Forced Periodic Solutions of a Stable Non-Linear System of Differential Equations

1998 ◽  
pp. 152-154
Author(s):  
A. B. Farnell ◽  
C. E. Langenhop ◽  
N. Levinson
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Periodic Solutions ◽  
System Of Differential Equations ◽  
Non Linear ◽  
Non Linear System

scholarly journals Bilateral approximations and periodic solutions of systems of differential equations with impulses

1985 ◽  
Vol 31 (2) ◽  
pp. 293-307
Author(s):  
S.G. Hristova ◽  
D.D. Bainov
Keyword(s):  
Periodic Solution ◽  
Linear System ◽  
Differential Equations ◽  
Periodic Solutions ◽  
System Of Differential Equations ◽  
Systems Of Differential Equations ◽  
Non Linear ◽  
Impulsive Perturbations ◽  
Non Linear System

The paper justifies a method of bilateral approximations for finding the periodic solution of a non-linear system of differential equations with impulsive perturbations at fixed moments of time.

scholarly journals ON APPROXIMATION OF ALMOST-PERIODIC SOLUTIONS FOR A NON-LINEAR COUNTABLE SYSTEM OF DIFFERENTIAL EQUATIONS BY QUASI-PERIODIC SOLUTIONS FOR SOME LINEAR SYSTEM

2021 ◽  
Vol 9 (2) ◽  
pp. 111-123
Author(s):  
Yu. Teplinsky
Keyword(s):  
Dynamical Systems ◽  
Linear System ◽  
Differential Equations ◽  
Periodic Solutions ◽  
System Of Equations ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
System Of Differential Equations ◽  
Dimensional Torus ◽  
Non Linear

It is well-known that many applied problems in different areas of mathematics, physics, and technology require research into questions of existence of oscillating solutions for differential systems, which are their mathematical models. This is especially true for the problems of celestial mechanics. Novadays, by oscillatory motions in dynamical systems, according to V. V. Nemitsky, we call their recurrent motions. As it is known from Birkhoff theorem, trajectories of such motions contain minimal compact sets of dynamical systems. The class of recurrent motions contains, in particular, both quasi-periodic and almost-periodic motions. There are renowned fundamental theorems by Amerio and Favard related to existence of almost-periodic solutions for linear and non-linear systems. It is also of interest to research the behavior of a dynamical system’s motions in a neighborhood of a recurrent trajectory. It became understood later, that the question of existence of such trajectories is closely related to existence of invariant tori in such systems, and the method of Green-Samoilenko function is useful for constructing such tori. Here we consider a non-linear system of differential equations defined on Cartesian product of the infinite-dimensional torus T∞ and the space of bounded number sequences m. The problem is to find sufficient conditions for the given system of equations to possess a family of almost-periodic in the sense of Bohr solutions, dependent on the parameter ψ ∈ T∞, every one of which can be approximated by a quasi-periodic solution of some linear system of equations defined on a finite-dimensional torus.

scholarly journals A parameter uniform almost first order convergent numerical method for non-linear system of singularly perturbed differential equations

BIOMATH ◽  
2016 ◽  
Vol 5 (2) ◽  
pp. 1608111
Author(s):  
Ishwariya Raj ◽  
Princy Mercy Johnson ◽  
John J.H Miller ◽  
Valarmathi Sigamani
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Numerical Method ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Initial Value ◽  
First Order ◽  
Non Linear ◽  
Perturbation Parameters ◽  
Non Linear System

In this paper an initial value problem for a non-linear system of two singularly perturbed first order differential equations is considered on the interval (0,1].The components of the solution of this system exhibit initial layers at 0. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be almost first order convergent in the maximum norm uniformly in the perturbation parameters.

scholarly journals Dynamics and Stability of Hopf Bifurcation for One Non-linear System

TEM Journal ◽  
2021 ◽  
pp. 820-824
Author(s):  
Vahidin Hadžiabdić ◽  
Midhat Mehuljić ◽  
Jasmin Bektešević ◽  
Adnan Mašić
Keyword(s):  
Hopf Bifurcation ◽  
Linear System ◽  
Differential Equations ◽  
Detailed Analysis ◽  
Limit Cycle ◽  
Local Stability ◽  
System Of Differential Equations ◽  
The One ◽  
Non Linear System ◽  
The Given

In this paper we will look at the one system of ODE and analyze it. We aim to determine the points of equilibrium; examine their character and establish the existence of a bifurcation for the corresponding parameter value. A detailed analysis of local stability was performed for all values of the given parameter. For a certain value of the parameter, the existence of supercritical Hopf bifurcation of the observed system of differential equations has been proved. Also, the existence of a limit cycle that is always stable has been proved.

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