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Prediction of period doubling bifurcations in harmonically forced memristor circuits

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Abstract

The paper studies bifurcations and complex dynamics in a class of nonautonomous oscillatory circuits with a flux-controlled memristor and harmonic forcing term. It is first shown that, as in the autonomous case, the state space of any memristor circuit of the class can be decomposed in invariant manifolds. It turns out that the memristor circuit dynamics is given by the collection of the dynamics of a family of circuits, with a nonlinear resistor in place of the memristor, which is parameterized by an additional constant input whose value depends on the initial conditions of the memristor circuit. This property makes it possible to employ the harmonic balance method in order to study the periodic solutions and their bifurcations due to changing the amplitude and the frequency of the harmonic input on a fixed manifold or due to changing the initial conditions for a fixed harmonic input. The main result is that in both of these cases the harmonic balance method is quite effective to accurately predict period doubling bifurcations of the periodic solutions. Analytical predictions are obtained in the cases of linear-plus-cubic and piecewise linear memristor flux–charge characteristics.

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Notes

  1. For space limitations, \(L_1(\mathcal {D})\) and \(L_2(\mathcal {D})\) are expressed in terms of the related \(L_3(\mathcal {D})\).

  2. Similar results can be obtained for other kinds of bifurcations along the lines developed in [43].

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Authors and Affiliations

  1. Department of Information Engineering and Mathematics, University of Siena, via Roma 56, 53100, Siena, Italy

    Mauro Di Marco & Mauro Forti

  2. Department of Information Engineering, University of Florence, via S. Marta 3, 50139, Florence, Italy

    Giacomo Innocenti & Alberto Tesi

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  1. Giacomo Innocenti
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  2. Mauro Di Marco
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  3. Mauro Forti
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  4. Alberto Tesi
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Correspondence to Giacomo Innocenti.

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Innocenti, G., Di Marco, M., Forti, M. et al. Prediction of period doubling bifurcations in harmonically forced memristor circuits. Nonlinear Dyn 96, 1169–1190 (2019). https://doi.org/10.1007/s11071-019-04847-4

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