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A study of the boundaries of stability regions in two-parameter dynamical systems

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Abstract

We consider dynamical systems defined by autonomous and periodic differential equations that depend on two scalar parameters. We study the problems of constructing boundaries of stability regions for equilibrium points in the plane of parameters. We identify conditions under which a point on the boundary of a stability region has one or more smooth boundary curves coming through it. We show schemes to find the basic scenarios of bifurcations when parameters transition over the boundaries of stability regions. We distinguish types of boundaries (dangerous or safe). The main formulas have been obtained in the terms of original equations and do not require to pass to normal forms and using theorems on a central manifold.

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Authors and Affiliations

  1. Bashkir State University, Ufa, Russia

    M. G. Yumagulov & I. Zh. Mustafina

  2. Bashkir State Agrarian University, Ufa, Russia

    L. S. Ibragimova

Authors
  1. M. G. Yumagulov
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  2. I. Zh. Mustafina
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  3. L. S. Ibragimova
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Correspondence to M. G. Yumagulov.

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Original Russian Text © M.G. Yumagulov, I.Zh. Mustafina, L.S. Ibragimova, 2017, published in Avtomatika i Telemekhanika, 2017, No. 10, pp. 74–89.

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Yumagulov, M.G., Mustafina, I.Z. & Ibragimova, L.S. A study of the boundaries of stability regions in two-parameter dynamical systems. Autom Remote Control 78, 1790–1802 (2017). https://doi.org/10.1134/S0005117917100046

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  • DOI: https://doi.org/10.1134/S0005117917100046

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