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Mixed-Mode Oscillations in a Modified Chua’s Circuit

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Abstract

We consider a singularly perturbed system of differential equations of the form ε u′=g(u,v,λ), v′=f(u,v,λ), where (u,v)∈R 3, 0<ε≪1, and λ is a set of parameters. Such a system describes a modified Chua’s circuit with mixed-mode oscillations (MMOs). MMOs consist of a series of small-amplitude oscillations (canard solutions) and large-amplitude relaxations. In the paper we provide a series of both numerical and analytical analyses of the singularly perturbed system for the modified Chua’s circuit with nonlinear f and g. In particular, we analyze the occurrence of the Farey sequence \(\it L^{s}\), where \(\it L\) and \(\it s\) are the numbers of large and small oscillations, respectively.

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References

  1. J. Abshagen, J.M. Lopez, F. Marques, G. Pfister, Bursting dynamics due to a homoclinic cascade in Taylor–Couette flow. J. Fluid Mech. 613, 357–384 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. M. Brøns, M. Krupa, M. Wechselberger, Mixed mode oscillations due to the generalized canard phenomenon. Fields Inst. Commun. 49, 39–63 (2006)

    Google Scholar 

  3. D. Cafagna, G. Grassi, Generation of chaotic beats in a modified Chua’s circuit, Part I: Dynamical behavior & Part II: Circuit design. Nonlinear Dyn. 44, 91–108 (2005)

    Article  Google Scholar 

  4. L.O. Chua, http://www.scholarpedia.org/article/Chua_circuit

  5. Focus issue, Mixed mode oscillations: experiment, computation, and analysis. Chaos 18 (2008)

  6. M.T.M. Koper, Bifurcation of mixed-mode oscillations in a three-variable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram. Physica D 80, 72–94 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Krupa, P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions. SIAM J. Math. Anal. 33, 286–314 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Krupa, P. Szmolyan, Relaxation oscillation and canard explosion. J. Differ. Equ. 174, 312–368 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Krupa, N. Popovic, N. Kopell, Mixed-mode oscillations in three time-scale systems: a prototypical example. SIAM J. Appl. Dyn. Syst. 7, 361–420 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  10. R. Li, Z. Duan, B. Wang, G. Chen, A modified Chua’s circuit with an attraction-repulsion function. Int. J. Bifurc. Chaos (IJBC) 18, 1865–1888 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. W. Marszalek, Fold points and singularities in Hall MHD differential-algebraic equations. IEEE Trans. Plasma Sci. 37, 254–260 (2009)

    Article  Google Scholar 

  12. W. Marszalek, Z.W. Trzaska, New solutions of resistive MHD systems with singularity induced bifurcations. IEEE Trans. Plasma Sci. 35, 509–515 (2007)

    Article  Google Scholar 

  13. W. Marszalek, T. Amdeberhan, R. Riaza, Singularity crossing phenomena in DAEs: a two-phase fluid flow application case study. Comput. Math. Appl. 49, 303–319 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. R. Riaza, S.L. Campbell, W. Marszalek, On singular equilibria of index-1 DAEs. Circuits Syst. Signal Process. 19, 131–157 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. P. Szmolyan, M. Wechselberger, Canards in R 3. J. Differ. Equ. 177, 419–453 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  16. F. Tang, L. Wang, An adaptive active control for the modified Chua’s circuit. Phys. Lett. A 346, 342–346 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Q. Wang, Y. Chen, Generalized Q-S (lag, anticipated and complete) synchronization in modified Chua’s circuit and Hindmarsh-Rose systems. Appl. Math. Comput. 181, 48–56 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Wechselberger, Existence and bifurcations of canards in R 3 in the case of a folded node. SIAM J. Appl. Dyn. Syst. 4, 101–139 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Engineering & Information Sciences, DeVry University, 630 US Highway 1, North Brunswick, NJ, 08902, USA

    Wieslaw Marszalek

  2. Department of Management and Production Engineering, Warsaw University of Ecology and Management, Wawelska 14, 02-061, Warsaw, Poland

    Zdzislaw Trzaska

Authors
  1. Wieslaw Marszalek
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  2. Zdzislaw Trzaska
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Correspondence to Wieslaw Marszalek.

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Marszalek, W., Trzaska, Z. Mixed-Mode Oscillations in a Modified Chua’s Circuit. Circuits Syst Signal Process 29, 1075–1087 (2010). https://doi.org/10.1007/s00034-010-9190-8

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  • DOI: https://doi.org/10.1007/s00034-010-9190-8

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