scholarly journals The sufficient conditions for polystability of solutions of nonlinear systems of ordinary differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 304-317
Author(s):  
P. A. Shamanaev ◽  
O. S. Yazovtseva
Keyword(s):  
Banach Space ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Approximation ◽  
Critical Case ◽  
Sufficient Conditions ◽  
Asymptotic Equivalence ◽  
Theorem Proof

The article states the sufficient polystability conditions for part of variables for nonlinear systems of ordinary differential equations with a sufficiently smooth right-hand side. The obtained theorem proof is based on the establishment of a local componentwise Brauer asymptotic equivalence. An operator in the Banach space that connects the solutions of the nonlinear system and its linear approximation is constructed. This operator satisfies the conditions of the Schauder principle, therefore, it has at least one fixed point. Further, using the estimates of the non-zero elements of the fundamental matrix, conditions that ensure the transition of the properties of polystability are obtained, if the trivial solution of the linear approximation system to solutions of a nonlinear system that is locally componentwise asymptotically equivalent to its linear approximation. There are given examples, that illustrate the application of proven sufficient conditions to the study of polystability of zero solutions of nonlinear systems of ordinary differential equations, including in the critical case, and also in the presence of positive eigenvalues.

scholarly journals The sufficient conditions for polystability of solutions of~nonlinear systems of ordinary differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 304-317
Author(s):  
Pavel A. Shamanaev ◽  
Olga S. Yazovtseva
Keyword(s):  
Banach Space ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Approximation ◽  
Critical Case ◽  
Sufficient Conditions ◽  
Asymptotic Equivalence ◽  
Theorem Proof

The article states the sufficient polystability conditions for part of variables for nonlinear systems of ordinary differential equations with a sufficiently smooth right-hand side. The obtained theorem proof is based on the establishment of a local componentwise Brauer asymptotic equivalence. An operator in the Banach space that connects the solutions of the nonlinear system and its linear approximation is constructed. This operator satisfies the conditions of the Schauder principle, therefore, it has at least one fixed point. Further, using the estimates of the non-zero elements of the fundamental matrix, conditions that ensure the transition of the properties of polystability are obtained, if the trivial solution of the linear approximation system to solutions of a nonlinear system that is locally componentwise asymptotically equivalent to its linear approximation. There are given examples, that illustrate the application of proven sufficient conditions to the study of polystability of zero solutions of nonlinear systems of ordinary differential equations, including in the critical case, and also in the presence of positive eigenvalues.

scholarly journals On the solvability of the modified Cauchy problem for linear systems of generalized ordinary differential equations with singularities

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Malkhaz Ashordia ◽  
Inga Gabisonia ◽  
Mzia Talakhadze
Keyword(s):  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
Unique Solvability ◽  
Sufficient Conditions ◽  
The Cauchy Problem ◽  
Generalized Ordinary Differential Equations

AbstractEffective sufficient conditions are given for the unique solvability of the Cauchy problem for linear systems of generalized ordinary differential equations with singularities.

scholarly journals The Ermakov equation: A commentary

2008 ◽  
Vol 2 (2) ◽  
pp. 146-157 ◽  
Author(s):  
P.G.L. Leach ◽  
S.K. Andriopoulos
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
English Translation ◽  
Singularity Theory ◽  
Discrete Math ◽  
Previous Article ◽  
Short History ◽  
History Of ◽  
Modern Context

We present a short history of the Ermakov equation with an emphasis on its discovery by thewest and the subsequent boost to research into invariants for nonlinear systems although recognizing some of the significant developments in the east. We present the modern context of the Ermakov equation in the algebraic and singularity theory of ordinary differential equations and applications to more divers fields. The reader is referred to the previous article (Appl. Anal. Discrete math., 2 (2008), 123-145) for an english translation of Ermakov's original paper.

scholarly journals On periodic-type solutions of systems of linear ordinary differential equations

2004 ◽  
Vol 2004 (5) ◽  
pp. 395-406 ◽  
Author(s):  
I. Kiguradze
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Sufficient Conditions ◽  
Type Solution ◽  
Linear Ordinary Differential Equations

We establish nonimprovable, in a certain sense, sufficient conditions for the existence of a unique periodic-type solution for systems of linear ordinary differential equations.

Heteroclinic Orbit, Forced Lorenz System, and Chaos

2009 ◽  
Vol 5 (1) ◽  
Author(s):  
Dibakar Ghosh ◽  
Anirban Ray ◽  
A. Roy Chowdhury
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lorenz System ◽  
Heteroclinic Orbit ◽  
Convergent Series ◽  
Heteroclinic Orbits ◽  
Onset Of Chaos ◽  
Uniformly Convergent ◽  
Analytic Expression

Forced Lorenz system, important in modeling of monsoonlike phenomena, is analyzed for the existence of heteroclinic orbit. This is done in the light of the suggested new mechanism for the onset of chaos by Magnitskii and Sidorov (2006, “Finding Homoclinic and Heteroclinic Contours of Singular Points of Nonlinear Systems of Ordinary Differential Equations,” Diff. Eq., 39, pp. 1593–1602), where heteroclinic orbits plays important and dominant roles. The analysis is performed based on the theory laid down by Shilnikov. An analytic expression in the form of uniformly convergent series is obtained. The same orbit is also obtained numerically by a technique enunciated by Magnitskii and Sidorov, reproducing the necessary important features.

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