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Non-observable chaos in piecewise smooth systems

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Abstract

In the present paper, we discuss bifurcations of chaotic attractors in piecewise smooth one-dimensional maps with a high number of switching manifolds. As an example, we consider models of DC/AC power electronic converters (inverters). We demonstrate that chaotic attractors in the considered class of models may contain parts of a very low density, which are unlikely to be observed, neither in physical experiments nor in numerical simulations. We explain how the usual bifurcations of chaotic attractors (merging, expansion and final bifurcations) in piecewise smooth maps with a high number of switching manifolds occur in a specific way, involving low-density parts of attractors, and how this leads to an unusual shape of the bifurcation diagrams.

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Notes

  1. Throughout this paper, the notation of a critical value refers to a value of the function at a kink point. The lower index in this notation determines whether the kink point is located on the left or on the right of the fixed point \({{\mathcal {O}}}_{{{\mathcal {M}}}^m}\). As for the upper index, it is not related to iterations (as it is typically the case in the literature, see, e.g., [26]) and has the only purpose to distinguish between different critical values.

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Acknowledgements

The work of V. Avrutin was supported by the German Research Foundation within the scope of the project “Generic bifurcation structures in piecewise smooth maps with extremely high number of borders in theory and applications for power converter systems”. The work of A. El Aroudi was supported by the Spanish Agencia Estatal de Investigacion (AEI) and the Fondo Europeo de Desarrollo Regional (FEDER) under Grant DPI2017-84572-C2-1-R.

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Authors and Affiliations

  1. Institute for Systems Theory and Automatic Control, University of Stuttgart, Pfaffenwaldring 9, 70550, Stuttgart, Germany

    Viktor Avrutin & Dan Suissa

  2. Department of Computer Science, Southwest State University, 50 Years of October Str. 94, Kursk, Russia, 305040

    Zhanybai T. Zhusubaliyev

  3. Departament d’Enginyeria Electronica, University Rovira i Virgili, Av. Paisos Catalans, no. 27, 43007, Tarragona, Spain

    Abdelali El Aroudi

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  1. Viktor Avrutin
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  2. Zhanybai T. Zhusubaliyev
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  3. Dan Suissa
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  4. Abdelali El Aroudi
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Correspondence to Viktor Avrutin.

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Avrutin, V., Zhusubaliyev, Z.T., Suissa, D. et al. Non-observable chaos in piecewise smooth systems. Nonlinear Dyn 99, 2031–2048 (2020). https://doi.org/10.1007/s11071-019-05406-7

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