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

Nikolai Sidorov and Denis Sidorov
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Nikolai Sidorov: Institute of Mathematics & IT, Irkutsk State University, 664033 Irkutsk, Russia
Denis Sidorov: Institute of Mathematics & IT, Irkutsk State University, 664033 Irkutsk, Russia

Mathematics, 2022, vol. 10, issue 12, 1-8

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.

Keywords: Cauchy problem; loaded differential equation; bifurcation; homotopy; Newton diagram (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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