Qualitative Theory of Dynamical Systems

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

Canards Existence in Memristor’s Circuits



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four-dimensional memristor canonical Chua’s circuits for which the classical piecewise-linear characteristic curve has been replaced by a smooth cubic nonlinear function according to the least squares method enables to show the existence of “canard solutions” in such Memristor Based Chaotic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 


  1. 1.
    Argémi, J.: Approche qualitative d’un problème de perturbations singulières dans \({\mathbb{R}}^4\). In: Conti, R., Sestini, G., Villari, G. (eds.) Equadiff, pp. 330–340 (1978)Google Scholar
  2. 2.
    E. Benoît, J.L. Callot, F., Diener and M. Diener, Chasse au canard, Collectanea Mathematica, 32(1), 37–74 (1981)Google Scholar
  3. 3.
    E. Benoît, J.L. Callot, F., Diener and M. Diener, Chasse au canard, Collectanea Mathematica, 32(2), 77–119 (1981)Google Scholar
  4. 4.
    Benoît, E.: Tunnels et entonnoirs. CR. Acad. Sc. Paris 292(Série I), 283–286 (1981)Google Scholar
  5. 5.
    Benoît, E.: Équations différentielles : relation entrée-sortie. CR. Acad. Sc. Paris 293(Série I), 293–296 (1981)MATHGoogle Scholar
  6. 6.
    Benoît, E., Lobry, C.: Les canards de \(\mathbb{R}^3\). CR. Acad. Sc. Paris 294(Série I), 483–488 (1982)MathSciNetMATHGoogle Scholar
  7. 7.
    Benoît, E.: Systèmes lents-rapides dans \({\mathbb{R}}^3\) et leurs canards. Société Mathématique de France Astérisque (109–110), 159–191 (1983)Google Scholar
  8. 8.
    Benoît, E.: Canards de \({\mathbb{R}}^3\), Thèse d’état (PhD), Université de Nice (1984)Google Scholar
  9. 9.
    Benoît, E.: Canards et enlacements. Publications de l’Institut des Hautes Etudes Scientifiques 72, 63–91 (1990)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Benoît, E.: Perturbation singulière en dimension trois : Canards en un point pseudo singulier noeud. Bulletin de la Société Mathématique de France 129–1, 91–113 (2001)MATHGoogle Scholar
  11. 11.
    Callot, J.L., Diener, F., Diener, M.: Le problème de la “chasse au canard”. CR. Acad. Sc. Paris 286(Série A), 1059–1061 (1978)MathSciNetMATHGoogle Scholar
  12. 12.
    Chua, L.O.: Memristor—the missing circuit element. IEEE Trans. Circuit Theory 18(5), 507–519 (1971)CrossRefGoogle Scholar
  13. 13.
    Di Ventra, M., Pershin, Y.V., Chua, L.O.: Circuit elements with memory: memristors, memcapacitors and meminductors. Proc. IEEE 97, 1717–1724 (2009)CrossRefGoogle Scholar
  14. 14.
    Diener, M.: The canard unchained or how fast/slow dynamical systems bifurcate. Math. Intellingencer 6(3), 38–49 (1984)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Ind. Univ. Math. J. 21, 193–225 (1971)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fenichel, N.: Asymptotic stability with rate conditions. Ind. Univ. Math. J. 23, 1109–1137 (1974)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Fenichel, N.: Asymptotic stability with rate conditions II. Ind. Univ. Math. J. 26, 81–93 (1977)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)Google Scholar
  19. 19.
    Fitch, A., Yu, D., Iu, H., Sreeram, V.: Hyperchaos in a memristor-based modified canonical Chua’s circuit. Int. J. Bifurc. Chaos 22(6), 1250133 (2012)CrossRefMATHGoogle Scholar
  20. 20.
    Fitch, A., Iu, H.: Development of memristor based circuits. In: World Scientific Series on Nonlinear Science, Series A 82. World Scientific, Singapore (2013)Google Scholar
  21. 21.
    Fruchard, A., Schäfke, R.: Sur le retard à la bifurcation. In: Sari, T. (ed.) Colloque de Saint Louis (Sénégal), vol. 9, pp. 431–468. ARIMA (2007)Google Scholar
  22. 22.
    Ginoux, J.M., Llibre, J.: Flow curvature method applied to canard explosion. J. Phys. A Math. Theor. 44(46), 465203 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Ginoux, J.M., Llibre, J., Chua, L.O.: Canards from Chua’s circuit. Int. J. Bifur. Chaos 23(4), 1330010 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ginoux, J.M., Rossetto, B.: The singing arc: the oldest memristor? In: Adamatsky, A., Chen, G. (eds.) Chaos, CNN, Memristors and Beyond: A Festschrift for Leon Chua, pp. 494–507. World Scientific Publishing, Singapore (2013)Google Scholar
  25. 25.
    Guckenheimer, J., Haiduc, R.: Canards at folded nodes. Mosc. Math. J. 5(1), 91–103 (2005)MathSciNetMATHGoogle Scholar
  26. 26.
    Hurwitz, A.: Über die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt. Math. Ann. 41, 403–442 (1893)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Itoh, M., Chua, L.O.: Canards and chaos in nonlinear systems. In: Proceedings of Circuits and Systems 1992 (ISCAS’92), vol. 6, pp. 2789–2792 (1992)Google Scholar
  28. 28.
    Itoh, M., Chua, L.O.: Memristors oscillators. Int. J. Bifurc. Chaos 18(11), 3183–3206 (2008)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Itoh, M., Chua, L.O.: Duality of memristors. Int. J Bifurc. Chaos 23(1), 1330001 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Jones, C.K.R.T.: Geometric singular perturbation theory in dynamical systems. In: Arnold, L. (ed.) Montecatini Terme, Lecture Notes in Mathematics, vol. 1609, pp. 44–118. Springer, Berlin (1994)Google Scholar
  31. 31.
    Kaper, T.: An introduction to geometric methods and dynamical systems theory for singular perturbation problems. In: Analyzing multiscale phenomena using singular perturbation methods, Baltimore, MD, pp. 85–131. Amer. Math. Soc, Providence (1998)Google Scholar
  32. 32.
    Muthuswamy, B., Kokate, P.P.: Memristorbased chaotic circuits. IETE Tech. Rev. 26, 417–429 (2009)CrossRefGoogle Scholar
  33. 33.
    Muthuswamy, B.: Implementing memristor based chaotic circuits. Int. J. Bifurc. Chaos 20, 1335–1350 (2010)CrossRefMATHGoogle Scholar
  34. 34.
    Muthuswamy, B., Chua, L.O.: Simplest chaotic circuit. Int. J. Bifurc. Chaos 20, 1567–1580 (2010)CrossRefGoogle Scholar
  35. 35.
    Nelson, E.: Internal set theory: a new approach to nonstandard analysis. Bull. Am. Math. Soc. 83(6), 1165–1198 (1977)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    O’Malley, R.E.: Introduction to Singular Perturbations. Academic Press, New York (1974)MATHGoogle Scholar
  37. 37.
    Pershin, Y.V., Di Ventra, M.: Experimental demonstration of associative memory with memristive neural networks. Available: arXiv:0905.2935 (2009)
  38. 38.
    Pontryagin, L.S.: The asymptotic behaviour of systems of differential equations with a small parameter multiplying the highest derivatives. Izv. Akad. Nauk. SSSR, Ser. Mat. 21(5), 605–626 (1957)MathSciNetGoogle Scholar
  39. 39.
    Robinson, A.: Nonstandard Analysis. North-Holland, Amsterdam (1966)Google Scholar
  40. 40.
    Routh, E.J.: A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion. Macmillan and co, New York (1877)Google Scholar
  41. 41.
    Strukhov, D.B., Snider, G.S., Stewart, G.R., Williams, R.S.: The missing memristor found. Nature 453, 80–83 (2008)CrossRefGoogle Scholar
  42. 42.
    Szmolyan, P., Wechselberger, M.: Canards in \({\mathbb{R}}^3\). J. Differ. Equ. 177, 419–453 (2001)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Takens, F.: Constrained equations, a study of implicit differential equations and their discontinuous solutions. In: Structural Stability, The Theory of Catastrophes and Applications in the Sciences, Springer Lecture Notes in Math., vol. 525, pp. 143–234 (1976)Google Scholar
  44. 44.
    Tikhonov, A.N.: On the dependence of solutions of differential equations on a small parameter. Mat. Sbornik N.S. 31, 575–586 (1948)MathSciNetGoogle Scholar
  45. 45.
    Tsuneda, A.: A gallery of attractors from smooth Chua’s equation. Int. J. Bifurc. Chaos 15(1), 1–49 (2005)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Van der Pol, B.: On relaxation-oscillations. Lond. Edinb. Dublin Philos. Mag. J. Sci. 7(2), 978–992 (1926)Google Scholar
  47. 47.
    Wechselberger, M.: Existence and bifurcation of canards in \({\mathbb{R}}^3\) in the case of a Folded node. SIAM J. Appl. Dyn. Syst. 4, 101–139 (2005)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Wechselberger, M.: À propos de canards. Trans. Am. Math. Soc. 364, 3289–3309 (2012)MathSciNetCrossRefMATHGoogle Scholar

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© 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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