Qualitative Theory of Dynamical Systems

, Volume 15, Issue 2, pp 383–431 | Cite as

Canards Existence in Memristor’s Circuits

  • Jean-Marc Ginoux
  • Jaume Llibre


The aim of this work is to propose an alternative method for determining the condition of existence of “canard solutions” for three and four-dimensional singularly perturbed systems with only one fast variable in the folded saddle case. This method enables to state a unique generic condition for the existence of “canard solutions” for such three and four-dimensional singularly perturbed systems which is based on the stability of folded singularities of the normalized slow dynamics deduced from a well-known property of linear algebra. This unique generic condition is perfectly identical to that provided in previous works. Application of this method to the famous three and four-dimensional memristor canonical Chua’s circuits for which the classical piecewise-linear characteristic curve has been replaced by a smooth cubic nonlinear function according to the least squares method enables to show the existence of “canard solutions” in such Memristor Based Chaotic Circuits.


Geometric singular perturbation theory Singularly perturbed dynamical systems Canard solutions 



We would like to thank to Ernesto Pérez Chavela for previous discussions related with this work. The authors are partially supported by a MINECO/FEDER grant number MTM2008-03437. The second author is partially supported by a MICINN/FEDER grants numbers MTM2009-03437 and MTM2013-40998-P, by an AGAUR grant number 2014SGR-568, by an ICREA Academia, two FP7+PEOPLE+2012+IRSES numbers 316338 and 318999, and FEDER-UNAB10-4E-378.


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Laboratoire LSIS, CNRS, UMR 7296Université de ToulonLa Garde cedexFrance
  2. 2.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterraSpain

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