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 The Bulletin of Irkutsk State University. Series Mathematics, 2018, Volume 23, Pages 46–63 (Mi iigum330)

Areas of attraction of equilibrium points of nonlinear systems: stability, branching and blow-up of solutions

N. A. Sidorova, D. N. Sidorovbc, Yong Lid

a Institute of Mathematics, Economics and Informatics, Irkutsk State University, 1, K. Marx st., Irkutsk, 664003, Russian Federation
b Melentiev Energy Systems Institute SB RAS, 130, Lermontov st., Irkutsk, 664033, Russian Federation
c Institute of Solar-Terrestrial Physics SB RAS, 126a, Lermontov st., Irkutsk, 664033, Russian Federation
d College of Electrical and Information Engineering, Hunan University, Changsha 410082, People’s Republic of China

Abstract: The dynamical model consisting of the differential equation with a nonlinear operator acting in Banach spaces and a nonlinear operator equation with respect to two elements from different Banach spaces is considered. It is assumed that the system has stationary solutions (rest points). The Cauchy problem with the initial condition with respect to one of the unknown functions is formulated. The second function playing the role of controlling the corresponding nonlinear dynamic process, the initial conditions are not set. Sufficient conditions are obtained for which the problem has the global classical solution stabilizing at infinity to the rest point. Under suitable sufficient conditions it is shown that a solution can be constructed by the method of successive approximations. If the conditions of the main theorem are not satisfied, then several solutions can exists. Some of them can blow-up in a finite time, while others stabilize to a rest point. Examples are given to illustrate the constructed theory.

Keywords: dynamical models, rest point, stability, blow-up, branching, Cauchy problem, bifurcation.

 Funding Agency Grant Number National Natural Science Foundation of China 61673398 Siberian Branch of Russian Academy of Sciences ÀÀÀÀ-À17-117030310442-8, íàó÷íûé ïðîåêò III.17.3.1 Ïðîãðàììà ìåæäóíàðîäíîãî íàó÷íî-òåõíè÷åñêîãî ñîòðóäíè÷åñòâà Êèòàÿ è Ðîññèè 2015DFR70850

DOI: https://doi.org/10.26516/1997-7670.2018.23.46

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Bibliographic databases:

UDC: 517.925
MSC: 34K18, 34D23

Citation: N. A. Sidorov, D. N. Sidorov, Yong Li, “Areas of attraction of equilibrium points of nonlinear systems: stability, branching and blow-up of solutions”, The Bulletin of Irkutsk State University. Series Mathematics, 23 (2018), 46–63

Citation in format AMSBIB
\Bibitem{SidSidLi18} \by N.~A.~Sidorov, D.~N.~Sidorov, Yong~Li \paper Areas of attraction of equilibrium points of nonlinear systems: stability, branching and blow-up of solutions \jour The Bulletin of Irkutsk State University. Series Mathematics \yr 2018 \vol 23 \pages 46--63 \mathnet{http://mi.mathnet.ru/iigum330} \crossref{https://doi.org/10.26516/1997-7670.2018.23.46}