Chaotic behavior in the unfolding of hopf-bogdanov-takens singularities

Discrete and Continuous Dynamical Systems - Series B

Volumn 25, Issue 2, 2020, Pages 599-615

  Drubi, Fátima ^a   Ibánez, Santiago ^a   Rivela, David ^a  

^a UNIVERSITY OF OVIEDO   (Spain)

Author keywords

chaotic behavior; Hopf Bogdanov Takens; Singularities; unfoldings

Indexed keywords

EID: 85076577337     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2019256     Document Type: Article

References (31)

1. A. Algaba, E. Freire and E. Gamero, Hypernormal form for the Hopf-zero bifurcation, Inter-nat. J. Bifur. Chaos Appl. Sci. Engrg., 8 (1988), 1857-1887.
2. A. Algaba, E. Freire, E. Gamero and A. J. Rodriguez-Luis, On a codimension-three unfolding of the interaction of degenerate Hopf and pitchfork bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 1333-1362.
3. A. Algaba, E. Freire, E. Gamero and A. J. Rodriguez-Luis, A three-parameter study of a degenerate case of the Hopf-pitchfork bifurcation, Nonlinearity, 12 (1999), 1177-1206.
4. A. Algaba, E. Freire, E. Gamero and A. J. Rodriguez-Luis, A tame degenerate Hopf-pitchfork bifurcation in a modified van der Pol-During oscillator, Nonlinear Dynam., 22 (2000), 249- 269.
5. A. Algaba, M. Merino, E. Freire, E. Gamero and A. J. Rodriguez-Luis, On the Hopf-pitchfork bifurcation in the Chua's equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 10 (2000), 291-305.
6. I. Baldoma, O. Castejon and T. M. Seara, Exponentially small heteroclinic breakdown in the generic Hopf-Zero singularity, J. Dyn. Diff. Equat., 25 (2013), 335-392.
7. I. Baldoma, O. Castejon and T. M. Seara, Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (I), J. Nonlinear Sci., 28 (2018), 1551-1627.
8. I. Baldoma, O. Castejon and T. M. Seara, Breakdown of a 2D heteroclinic connection in the Hopf-zero singularity (II): The generic case, J. Nonlinear Sci., 28 (2018), 1489-1549.
9. I. Baldoma, S. Iba~nez and T. M. Seara, Hopf-Zero singularities truly unfold chaos, preprint, arXiv:1903.09023.
10. P. G. Barrientos, S. Iba~nez and J. A. Rodriguez, Heteroclinic cycles arising in generic unfoldings of nilpotent singularities, J. Dyn. Diff. Equat., 23 (2011), 999-1028.
11. P. G. Barrientos, S. Iba~nez and J. A. Rodriguez, Robust cycles unfolding from conservative bifocal homoclinic orbits, Dyn. Syst., 31 (2016), 546-579.
12. H. W. Broer and G. Vegter, Subordinate Sil'nikov bifurcations near some singularities of vector fields having low codimension, Ergodic Theory Dynam. Systems, 4 (1984), 509-525.
13. A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer and B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14 (2008), 147-175.
14. F. Drubi, S. Iba~nez and J. A. Rodriguez, Coupling leads to chaos, J. Differential Equations, 239 (2007), 371-385.
15. F. Drubi, S. Iba~nez and J. A. Rodriguez, Hopf-pitchfork singularities in coupled systems, Phys. D, 240 (2011), 825-840.
16. F. Drubi, S. Iba~nez and D. Rivela, A formal classification of Hopf-Bogdanov-Takens singularities of codimension three, J. Math. Anal. Appl., 480 (2019), 123408.
17. F. Dumortier, S. Iba~nez, H. Kokubu and C. Simo, About the unfolding of a Hopf-zero singularity, Discrete Contin. Dyn. Syst., 33 (2013), 4435-4471.
18. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurca-tions of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983.
19. A. J. Homburg, Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbits to saddle-focus equilibria, Nonlinearity, 15 (2002), 1029-1050.
20. S. Iba~nez and J. A. Rodriguez, Shilnikov bifurcations in generic 4-unfoldings of a codimension-4 singularity, J. Differential Equations, 120 (1995), 411-428.
21. S. Iba~nez and J. A. Rodriguez, Shilnikov confirgurations in any generic unfolding of the nilpotent singularity of codimension three on R3, J. Differential Equations, 208 (2005), 147-175.
22. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998.
23. W. F. Langford and K. J. Zhan, Interactions of Andronov-Hopf and Bogdanov-Takens bifurcations, The Arnoldfest, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 24 (1999), 365-383.
24. L. Mora and M. Viana, Abundance of strange attractors, Acta Math., 171 (1993), 1-71.
25. A. Pumari~no and J. A. Rodriguez, Coexistence and Persistence of Strange Attractors, Lecture Notes in Mathematics, 1658. Springer-Verlag, Berlin, 1997.
26. A. Pumari~no and J. A. Rodriguez, Coexistence and persistence of infinitely many strange attractors, Ergodic Theory Dynam. Systems, 21 (2001), 1511-1523.
27. L. P. Sil'nikov, A case of the existence of a denumerable set of periodic motions, Dokl. Akad. Nauk SSSR, 160 (1965), 558-561.
28. L. P. Sil'nikov, A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type, Math. USSR Sb., 10 (1970), 91-102.
29. A. Steindl, Numerical investigation of the Hopf-Bogdanov-Takens mode interaction for a uidconveying tube, Procedia Engineering, 199 (2017), 857-862.
30. F. Takens, Singularities of vector fields, Publ. Math. IHES, (1974), 47-100.
31. C. Tresser, About some theorems by L. P. Sil'nikov, Ann. Inst. H. Poincare Phys. Theor., 40 (1984), 441-461.