Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Nonlinear Dynamic and Chaotic Saddle in Rectifier Circuit

Discontinuity, Nonlinearity, and Complexity 1(4) (2012) 387--398 | DOI:10.5890/DNC.2012.11.002

Luiz F.R. Turci$^{1}$; Elbert E.N. Macau$^{2}$; Takashi Yoneyama$^{3}$

$^{1}$ Institute of Science and Technology - ICT, Federal University of Alfenas - UNIFAL-MG, Poços de Caldas - MG, Brazil

$^{2}$ Computation and AppliedMathematics Associated Laboratory - LAC, National Institute of Space Research - Inpe, São José dos Campos - SP, Brazil

$^{3}$ Technological Institute of Aeronautics - ITA, Aerospace Technical Center - CTA, São José dos Campos - SP, Brazil

Download Full Text PDF



Chaotic systems are recognized by presenting an evolved dynamics in which sophisticated phenomena as crisis, metamorphoses and transitions may take places. In this work we show that these phenomena are present even in very simple system. Here we analyze a simple rectifier electronic circuit and show the mechanisms that mediate its transition from a simple to a rich dynamics. Furthermore, we identified the presence of a chaotic saddle in the system, which also implies the occurrence of transient chaos even for parameters for which the systems eventually sets in a regular behavior.


We would like to thank Fapesp and CNPq for their financial support on this work.


  1. [1]  Hayes, S. and, Grebogi, C., and Ott, E. (1993), Communicating with chaos, Phys. Rev. Lett., 70, 3031.
  2. [2]  Baptista, M. S., Macau, E. E. N., Grebogi, C., Lai, Y.-C., and Rosa, E. (2000), Integrated chaotic communication scheme, Phys. Rev. E, 62, 4835.
  3. [3]  Witkowski, F. X., Kavanagh, K. M., Penkoske, P.A., Plonsey, R., Span,M.L., Ditto,W.L. and Kaplan, D.T. (1995), Evidence for determinism in ventricular fibrillation, Phys. Rev. Lett., 75, 1230.
  4. [4]  Tse, C.K. and di Bernardo, M. (2002), Complex behavior in switching power converters, Proceeding of the IEEE, 90, 768.
  5. [5]  di Bernardo, M., Garofalo, F., Glielmo, L., and Vasca, F. (1998), Switching, bifurcations, and chaos in DC/DC converters, IEEE Trans. on Circuits and Systems, 45, 133.
  6. [6]  Rempel, E.L. and Chian, A.C.-L. (2003), High-dimensional chaotic saddles in the Kuramoto-Sivashinky Equation, Physics Letters A., 319, 104.
  7. [7]  Ott, E. (1993), Chaos in Dynamical Systems, Cambridge Univ. Press, Cambridge.
  8. [8]  Ziemniak, E.M., Jung, C., and Tél, T. (1989), Tracer dynamics in open hydrodynamical flows as chaotic scattering, Physica D, 36, 137.
  9. [9]  Szabó, K.G., Lai, Y.-C., Tel, T. and Grebogi, C. (2000), Topological gap filling at crisis, Physica D, Ver. E 61, 5019.
  10. [10]  Sedra, A. and Smith, K. (2000),Microeletronics, Ed. Makron Books, Ed. 4.
  11. [11]  Pierce, J.F. (1972), Dispositivos de Jun o Semicondutores, Ed. USP.
  12. [12]  Gray, P.E. and Searle, C.L. (1977), Princípios de Eletrônica, Vol. 1, Livros Técnicos e Científicos Ed. S.A.
  13. [13]  Gray, P.E. and Searle, C.L.(2003) Silicon Power Rectifier Datasheet S/R25 Serie, Microsemi.
  14. [14]  Alligood, K.I., Sauer, T.D., and Yorke, J.A. (1996), Chaos: an Introduction to Dynamical Systems, Springer, New Yorke.
  15. [15]  Prucha, B. (1997),Mearusing Feigenbaum’s δ in a bifurcating electric circuit.
  16. [16]  Rauch, A.C. (1998), Chaos in a driven nonlinear electrical oscillator: Determinig Feigenbaum’s delta.
  17. [17]  Arrowsmith, D.K. and Place, C. M. (1994), An Introduction to Dynamical Systems, Cambridge.
  18. [18]  Guckenheimer, J. and Holmes, P. (1983), Nonlinear Oscilations, Dynamical Systems, and Biffurcations of Vector Fields, Springer.
  19. [19]  Smale, S. (1967), Differentiable dynamical system, Bull. Amer. Math. Soc., 73, 747.
  20. [20]  Grebogi, C. and Yorke, J.A. (1983), Crises, sudden changes in chaotic attractors and chaotic transients, Physica, 7, 183.
  21. [21]  Kantz, K. and Grassberger, P. (1985), Repellers, semi-attractors, and long-lived chaotic transients, Physica D, 17, 15.
  22. [22]  Lai, Y.C., Zyczkowski, K., and Grebogi, C. (1999), Universal behavior in the parametric evolution of chaotic saddles, Phys. Rev. E, 59, 5261.
  23. [23]  McDonalds, S.W., Grebogi, C., Ott, E., and Yorke, J.A. (1985), Fractal basin boundaries, Physica D, 17, 125.
  24. [24]  Eckhardt, B. (1987), Fractal properties of scattering singularities, J. Phys. A, 20, 5971.
  25. [25]  Eckhardt, B. (1988), Irregulas scattering, Physica D, 33, 89.
  26. [26]  Hsu, G.H. and Ott, E., and Grebogi, C. (1988), Strange saddles and the dimensions of their invariant manifolds, Phys. Lett. A, 127, 199.
  27. [27]  Sweet, D., Nusse, H.E. and Yorke, J.A. (2001), Stagger-and-Step Method: Detecting and Computing Chaotic Saddles in High Dimensions, Phys. Rev. Lett., 86, 2261.