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Analysis of Hopf and Takens–Bogdanov Bifurcations in a Modified van der Pol–Duffing Oscillator

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Abstract

We analyze a modified van der Pol–Duffing electronic circuit, modeled by a tridimensional autonomous system of differential equations with Z2-symmetry. Linear codimension-one and two bifurcations of equilibria give rise to several dynamical behaviours, including periodic, homoclinic and heteroclinic orbits. The local analysis provides, in first approximation, the different bifurcation sets. These local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Hopf bifurcation shows the presence of cusps of saddle-node bifurcations of periodic orbits. The existence of a codimension-four Hopf bifurcation is also pointed out. In the case of the Takens–Bogdanov bifurcation, several degenerate situations of codimension-three are analyzed in both homoclinic and heteroclinic cases. The existence of a Hopf–Shil'nikov singularity is also shown.

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References

  1. Algaba A., Gamero, E., Freire, E., and Rodríguez-Luis, A. J., ‘On a codimension-three unfolding of the interaction of degenerate Hopf and pitchfork bifurcations’, Preprint, 1998.

  2. Algaba A., Gamero, E., Freire, E., and Rodríguez-Luis, A. J., ‘A three-parameter study of a degenerate case of the Hopf-pitchfork bifurcation’, Preprint, 1998.

  3. Champneys, A. R. and Kuznetsov, Y. A., ‘Numerical detection and continuation of codimension-two homoclinic bifurcations’, International Journal of Bifurcation and Chaos 4(4), 1994, 785-822.

    Google Scholar 

  4. Doedel, E., Wang, X., and Fairgrieve, T., Auto94: Software for Continuation and Bifurcation Problems in ODE. User's Manual, Applied Mathematics Report, California Institute of Technology, Pasadena, 1995.

    Google Scholar 

  5. Freire, E., Franquelo, L. G., and Aracil, J., ‘Periodicity and chaos in an autonomous electronic system’, IEEE Transactions on Circuits and Systems 31, 1984, 237-247.

    Google Scholar 

  6. Freire, E., Gamero, E., and Ponce, E., ‘An algorithm for symbolic computation of Hopf bifurcation’, in Computers and Mathematics, E. Kaltofen and S. M. Watt (eds.), Springer-Verlag, New York, 1989, pp. 109-118.

    Google Scholar 

  7. Freire, E., Gamero, E., Ponce, E., and Franquelo, L. G., ‘An algorithm for symbolic computation of center manifolds’, in Symbolic And Algebraic Computation, P. Gianni (ed.), Lecture Notes in Computer Science, Vol. 358, Springer-Verlag, Berlin, 1989, pp. 218-230.

    Google Scholar 

  8. Freire, E., Rodríguez-Luis, A. J., Gamero, E., and Ponce, E., ‘A case study for homoclinic chaos in an autonomous electronic circuit. A trip from Takens-Bogdanov to Hopf-Shil'nikov', Physica D 62, 1993, 230-253.

    Google Scholar 

  9. Gamero, E., Freire, E., and Ponce, E., ‘Normal forms for planar systems with nilpotent linear part’, in Bifurcation and Chaos: Analysis, Algorithms, Applications, R. Seydel, F. W. Schneider, T. Küpper, and H. Troger (eds.), International Series of Numerical Mathematics, Vol. 97, Birkhäuser, Basel, 1991, pp. 123-127.

    Google Scholar 

  10. Gamero, E., Freire, E., Rodríguez-Luis, A. J., Ponce, E., and Algaba A., ‘Hypernormal form calculation for triple zero degeneracies’, Bulletin of the Belgian Mathematical Society - Simon Stevin, 1998, to appear.

  11. Gomes, M. G. M. and King, G. P., ‘Bistable chaos II: Bifurcation analysis’, Physical Review A 46, 1992, 3100-3110.

    Google Scholar 

  12. Guckenheimer, J. and Holmes, P. J., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, Berlin, 1983.

    Google Scholar 

  13. Healey, J. J., Broomhead, D. S., Cliffe, K. A., Jones, R., and Mullin, T., ‘The origins of chaos in a modified van der Pol oscillator’, Physica D 48, 1991, 322-339.

    Google Scholar 

  14. Hirschberg, P. and Knobloch, E., ‘Shilnikov-Hopf bifurcation’, Physica D 62, 1993, 202-216.

    Google Scholar 

  15. Huang, A., Pivka, L., and Wu, C. W., ‘Chua's equation with cubic nonlinearity’, International Journal of Bifurcation and Chaos 6(12A), 1996, 2175-2222.

    Google Scholar 

  16. Huertas, J. L. and Rodríguez-Vázquez, A., eds., Proceedings of 4th International Workshop on Nonlinear Dynamics of Electronic Systems NDES’96, Sevilla, Spain, June 27-28, 1996.

  17. Madan, R. A., Chua's Circuit: A Paradigm for Chaos, World Scientific, Singapore, 1993.

    Google Scholar 

  18. Nayfeh, A. H., Method of Normal Forms, Wiley, New York, 1993.

    Google Scholar 

  19. Rodríguez-Luis, A. J., Fernández-Sánchez, F., Freire, E., and Gamero, E., ‘Homoclinic-Hopf interactions in an electronic oscillator’, in Proceedings of 4th Workshop on Nonlinear Dynamics of Electronic Systems NDES'96, Sevilla, Spain, June 27-28, 1996, pp. 429-434.

  20. Rodríguez-Luis, A. J., Freire, E., and Ponce, E., ‘A method for homoclinic and heteroclinic continuation in two and three dimensions’, in Continuation and Bifurcations: Numerical Techniques and Applications, D. Roose, A. Spence, and B. De Dier (eds.), NATO ASI Series C, Vol. 313, Kluwer, Dordrecht, 1990, pp. 197-210.

    Google Scholar 

  21. Rodríguez-Luis, A. J., Freire, E., and Ponce, E., ‘On a codimension 3 bifurcation arising in an autonomous electronic circuit’, in Bifurcation and Chaos: Analysis, Algorithms, Applications, R. Seydel, F.W. Schneider, T. Küpper, and H. Troger (eds.), International Series of Numerical Mathematics, Vol. 97, Birkhäuser, Basel, 1991, pp. 301-306.

    Google Scholar 

  22. Rousseau, C., ‘Universal unfolding of a singularity of a symmetric vector field with C∞-equivalent to y(∂/∂x) + (±x3 ± x6y)(∂/∂y)’, in Bifurcations of Planar Vector Fields, J. P. Françoise and R. Roussarie (eds.), Lecture Notes in Mathematics, Vol. 1455, Springer-Verlag, Berlin, 1990, pp. 334-355.

    Google Scholar 

  23. Rousseau, C. and Li, C., ‘A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: The cusp of order 4’, Journal of Differential Equations 79, 1989, 132-167.

    Google Scholar 

  24. Shinriki, R., Yamamoto, M., and Mori, S., ‘Multimode oscillations in a modified van der Pol oscillator containing a positive nonlinear conductance’, IEEE Proceedings 69, 1981, 394-395.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Escuela Politécnica Superior, University of Huelva, 21819, Huelva, Spain

    A. Algaba

  2. Department of Applied Mathematics II, Escuela Superior Ingenieros, University of Sevilla, Camino de los Descubrimientos s/n, 41092, Sevilla, Spain

    E. Freire, E. Gamero & A. J. Rodríguez-Luis

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  1. A. Algaba
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  2. E. Freire
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  3. E. Gamero
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  4. A. J. Rodríguez-Luis
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Algaba, A., Freire, E., Gamero, E. et al. Analysis of Hopf and Takens–Bogdanov Bifurcations in a Modified van der Pol–Duffing Oscillator. Nonlinear Dynamics 16, 369–404 (1998). https://doi.org/10.1023/A:1008294110873

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