On Nonlinear Systems of Ordinary Differential Equations

1988 ◽  
pp. 99-111
Author(s):  
Lothar Berg
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations

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.

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.

On the asymptotic behaviour of nonlinear systems of ordinary differential equations

1985 ◽  
Vol 26 (2) ◽  
pp. 161-170
Author(s):  
Zhivko S. Athanassov
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Ordinary Differential Equations ◽  
Asymptotically Stable ◽  
Future Time ◽  
Zero Function ◽  
Perturbed Systems ◽  
Approach Zero ◽  
Image Position

In this paper we study the asymptotic behaviour of the following systems of ordinary differential equations:where the identically zero function is a solution of (N) i.e. f(t, 0)=0 for all time t. Suppose one knows that all the solutions of (N) which start near zero remain near zero for all future time or even that they approach zero as time increases. For the perturbed systems (P) and (P1) the above property concerning the solutions near zero may or may not remain true. A more precise formulation of this problem is as follows: if zero is stable or asymptotically stable for (N), and if the functions g(t, x) and h(t, x) are small in some sense, give conditions on f(t, x) so that zero is (eventually) stable or asymptotically stable for (P) and (P1).

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 DEGENERATE MATRIX METHOD FOR SOLVING NONLINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

1998 ◽  
Vol 3 (1) ◽  
pp. 45-56
Author(s):  
T. Cîrulis ◽  
O. Lietuvietis
Keyword(s):  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Matrix Method ◽  
Iteration Procedure ◽  
Kutta Method ◽  
Runge Kutta Method ◽  
Systems Of Differential Equations ◽  
Runge Kutta ◽  
Degenerate Matrix

Degenerate matrix method for numerical solving nonlinear systems of ordinary differential equations is considered. The method is based on an application of special degenerate matrix and usual iteration procedure. The method, which is connected with an implicit Runge‐Kutta method, can be simply realized on computers. An estimation for the error of the method is given.

Sign in / Sign up

Export Citation Format

Share Document