolds in multidimensional near-integrable Hamiltonian systems. Mem. Am. Math. Soc. 163(775), viii+145 (2003)MathSciNetMATHGoogle Scholar
  55. Lombardi, E.: Non-persistence of homoclinic connections for perturbed integrable reversible systems. J. Dyn. Differ. Equ. 11(1), 129–208 (1999)MathSciNetCrossRefMATHGoogle Scholar
  56. Lombardi, E.: Oscillatory Integrals and Phenomena Beyond All Algebraic Orders. Lecture Notes in Mathematics, vol. 1741. Springer, Berlin (2000). (With applications to homoclinic orbits in reversible systems)CrossRefMATHGoogle Scholar
  57. Martín, P., Sauzin, D., Seara, T.M.: Exponentially small splitting of separatrices in the perturbed McMillan map. Discrete Contin. Dyn. Syst. 31(2), 301–372 (2011)MathSciNetCrossRefMATHGoogle Scholar
  58. Mel’nikov, V.K.: On the stability of a center for time-periodic perturbations. Trudy Moskov. Mat. Obšč. 12, 3–52 (1963)MathSciNetMATHGoogle Scholar
  59. Miguel, N., Simó, C., Vieiro, A.: From the Hénon conservative map to the chirikov standard map for large parameter values. Regul. Chaotic Dyn. 18(5), 469–489 (2013)MathSciNetCrossRefMATHGoogle Scholar
  60. Neĭshtadt, A.I.: The separation of motions in systems with rapidly rotating phase. Prikl. Mat. Mekh. 48(2), 197–204 (1984)MathSciNetGoogle Scholar
  61. Olivé, C., Sauzin, D., Seara, T.M.: Resurgence in a Hamilton-Jacobi equation. In: Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002). Ann. Inst. Fourier (Grenoble), vol. 53(4), pp. 1185–1235 (2003)Google Scholar
  62. Poincaré, H.: Sur le problème des trois corps et les équations de la dynamique. Acta Math. 13(1), A3–A270 (1890)Google Scholar
  63. Sauzin, D.: A new method for measuring the splitting of invariant manifolds. Ann. Sci. École Norm. Sup. (4) 34(2), 159–221 (2001)MathSciNetCrossRefMATHGoogle Scholar
  64. Šil’nikov, L.P.: A case of the existence of a denumerable set of periodic motions. Dokl. Akad. Nauk SSSR 160, 558–561 (1965)MathSciNetGoogle Scholar
  65. Simó, C., Vieiro, A.: Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps. Nonlinearity 22(5), 1191 (2009)MathSciNetCrossRefMATHGoogle Scholar
  66. Simó, C., Broer, H., Roussarie, R.: A numerical survey on the Takens–Bogdanov bifurcation for diffeomorphisms. In: European Conference on Iteration Theory (Batschuns, 1989), pp. 320–334. World Science Publications, River Edge, NJ (1991)Google Scholar
  67. Takens, F.: A nonstabilizable jet of a singularity of a vector field. In: Dynamical Systems (Proceedings Symposium, University Bahia, Salvador, 1971), pp. 583–597. Academic Press, New York (1973a)Google Scholar
  68. Takens, F.: Normal forms for certain singularities of vectorfields. Ann. Inst. Fourier (Grenoble) 23(2):163–195 (1973b) (Colloque International sur l’Analyse et la Topologie Différentielle (Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1972))Google Scholar
  69. Takens, F.: Singularities of vector fields. Publ. Math. l’IHES 43(1), 47–100 (1974)MathSciNetCrossRefMATHGoogle Scholar
  70. Treschev, D.V.: Splitting of separatrices for a pendulum with rapidly oscillating suspension point. Russ. J. Math. Phys. 5(1), 63–98 (1998). 1997MathSciNetMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain

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