Reducibility for a class of nonlinear quasi-periodic differential equations with degenerate equilibrium point under small perturbation

2010 ◽  
Vol 31 (2) ◽  
pp. 599-611 ◽  
Author(s):  
JUNXIANG XU ◽  
SHUNJUN JIANG
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Differential Equations ◽  
Equilibrium Point ◽  
Small Perturbation ◽  
Periodic Differential Equation ◽  
Degenerate Equilibrium

AbstractIn this paper, using the Kolmogorov–Arnold–Moser method we prove reducibility of a class of nonlinear quasi-periodic differential equation with degenerate equilibrium point under small perturbation and obtain a quasi-periodic solution near the equilibrium point.

scholarly journals Reducibility for a Class of Almost-Periodic Differential Equations with Degenerate Equilibrium Point under Small Almost-Periodic Perturbations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Wenhua Qiu ◽  
Jianguo Si
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Equilibrium Point ◽  
Almost Periodic Solution ◽  
Time Dependent ◽  
Almost Periodic ◽  
Degenerate Equilibrium ◽  
Periodic Transformation ◽  
Perturbed Equation ◽  
Kam Method

This paper focuses on almost-periodic time-dependent perturbations of an almost-periodic differential equation near the degenerate equilibrium point. Using the KAM method, the perturbed equation can be reduced to a suitable normal form with zero as equilibrium point by an affine almost-periodic transformation. Hence, for the equation we can obtain a small almost-periodic solution.

On the existence of periodic solutions for autonomous retarded functional differential equations on R2

1986 ◽  
Vol 102 (3-4) ◽  
pp. 259-262 ◽  
Author(s):  
J. G. Dos Reis ◽  
R. L. S. Baroni
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Functional Differential Equations ◽  
Continuous Functions ◽  
Retarded Functional Differential Equations ◽  
Existence Of Periodic Solutions ◽  
Functional Differential

SynopsisLet Ca be the set of all the continuous functions from the interval [−r, 0] on the sphere of radius a, on the plane. We prove, under certains conditions, that a retarded autonomous differential equation that leaves Ca invariant has a non-constant periodic solution.

scholarly journals Positive Periodic Solution for Second-Order Singular Semipositone Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xiumei Xing
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Lower Bound ◽  
Differential Equations ◽  
Positive Periodic Solution ◽  
Second Order ◽  
Nonlinear Alternative ◽  
Alternative Principle

We study the existence of a positive periodic solution for second-order singular semipositone differential equation by a nonlinear alternative principle of Leray-Schauder. Truncation plays an important role in the analysis of the uniform positive lower bound for all the solutions of the equation. Recent results in the literature (Chu et al., 2010) are generalized.

scholarly journals The frequencies of almost periodic solutions of almost periodic differential equations

1974 ◽  
Vol 17 (3) ◽  
pp. 332-344
Author(s):  
G. C. O'Brien
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
First Order ◽  
Periodic Differential Equation ◽  
Right Hand

AbstractAlmost periodic solutions of a first order almost periodic differential equation in Rp are shown to have less than p basic frequencies additional to the basic frequencies of the almost periodic right hand of the equation.

PERIODIC SOLUTION FOR GENERALIZED HIGH-ORDER DELAY DIFFERENTIAL EQUATIONS

2010 ◽  
Vol 03 (01) ◽  
pp. 31-43
Author(s):  
Zhibo Cheng ◽  
Jingli Ren ◽  
Stefan Siegmund
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Sufficient Conditions ◽  
Existence Of Periodic Solutions ◽  
Delay Differential ◽  
New Inequalities

In this paper we consider a generalized n-th order delay differential equation, by applying Mawhin's continuation theory and some new inequalities, we obtain sufficient conditions for the existence of periodic solutions. Moreover, an example is given to illustrate the results.

scholarly journals A Criterion for the Existence of the Unique Periodic Solution of One-Dimensional Periodic Differential Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ni Hua
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Fixed Point ◽  
Lyapunov Method ◽  
Fixed Point Theory ◽  
Point Theory ◽  
One Dimensional ◽  
Periodic Differential Equation

In this paper, we discuss one-dimensional differential equation with ω -period. By using the fixed point theory, the existence of a periodic solution is obtained; by using the second Lyapunov method, the uniqueness and stability of the periodic solution are obtained.

Stability Analysis and Application for Equilibrium Point of Differential Equation

2014 ◽  
Vol 631-632 ◽  
pp. 261-264
Author(s):  
Dan Li
Keyword(s):  
Social Sciences ◽  
Differential Equation ◽  
Differential Equations ◽  
Equilibrium Point ◽  
Species Coexistence ◽  
Order Differential Equation ◽  
Stable Equilibrium ◽  
Stable Equilibrium Point ◽  
First Order ◽  
The Social

A large number of problems in natural sciences (physics, chemistry, biology, astronomy, etc.) and the social sciences (engineering, economic, military, etc.) can be described by using differential equations. In this paper, we study two kinds of differential equations, i.e., the first order differential equation and the second order differential equation, and give the definitions of equilibrium point and stable equilibrium point. Moreover, we discuss the judgment methods of stable equilibrium point, and give the application of stable equilibrium point in the species coexistence problem.

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