Some results on Chua's equation near a triple-zero linear degeneracy

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 13, Issue 3, 2003, Pages 583-608

  Algaba, A ^a   Merino, M ^a   Freire, E ^b   Gamero, E ^b   Rodriguez Luis A J ^b  

^a UNIVERSITY OF HUELVA   (Spain)

^b UNIVERSITY OF SEVILLE   (Spain)

Author keywords

Chua's equation; Homoclinic connections; Local bifurcations; Triple zero degeneracy

Indexed keywords

BIFURCATION (MATHEMATICS); CIRCUIT THEORY; COMPUTATIONAL COMPLEXITY; CONTROL SYSTEM ANALYSIS; PARAMETER ESTIMATION;

TRIPLE-ZERO BIFURCATIONS;

LINEAR EQUATIONS;

EID: 0037598810     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127403006741     Document Type: Article

References (35)

1. Algaba, A., Freire, E., Gamero, E. & Rodríguez-Luis, A. J. [1999] "On the Takens-Bogdanov bifurcation in the Chua's equations," IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E82-A, 1722-1728.
2. Algaba, A., Merino, M., Freire, E., Gamero, E. & Rodríguez-Luis, A. J. [2000] "On the Hopf-pitchfork bifurcation in the Chua's equation," Int. J. Bifurcation and Chaos 10, 291-305.
3. Algaba, A., Freire, E. & Gamero, E. [2001a] "Characterizing and computing normal forms using Lie transforms: A survey," Dyn. Contin. Discr. Impul. Syst. 8, 449-475.
4. Algaba, A., Merino, M. & Rodríguez-Luis, A. J. [2001b] "Takens-Bogdanov bifurcations of periodic orbits and Arnold's tongues in a three-dimensional electronic model," Int. J. Bifurcation and Chaos 11, 513-531.
5. Algaba, A., Merino, M., Preire, E., Gamero, E. & Rodríguez-Luis, A. J. [2003] "A new step in the comprehension of global bifurcations in the vicinity of a triple-zero linear degeneracy in Chua's equation," in preparation.
6. Arrowsmith, D. & Place, C. [1990] An Introduction to Dynamical Systems (Cambridge University Press, Cambridge).
7. Barnett, S. & Cameron, R. G. [1985] Introduction to Mathematical Control Theory (Clarendon Press, Oxford).
8. Belyakov, L. A. [1980] "The bifurcation set in a system with homoclinic saddle curve," Matematicheskie Zametki 28, 910-916.
9. Belyakov, L. A. [1984] "Bifurcation of systems with homoclinic curve of a saddle-focus with saddle quantity zero," Matematicheskie Zametki 36, 838-843.
10. Champneys, A. R. & Kuznetsov, Y. A. [1994] "Numerical detection and continuation of codimension-two homoclinic bifurcations," Int. J. Bifurcation and Chaos 4, 785-822.
11. Champneys, A. R. & Rodríguez-Luis, A. J. [1999] "The non-transverse Shil'nikov-Hopf bifurcation: Uncoupling of homoclinic orbits and homoclinic tangencies," Physica D128, 130-158.
12. Deng, B. [1993] "Homoclinic twisting bifurcations and cusp horseshoe map," J. Dyn. Diff. Eqs. 5, 417-467.
13. Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Y. A., Sandstede, B. & Wang, X. [1998] "AUTO97: Continuation and bifurcation software for ordinary differential equations (with HomCont)".
14. Dumortier, F., Ibáñez, S. & Kokubu, H. [2001] "New aspects in the unfolding of the nilpotent singularity of codimension three," Dyn. Syst. 16, 63-95.
15. 2-symmetric electronic oscillator. (I) Analysis," Nonlin. Dyn. 28, 53-69.
16. 2-symmetric electronic oscillator. (II) Trip between a triple-zero and a Hopf bifurcation," preprint.
17. Freire, E., Gamero, E., Ponce, E. & Franquelo, L. G. [1989] "An algorithm for symbolic computation of center manifolds," Symbolic and Algebraic Computation, ed. Gianni, P., Lecture Notes in Computer Science, Vol. 358 (Springer-Verlag, Berlin), pp. 218-230.
18. Freire, E., Pizarro, L. & Rodríguez-Luis, A. J. [1999] "Analytical and numerical study of a four-parameter family of mechanical oscillators," Mech. Res. Commun. 26, 253-259.
19. Freire, E., Gamero, E., Rodríguez-Luis, A. J. & Algaba, A. [2002] "A note on the triple-zero linear degeneracy: Normal forms, dynamical and bifurcation behaviors of an unfolding," Int. J. Bifurcation and Chaos 12(12), 2799-2820.
20. Gamero, E., Freire, E., Rodríguez-Luis, A. J., Ponce, E. & Algaba, A. [1999] "Hypernormal form for triple-zero degeneracy," Bull. Belgian Math. Soc. 6, 357-368.
21. Glendinning, P. [1984] "Bifurcations near homoclinic orbits with symmetry," Phys. Lett. A103, 163-166.
22. Glendinning, P. & Sparrow, C. [1984] "Local and global behavior near homoclinic orbits," J. Statist. Phys. 35, 645-696.
23. Gonchenko, S., Turaev, D., Gaspard, P. & Nicolis, G. [1997] "Complexity in the bifurcation structure of homoclinic loops to a saddle-focus," Nonlinearity 10, 409-423.
24. Guckenheimer, J. & Holmes, P. J. [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, Berlin).
25. Khibnik, A., Roose, D. & Chua, L. O. [1993] "On periodic orbits and homoclinic bifurcations in Chua's circuit with a smooth nonlinearity," Int. J. Bifurcation and Chaos 3, 363-384.
26. Kuznetsov, Y. A. [1995] Elements of Applied Bifurcation Theory (Springer, NY).
27. Nayfeh, A. H. & Balachandran, B. [1995] Applied Nonlinear Dynamics (John Wiley, NY).
28. Oldeman, B., Krauskopf, B. & Champneys, A. R. [2001] "Numerical unfoldings of codimension-three resonant homoclinic flip bifurcations," Nonlinearity 14, 597-621.
29. Pivka, L., Wu, C. W. & Huang, A. [1996] "Lorenz equation and Chua's equation," Int. J. Bifurcation and Chaos 6, 2443-2489.
30. Rodríguez-Luis, A. J., Freire, E. & Ponce, E. [1991] "On a codimension 3 bifurcation arising in an autonomous electronic circuit," Bifurcation and Chaos: Analysis, Algorithms, Applications, eds. Seydel, R., Schneider, F. W., Küpper, T. & Troger, H. International Series of Numerical Mathematics, Vol. 97 (Birkäuser, Basel), pp. 301-306.
31. 3," J. Diff. Eqs. 126, 1-47.
32. Rousseau, C. & Li, C. [1989] "A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: The cusp of order 4," J. Diff. Eqs. 79, 132-167.
33. 6y)(∂/∂y)," Bifurcations of Planar Vector Fields, eds. Françoise, J.-P. & Roussarie, R., Lecture Notes in Mathematics, Vol. 1455 (Springer, Berlin), pp. 334-355.
34. Shil'nikov, L. P. [1969] "On a new type of bifurcation of multidimensional dynamical systems," Sov. Math. Dokl. 10, 1368-1371.
35. Wiggins, S. [1996] Introduction to Applied Dynamical Systems and Chaos (Springer, NY).