Breakdown of heteroclinic orbits for some analytic unfoldings of the hopf-zero singularity

Journal of Nonlinear Science

Volumn 16, Issue 6, 2006, Pages 543-582

  Baldoma, I ^a   Seara, T M ^b  

^a UNIVERSITY OF BARCELONA   (Spain)

^b UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

Borel transform; Exponentially small splitting; Hopf zero bifurcation; Melnikov function

Indexed keywords

EID: 84867945302     PISSN: 09388974     EISSN: 14321467     Source Type: Journal    
DOI: 10.1007/s00332-005-0736-z     Document Type: Article

References (42)

1. A. Algaba, M. Merino, E. Freire, E. Gamero, and A. J. Rodríguez-Luis. Some results on Chua's equation near a triple-zero linear degeneracy. Int. J. Bifur. Chaos Appl. Sci. Eng., 13(3):583-608, 2003. (Pubitemid 36591056)
2. S. Angenent. A variational interpretation of Melnikov's function and exponentially small separatrix splitting. In Symplectic Geometry, volume 192 of London Math. Soc. Lecture Note Ser., pages 5-35. Cambridge Univ. Press, Cambridge, 1993.
3. M. Abramowitz and I. A. Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications Inc., New York, 1992. Reprint of the 1972 edition.
4. I. Baldom'a. The inner equation for one and a half degrees of freedom rapidly forced Hamiltonian systems. Nonlinearity, 19:1415-1445, 2006.
5. I. Baldom'a and E. Fontich. Exponentially small splitting of invariant manifolds of parabolic points. Mem. Amer. Math. Soc., 167(792):x-83, 2004.
6. I. Baldoḿa and T.M. Seara. The inner equation for analytic unfoldings of the hopfzero singularity. preprint.
7. H. Broer. Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case. In Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), volume 898 of Lecture Notes in Math., pages 54-74. Springer, Berlin, 1981.
8. H. Broer. Quasiperiodic flow near a codimension one singularity of a divergence free vector field in dimension three. In Dynamical Systems and Turbulence,Warwick 1980 (Coventry, 1979/1980), volume 898 of Lecture Notes in Math., pages 75-89. Springer, Berlin, 1981.
9. H.W. Broer and G.Vegter. Subordinate Silnikov bifurcations near some singularities of vector fields having low codimension. Ergodic Theory Dynam. Syst., 4(4):509-525, 1984.
10. A. Delshams and P. Gutíerrez. Splitting potential and the Poincaŕe-Melnikov method for whiskered tori in Hamiltonian systems. J. Nonlin. Sci., 10(4):433-476, 2000.
11. A. Delshams,V. Gelfreich, A. Jorba, and T. M. Seara. Exponentially small splitting of separatrices under fast quasiperiodic forcing. Commun. Math. Phys., 189(1):35-71, 1997. (Pubitemid 127358543)
12. F. Dumortier and S. Ib'añez. Singularities of vector fields on R3. Nonlinearity, 11(4):1037-1047, 1998.
13. A. Delshams and R. Ramírez-Ros. Exponentially small splitting of separatrices for perturbed integrable standard-like maps. J. Nonlin. Sci., 8(3):317-352, 1998. (Pubitemid 128461157)
14. A. Delshams and T. M. Seara. Splitting of separatrices in Hamiltonian systems with one and a half degrees of freedom. Math. Phys. Electron. J., 3:Paper 4, 40 pp. (electronic), 1997.
15. L. H. Eliasson. Biasymptotic solutions of perturbed integrable Hamiltonian systems. Bol. Soc. Brasil. Mat. (N.S.), 25(1):57-76, 1994.
16. A. Erd'elyi,W. Magnus, F. Oberhettinger, and F. G. Tricomi. Higher Transcendental Functions. Vol. I. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman.
17. E. Freire, E. Gamero, A. J. Rodríguez-Luis, and A. Algaba. A note on the triplezero linear degeneracy: Normal forms, dynamical and bifurcation behaviors of an unfolding. Int. J. Bifur. Chaos Appl. Sci. Eng., 12(12):2799-2820, 2002. (Pubitemid 36261586)
18. V. G. Gelfreich. Melnikov method and exponentially small splitting of separatrices. Physica D, 101(3-4):227-248, 1997. (Pubitemid 127701364)
19. V. G. Gelfreich. Reference systems for splittings of separatrices. Nonlinearity, 10(1):175-193, 1997.
20. J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, volume 42 of Applied Mathematical Sciences. Springer-Verlag, New York, 1983.
21. V. Gelfreich and D. Sauzin. Borel summation and splitting of separatrices for the H'enon map. Ann. Inst. Fourier (Grenoble), 51(2):513-567, 2001.
22. J. Guckenheimer. On a codimension two bifurcation. In Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), volume 898 of Lecture Notes in Math., pages 99-142. Springer, Berlin, 1981.
23. P. Holmes, J. Marsden, and J. Scheurle. Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations. In Hamiltonian Dynamical Systems (Boulder, CO, 1987), volume 81 of Contemp. Math., pages 213-244. Amer. Math. Soc., Providence, RI, 1988.
24. J. Jones, W. C. Troy, and A. D. MacGillivray. Steady solutions of the Kuramoto-Sivashinsky equation for small wave speed. J. Diff. Eq., 96(1):28-55, 1992.
25. Y. Kuramoto and T. Tsuzuki. Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys., 55:356-369, 1976.
26. V. F. Lazutkin. 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. Translated from the Russian and with a preface by V. Gelfreich, Workshop on Differential Equations (Saint Petersburg, 2002).
27. E. Lombardi. Nonpersistence of homoclinic connections for perturbed integrable reversible systems. J. Dynam. Diff. Eq., 11(1):129-208, 1999.
28. J.W. Lamb, M.A. Teixeira, and K.Webster. Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in R3. J. Diff. Eq., 219(1):78-115.
29. V.F. Melnikov. On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc., 12:3-56, 1963.
30. D. Michelson. Steady solutions of the Kuramoto-Sivashinsky equation. Physica D, 19(1):89-111, 1986.
31. C. Oliv'e, D. Sauzin, and T. M. Seara. Resurgence in a Hamilton-Jacobi equation. In Proceedings of the International Conference in Honor of Fŕed'eric Pham (Nice, 2002), volume 53, pages 1185-1235, 2003.
32. H. Poincaŕe. Sur le probl̀eme des trois corps et les ́equations de la dynamique. Acta Math., 13:1-271, 1890.
33. S. V. Raghavan, J. B. McLeod, and W. C. Troy. A singular perturbation problem arising from the Kuramoto-Sivashinsky equation. Diff. Int. Eq., 10(1):1-36, 1997. (Pubitemid 24750434)
34. D. Sauzin. A new method for measuring the splitting of invariant manifolds. Ann. Sci. École Norm. Sup. (4), 34(2):159-221, 2001. (Pubitemid 33631318)
35. J. Scheurle. Chaos in a rapidly forced pendulum equation. In Dynamics and Control of Multibody Systems (Brunswick, ME, 1988), volume 97 of Contemp. Math., pages 411-419. Amer. Math. Soc., Providence, RI, 1989.
36. L. P. Silnikov. A case of the existence of a countable number of periodic motions. Soviet Math. Dokl, 6:163-166, 1965.
37. L. P. Silnikov. On the question of the structure of an extended neighborhood of a structurally stable state of equilibrium of saddle-focus type. Mat. Sb. (N.S.), 81 (123):92-103, 1970.
38. L. P. Shilnikov, A. L. Shilnikov, D. Turaev, and L. O. Chua. Methods of Qualitative Theory in Nonlinear Dynamics. Part II, volume 5 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises.World Scientific Publishing Co. Inc., River Edge, NJ, 2001.
39. F. Takens. A nonstabilizable jet of a singularity of a vector field. In Dynamical Systems (Proc. Sympos., Univ. Bahia, Salvador, 1971), pages 583-597. Academic Press, New York, 1973.
40. F. Takens. Normal forms for certain singularities of vector fields. Ann. Inst. Fourier (Grenoble), 23(2):163-195, 1973. Colloque International sur l'Analyse et la Topologie Diff́erentielle (Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1972).
41. F.Takens. Singularities of vector fields. Inst.HautesÉtudes Sci.Publ.Math., (43):47-100, 1974.
42. D. V. Treschev. Splitting of separatrices for a pendulum with rapidly oscillating suspension point. Russian J. Math. Phys., 5(1):63-98 (1998), 1997.