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Branching Solutions of the Cauchy Problem for Nonlinear Loaded Differential Equations with Bifurcation Parameters

Author

Listed:
  • Nikolai Sidorov

    (Institute of Mathematics & IT, Irkutsk State University, 664033 Irkutsk, Russia
    These authors contributed equally to this work.)

  • Denis Sidorov

    (Institute of Mathematics & IT, Irkutsk State University, 664033 Irkutsk, Russia
    Industrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, Russia
    Energy Systems Institute, Siberian Branch of Russian Academy of Science, 664033 Irkutsk, Russia
    These authors contributed equally to this work.)

Abstract

The Cauchy problem for a nonlinear system of differential equations with a Stieltjes integral (loads) of the desired solution is considered. The equation contains bifurcation parameters where the system has a trivial solution for any values. The necessary and sufficient conditions are derived for those parameter values (bifurcation points) in the neighborhood of which the Cauchy problem has a non-trivial real solution. The constructive method is proposed for the solution of real solutions in the neighborhood of those points. The method uses successive approximations and builds asymptotics of the solution. The theoretical results are illustrated by example. The Cauchy problem with loads and bifurcation parameters has not been studied before.

Suggested Citation

  • Nikolai Sidorov & Denis Sidorov, 2022. "Branching Solutions of the Cauchy Problem for Nonlinear Loaded Differential Equations with Bifurcation Parameters," Mathematics, MDPI, vol. 10(12), pages 1-8, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:12:p:2134-:d:842457
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    References listed on IDEAS

    as
    1. Agarwal, Praveen & Baltaeva, Umida & Alikulov, Yolqin, 2020. "Solvability of the boundary-value problem for a linear loaded integro-differential equation in an infinite three-dimensional domain," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
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