Nonlinear Dynamics

, Volume 63, Issue 3, pp 455–476 | Cite as

Structure of saddle-node and cusp bifurcations of periodic orbits near a non-transversal T-point

  • Antonio Algaba
  • Fernando Fernández-Sánchez
  • Manuel Merino
  • Alejandro J. Rodríguez-Luis
Original Paper


Non-transversal T-points have been recently found in problems from many different fields: electronic circuits, pendula, and laser problems. In this work, we study a model based on the construction of a Poincaré map that describes the behaviour of curves of saddle-node and cusp bifurcations in the vicinity of such a non-transversal T-point. This model is also able to predict, reproduce, and explain the numerical results previously obtained in Chua’s equation.


Periodic orbits Cusp bifurcation Saddle-node T-point Global bifurcations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Algaba, A., Merino, M., Rodríguez-Luis, A.J.: Takens-Bogdanov bifurcations of periodic orbits and Arnold’s tongues in a three-dimensional electronic model. Int. J. Bifurc. Chaos Appl. Sci. Eng. 11, 513–531 (2001) MATHCrossRefGoogle Scholar
  2. 2.
    Algaba, A., Fernández-Sánchez, F., Freire, E., Merino, M., Rodríguez-Luis, A.J.: Nontransversal curves of T-points: a source of closed curves of global bifurcations. Phys. Lett. A 303, 204–211 (2002) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Algaba, A., Merino, M., Freire, E., Gamero, E., Rodríguez-Luis, A.J.: Some results on Chua’s equation near a triple-zero linear degeneracy. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13, 583–608 (2003) MATHCrossRefGoogle Scholar
  4. 4.
    Algaba, A., Merino, M., Fernández-Sánchez, F., Rodríguez-Luis, A.J.: Closed curves of global bifurcations in Chua’s equation: a mechanism for their formation. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13, 609–616 (2003) MATHCrossRefGoogle Scholar
  5. 5.
    Algaba, A., Merino, M., Fernández-Sánchez, F., Rodríguez-Luis, A.J.: Open-to-closed curves of saddle-node bifurcations of periodic orbits near a nontransversal T-point in Chua’s equation. Int. J. Bifurc. Chaos Appl. Sci. Eng. 16, 2637–2647 (2006) MATHCrossRefGoogle Scholar
  6. 6.
    Arnold, V.I.: Singularity Theory. London Mathematical Society. Lecture Note Series, vol. 53. Cambridge University Press, Cambridge (1981) Google Scholar
  7. 7.
    Arrowsmith, D.K., Place, C.M.: An introduction to Dynamical Systems. Cambridge University Press, Cambridge (1994) MATHGoogle Scholar
  8. 8.
    Bykov, V.V.: The bifurcations of separatrix contours and chaos. Physica D 62, 290–299 (1993) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Champneys, A.R., Kirk, V., Knobloch, E., Oldeman, B.E., Rademacher, J.D.M.: Unfolding a tangent equilibrium-to-periodic heteroclinic cycle. SIAM J. Appl. Dyn. Syst. 8, 1261–1304 (2009) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Champneys, A.R., Rodríguez-Luis, A.J.: The non-transverse Shil’nikov-Hopf bifurcation; uncoupling of homoclinic orbits and homoclinic tangencies. Physica D 128, 130–158 (1999) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Doedel, E.J., Keller, H.B., Kernévez, J.P.: Numerical analysis and control of bifurcation problems (I) Bifurcation in finite dimensions. Int. J. Bifurc. Chaos Appl. Sci. Eng. 1, 493–520 (1991) MATHCrossRefGoogle Scholar
  12. 12.
    Fernández-Sánchez, F.: Comportamiento dinámico y de bifurcaciones en algunas conexiones globales de equilibrios en sistemas tridimensionales (PhD Thesis) (2002) Google Scholar
  13. 13.
    Fernández-Sánchez, F., Freire, E., Rodríguez–Luis, A.J.: Isolas, cusps and global bifurcations in an electronic oscillator. Dyn. Stab. Syst. 12, 319–336 (1997) MATHGoogle Scholar
  14. 14.
    Fernández-Sánchez, F., Freire, E., Rodríguez–Luis, A.J.: T-points in a ℤ2-symmetric electronic oscillator. (I) Analysis. Nonlinear Dyn. 28, 53–69 (2002) MATHCrossRefGoogle Scholar
  15. 15.
    Gaspard, P., Kapral, R., Nicolis, G.: Bifurcation phenomena near homoclinic systems: a two-parameter analysis. J. Stat. Phys. 35, 697–727 (1984) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Glendinning, P., Sparrow, C.: T-points: A codimension two heteroclinic bifurcation. J. Stat. Phys. 43, 479–488 (1986) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Golubitsky, M., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. I. Springer, Berlin (1985) Google Scholar
  18. 18.
    Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983) MATHGoogle Scholar
  19. 19.
    Hirschberg, P., Laing, C.: Sucessive homoclinic tangencies to a limit cycle. Physica D 89, 1–14 (1995) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Homburg, A., Natiello, M.A.: Accumulations of T-points in a model for solitary pulses in an excitable reaction-diffusion medium. Physica D 201, 212–229 (2005) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Knobloch, J., Lamb, J.S.W., Webster, K.N.: Shift dynamics near T-point heteroclinic cycles. Preprint (2007) Google Scholar
  22. 22.
    Krauskopf, B., Oldeman, B.: Bifurcation of global reinjection orbits near a saddle-node Hopf bifurcation. Nonlinearity 19, 2149–2167 (2006) MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Krauskopf, B., Sieber, J.: Bifurcation analysis of an inverted pendulum with delayed feedback control near a triple-zero eigenvalue singularity. Nonlinearity 17, 85–103 (2004) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Kuznetsov, Yu.: Elements of Applied Bifurcation Theory. Springer, New York (2004) MATHGoogle Scholar
  25. 25.
    Lamb, J.S.W., Teixeira, M.A., Webster, K.: Heteroclinic cycle bifurcations near Hopf-zero bifurcation in reversible vector fields in R 3. J. Differ. Equ. 219, 78–115 (2005) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Lin, X.B.: Using Melnikov’s method to solve Shilnikov’s problems. Proc. R. Soc. Edinb. A 116, 295–325 (1990) MATHGoogle Scholar
  27. 27.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995) MATHCrossRefGoogle Scholar
  28. 28.
    Pivka, L., Wu, C.W., Huang, A.: Lorenz equation and Chua’s equation. Int. J. Bifurc. Chaos Appl. Sci. Eng. 6, 2443–2489 (1996) CrossRefMathSciNetGoogle Scholar
  29. 29.
    Thom, R.: Structural Stability and Morphogenesis. Benjamin/Cummings, Reading (1980) Google Scholar
  30. 30.
    Wieczorek, S., Krauskopf, B.: Bifurcations of n-homoclinic orbits in optically injected lasers. Nonlinearity 18, 1095–1120 (2005) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Wiggins, S.: Introduction to Applied Dynamical Systems and Chaos. Springer, New York (2003) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Antonio Algaba
    • 1
  • Fernando Fernández-Sánchez
    • 2
  • Manuel Merino
    • 1
  • Alejandro J. Rodríguez-Luis
    • 2
  1. 1.Dept. Mathematics, Fac. Ciencias ExperimentalesUniv. Huelva Avda. Tres de Marzo s/nHuelvaSpain
  2. 2.Dept. Applied Mathematics II, E.S. IngenierosUniv. Sevilla Camino de los Descubrimientos s/nSevillaSpain

Personalised recommendations