Takens-Bogdanov bifurcations of periodic orbits and Arnold's tongues in a three-dimensional electronic model

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 11, Issue 2, 2001, Pages 513-531

  Algaba, A ^a   Merino, M ^a   Rodriguez Luis A J ^b  

^a UNIVERSITY OF HUELVA   (Spain)

^b UNIVERSITY OF SEVILLE   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035590087     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127401002286     Document Type: Article

References (35)

1. Algaba, A., Freire, E. & Gamero, E. [1998a] "Hyper-normal form for the Hopf-zero bifurcation," Int. J. Bifurcation and Chaos, 8, 1857-1887.
2. Algaba, A., Freire, E., Gamero, E. & Rodríguez-Luis, A. J. [1998b] "Analysis of Hopf and Takens-Bogdanov bifurcations in a modified van der Pol-Duffing oscillator," Nonlin. Dyn. 16, 369-404.
3. Algaba, A., Freire, E., Gamero, E. & Rodríguez-Luis, A. J. [1999a] "On a codimension-three unfolding of the interaction of degenerate Hopf and pitchfork bifurcations," Int. J. Bifurcation and Chaos 9, 1333-1362.
4. Algaba, A., Freire, E., Gamero, E. & Rodríguez-Luis, A. J. [1999b] "A three-parameter study of a degenerate case of the Hopf-pitchfork bifurcation," Nonlinearity 12, 1177-1206.
5. Algaba, A., Freire, E., Gamero, E. & Rodríguez-Luis, A. J. [2000] "A tame degenerate Hopf-pitchfork bifurcation in a modified van der Pol-Duffing oscillator," Nonlin. Dyn. 22, 249-269.
6. Andronov, A. A., Vitt, A. A. & Khaikin, S. E. [1966] Theory of Oscillators (Pergamon, London).
7. Arnold, V. I. [1983] Geometrical Methods in the Theory of Ordinary Differential Equations (Springer-Verlag, Berlin).
8. Aronson, D. G., Chory, M. A., Hall, G. R. & McGeehee, R. P. [1982] "Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer assisted study," Commun. Math. Phys. 83, 303-354.
9. Arrowsmith, D. K., Cartwright, J. H. E., Lansbury, A. N. & Place, C. M. [1993] "The Bogdanov map: Bifurcations, mode locking, and chaos in a dissipative system," Int. J. Bifurcation and Chaos 3, 803-842.
10. Arrowsmith, J. & Place, C. M. [1990] An Introduction to Dynamical Systems (Cambridge University Press, Cambridge).
11. Belyakov, L. [1984] "Bifurcation of systems with homoclinic curve of a saddle-focus with saddle quantity zero," Mat. Zametki 36, 838-843.
12. Broer, H., Roussarie, R. & Simó, C. [1993] "On the Bogdanov-Takens bifurcation for planar diffeomorphisms," Proc. Equadiff '91, eds. Perelló, C. et al. (World Scientific, Singapore), pp. 81-92.
13. Broer, H., Simó, C. & Tatjer, J. C. [1998] "Towards global models near homoclinic tangencies of dissipative diffeomorphisms," Nonlinearity 11, 667-770.
14. Bykov, V. V. [1993] "The bifurcations of separatrix contours and chaos," Physica D62, 290-299.
15. Champneys, A. R. & Rodríguez-Luis, A. J. [1999] "The nontransverse Shil'nikov-Hopf bifurcation; uncoupling of homoclinic orbits and homoclinic tangencies," Physica D128, 130-158.
16. Chow, S. N., Li, C. & Wang, D. [1994] Normal Forms and Bifurcations of Planar Vector Fields (Cambridge University Press, Cambridge).
17. Devaney, R. L. [1986] An Introduction to Chaotic Dynamics (Benjamin/Cummings, Menlo Park).
18. Doedel, E., Wang, X. & Fairgrieve, T. [1995] "AUTO94: Software for continuation and bifurcation problems in ordinary differential equations," Applied Mathematics Report, California Institute of Technology.
19. 2 electronic oscillator. (I) Analysis," preprint, University of Seville.
20. Freire, E., Rodríguez-Luis, A. J., Gamero, E. & Ponce, E. [1993] "A case study for homoclinic chaos in an autonomous electronic circuit. A trip from Takens-Bogdanov to Hopf-Shil'nikov," Physica D62, 230-253.
21. Freire, E., Gamero, E., Rodríguez-Luis, A. J. & Algaba, A. [2000a] "A note on the triple-zero linear degeneracy: Normal forms, dynamical and bifurcation behavior of an unfolding," preprint.
22. Freire, E., Gamero, E., Rodríguez-Luis, A. J. & Algaba, A. [2000b] "On the global bifurcations of an unfolding of the triple-zero linear degeneracy," preprint.
23. Frouzakis, C. E., Adomaitis, R. A., & Kevrekidis, I. G. [1991] "Resonance phenomena in an adaptively-controlled system," Int. J. Bifurcation and Chaos 1, 83-106.
24. Gaspard, P., Kapral, R. & Nicolis, G. [1984] "Bifurcation phenomena near homoclinic orbits: A two-parameter analysis," J. Stat. Phys. 35, 697-727.
25. Glendinning, P. & Sparrow, C. T. [1984] "Local and global behaviour near homoclinic orbits," J. Stat. Phys. 35, 645-696.
26. Glendinning, P. & Sparrow, C. T. [1986] "T-points: A codimension two heteroclinic bifurcation," J. Stat. Phys. 43, 479-488.
27. Guckenheimer, J. & Holmes, P. J. [1986] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci. Series, Vol. 42 (Springer-Verlag, Berlin).
28. Guckenheimer, J. & Kim, S. [1992] "Dstool: A dynamical system toolkit with an interactive graphical interface," Applied Mathematics Report, Center for Applied Mathematics, Cornell University, Ithaca, NY.
29. Hirschberg, P. & Knobloch, E. [1993] "Sil'nikov-Hopf bifurcation," Physica D62, 202-216.
30. Kirk, V. [1993] "Merging of resonance tongues," Physica D66, 267-281.
31. Kuznetsov, Y. A., Muratori, S. & Rinaldi, S. [1992] "Bifurcations and chaos in a periodic predator-prey model," Int. J. Bifurcation and Chaos 2, 117-128.
32. Kuznetsov, Y. A. [1998] Elements of Applied Bifurcation Theory, Appl. Math. Sci. Series, Vol. 112 (Springer-Verlag, Berlin).
33. Matsumoto, T., Chua, L. O. & Tokunaga, R. [1987] "Chaos via torus breakdown," IEEE Trans. Circuits Syst. CAS-34, 240-253.
34. Peckman, B. B., Frouzakis, C. E. & Kevrekidis, I. [1995] "Bananas and bananas splits: A parametric degeneracy in the Hopf bifurcation for maps," SIAM J. Math. Anal. 26, 190-217.
35. Wiggins, S. [1988] Global Bifurcations and Chaos, Appl. Math. Sci. Series, Vol. 73 (Springer-Verlag, Berlin).