The inner equation for generic analytic unfoldings of the Hopf-zero singularity

Discrete and Continuous Dynamical Systems - Series B

Volumn 10, Issue 2-3, 2008, Pages 323-347

  Baldoma I ^a   Seara, T M ^a  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

Inner system; Unfoldings of Hopf zero singularity

Indexed keywords

EID: 73149112791     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2008.10.323     Document Type: Article

References (19)

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, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 583-608.
2. S. Angenent, A variational interpretation of Mel'nikov's function and exponentially small separatrix splitting, in "Symplectic Geometry," London Math. Soc, Lecture Note Ser., Cambridge Univ. Press, 192 (1993), 5-35.
3. I. Baldomá, The inner equation for one and a half degrees of freedom rapidly forced hamiltonian systems, Nonlinearity, 19 (2006), 1415-1445.
4. I. Baldomá and T. M. Seara, Breakdown of heteroclinic orbits for some analytic unfoldings of the Hopf-zero singularity, 3. Nonlinear Sci., 16 (2006), 543-582.
5. H. W. Broer and G. Vegter, Subordinate Šil nikov bifurcations near some singularities of vector fields having low codimension, Ergodic Theory Dynam. Systems, 4 (1984), 509-525.
6. A. R. Champneys and V. Kirk, The entwined wiggling of homoclinic curves emerging from saddle-node/Hopf instabilities, Phys. D, 195 (1-2) (2004), 77-105.
7. F. Dumortier and S. Ibáñez, Singularities of vector fields on R3, Nonlinearity, 11 (1998), 1037-1047.
8. A. Delshams and R. Ramfrez-Ros, Exponentially small splitting of separatrices for perturbed integrable standard-like maps, J. Nonlinear Sci., 8 (1998), 317-352.
9. A. Delshams and R. Ramírez-Ros, Singular separatrix splitting and the Melnikov method: an experimental study, Experiment. Math., 8 (1999), 29-48.
10. E. Freire, E. Gamero, A. J. Rodríguez-Luis and A. Algaba, A note on the triple-zero linear degeneracy: normal forms, dynamical and bifurcation behaviors of an unfolding, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 2799-2820.
11. 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.
12. V. Gelfreich and D. Sauzin, Borel summation and splitting of separatrices for the Hénon map, Ann. Inst. Fourier (Grenoble), 51 (2001), 513-567.
13. V. Gelfreich and C. Simó, High-precision computations of divergent asymptotic series and homoclinic phenomena, preprint.
14. J. Guckenheimer, On a codimension two bifurcation, in "Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980)," Lecture Notes in Math., 898 (1981), 99-142.
15. J. W. Lamb, M. A. Teixeira and K. Webster, Heteroclinic bifurcations near hopf-zero bifurcation in reversible vector fields in R3, J. Differential Equations, 219 (2005), 78-115.
16. C. Olivé, D. Sauzin and T. M. Seara, Resurgence in a Hamilton-Jacobi equation, in "Proceedings of the International Conference in Honor of Frédéric Pham (Nice, 2002)," 53 (2003), 1185-1235.
17. F. Takens, A nonstabilizable jet of a singularity of a vector field, in "Dynamical Systems" (Proc. Sympos., Univ. Bahia, Salvador, 1971), Academic Press, New York, (1973), 583-597.
18. F. Takens, Normal forms for certain singularities of vector fields, Ann. Inst. Fourier (Grenoble), 23 (1973), 163-195. Colloque International sur l'Analyse et la Topologie Différentielle (Colloques Internationaux du Centre National de la Recherche Scientifique, Strasbourg, 1972).
19. F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 47-100.