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Border Collision and Heteroclinic Bifurcations in a 2D Piecewise Smooth Map

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Advances in Discrete Dynamical Systems, Difference Equations and Applications (ICDEA 2021)

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Abstract

We consider a 2D piecewise smooth map originating from an application (acting as a discrete-time model of a DC/DC converter with pulse-width modulated multilevel control). We focus on several non-trivial transformations occurring in the phase space of this map under parameter variation. In particular, we describe the effect of a fold border collision bifurcation leading to the appearance of a pair of cycles, an attracting and a saddle one, a sequence of transformations of the basins of coexisting attractors, as well as heteroclinic bifurcations which result first in the destruction of an attracting closed resonant curve and then in the creation of another one such curve.

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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 – 2”. Zh. T. Zhusubaliyev and U. A. Sopuev acknowledge the support by the grant 14-22 of the Osh State University.

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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

  2. Department of Economics, Society, Politics, University of Urbino, A. Saffi st. 42, 61029, Urbino, Italy

    Laura Gardini

  3. Institute of Mathematics, NAS of Ukraine, Tereshchenkivska st. 3, Kyiv, 01024, Ukraine

    Iryna Sushko

  4. Kyiv School of Economics, Mykoly Shpaka st. 3, Kyiv, 03113, Ukraine

    Iryna Sushko

  5. Department of Mathematics, Osh State University, Lenin st. 331, 723500, Osh, Kyrgyzstan

    Zhanybai T. Zhusubaliyev & Ulanbek A. Sopuev

Authors
  1. Viktor Avrutin
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  2. Laura Gardini
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  3. Iryna Sushko
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  4. Zhanybai T. Zhusubaliyev
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  5. Ulanbek A. Sopuev
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Corresponding author

Correspondence to Laura Gardini .

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Editors and Affiliations

  1. Department of Mathematics, Trinity University, San Antonio, TX, USA

    Saber Elaydi

  2. University of Rhode Island, Kingston, RI, USA

    Mustafa R. S. Kulenović

  3. University of Sarajevo, Sarajevo, Bosnia and Herzegovina

    Senada Kalabušić

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Avrutin, V., Gardini, L., Sushko, I., Zhusubaliyev, Z.T., Sopuev, U.A. (2023). Border Collision and Heteroclinic Bifurcations in a 2D Piecewise Smooth Map. In: Elaydi, S., Kulenović, M.R.S., Kalabušić, S. (eds) Advances in Discrete Dynamical Systems, Difference Equations and Applications. ICDEA 2021. Springer Proceedings in Mathematics & Statistics, vol 416. Springer, Cham. https://doi.org/10.1007/978-3-031-25225-9_3

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