Areas of Attraction of Equilibrium Points of Nonlinear Systems: Stability, Branching and Blow-up of Solutions
The dynamical model consisting of the differential equation with a non- linear 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...
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| Format: | Article |
| Language: | English |
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Irkutsk State University
2018-03-01
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| Series: | Известия Иркутского государственного университета: Серия "Математика" |
| Subjects: | |
| Online Access: | http://mathizv.isu.ru/journal/downloadArticle?article=_a8c81f7734724a34930bac373803cb8b&lang=rus |
| _version_ | 1811299124458291200 |
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| author | N.A. Sidorov D.N. Sidorov Li Yong |
| author_facet | N.A. Sidorov D.N. Sidorov Li Yong |
| author_sort | N.A. Sidorov |
| collection | DOAJ |
| description | The dynamical model consisting of the differential equation with a non- linear 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. |
| first_indexed | 2024-04-13T06:30:33Z |
| format | Article |
| id | doaj.art-f23483f5e4314e69b7bb4d80950654c5 |
| institution | Directory Open Access Journal |
| issn | 1997-7670 2541-8785 |
| language | English |
| last_indexed | 2024-04-13T06:30:33Z |
| publishDate | 2018-03-01 |
| publisher | Irkutsk State University |
| record_format | Article |
| series | Известия Иркутского государственного университета: Серия "Математика" |
| spelling | doaj.art-f23483f5e4314e69b7bb4d80950654c52022-12-22T02:58:11ZengIrkutsk State UniversityИзвестия Иркутского государственного университета: Серия "Математика"1997-76702541-87852018-03-012314663https://doi.org/10.26516/1997-7670.2018.23.46Areas of Attraction of Equilibrium Points of Nonlinear Systems: Stability, Branching and Blow-up of SolutionsN.A. SidorovD.N. SidorovLi YongThe dynamical model consisting of the differential equation with a non- linear 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.http://mathizv.isu.ru/journal/downloadArticle?article=_a8c81f7734724a34930bac373803cb8b&lang=rusdynamical modelsrest pointstabilityblow-upbranchingCauchy problembifurcation |
| spellingShingle | N.A. Sidorov D.N. Sidorov Li Yong Areas of Attraction of Equilibrium Points of Nonlinear Systems: Stability, Branching and Blow-up of Solutions Известия Иркутского государственного университета: Серия "Математика" dynamical models rest point stability blow-up branching Cauchy problem bifurcation |
| title | Areas of Attraction of Equilibrium Points of Nonlinear Systems: Stability, Branching and Blow-up of Solutions |
| title_full | Areas of Attraction of Equilibrium Points of Nonlinear Systems: Stability, Branching and Blow-up of Solutions |
| title_fullStr | Areas of Attraction of Equilibrium Points of Nonlinear Systems: Stability, Branching and Blow-up of Solutions |
| title_full_unstemmed | Areas of Attraction of Equilibrium Points of Nonlinear Systems: Stability, Branching and Blow-up of Solutions |
| title_short | Areas of Attraction of Equilibrium Points of Nonlinear Systems: Stability, Branching and Blow-up of Solutions |
| title_sort | areas of attraction of equilibrium points of nonlinear systems stability branching and blow up of solutions |
| topic | dynamical models rest point stability blow-up branching Cauchy problem bifurcation |
| url | http://mathizv.isu.ru/journal/downloadArticle?article=_a8c81f7734724a34930bac373803cb8b&lang=rus |
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