Smooth conjugacy of centre manifolds

1992 ◽  
Vol 120 (1-2) ◽  
pp. 61-77 ◽  
Author(s):  
Almut Burchard ◽  
Bo Deng ◽  
Kening Lu
Keyword(s):  
Differential Equations ◽  
Equilibrium Point ◽  
Ordinary Differential Equations ◽  
Parabolic Equations ◽  
Geometric Theory ◽  
Unstable Manifolds ◽  
Centre Manifolds ◽  
Smooth Conjugacy

SynopsisIn this paper, we prove that for a system of ordinary differential equations of class Cr+1,1, r≧0 and two arbitrary Cr+1, 1 local centre manifolds of a given equilibrium point, the equations when restricted to the centre manifolds are Cr conjugate. The same result is proved for similinear parabolic equations. The method is based on the geometric theory of invariant foliations for centre-stable and centre-unstable manifolds.

ESTIMATES OF SOLUTIONS OF IMPULSIVE PARABOLIC EQUATIONS AND APPLICATION

2008 ◽  
Vol 01 (02) ◽  
pp. 257-266
Author(s):  
GUOHUA SONG
Keyword(s):  
Boundary Condition ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Parabolic Equations ◽  
Population Biology ◽  
Model Problem ◽  
General Boundary ◽  
General Boundary Condition ◽  
Estimates Of Solutions

This paper is concerned with the estimates of solutions for an impulsive parabolic equations under general boundary condition. We prove that the solutions of impulsive parabolic equations can be controlled and estimated by the solutions of dominating impulsive ordinary differential equations. We also apply the above results to a model problem arising from population biology.

scholarly journals Quenching Time Optimal Control for Some Ordinary Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Ping Lin
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Parabolic Equations ◽  
Time Optimal Control ◽  
Optimal Controls ◽  
Time Optimal ◽  
Differential Control ◽  
Quenching Time ◽  
The Pontryagin Maximum Principle

This paper concerns time optimal control problems of three different ordinary differential equations inℝ2. Corresponding to certain initial data and controls, the solutions of the systems quench at finite time. The goal to control the systems is to minimize the quenching time. The purpose of this study is to obtain the existence and the Pontryagin maximum principle of optimal controls. The methods used in this paper adapt to more general and complex ordinary differential control systems with quenching property. We also wish that our results could be extended to the same issue for parabolic equations.

Bifurcation from a homoclinic orbit in parabolic differential equations

1986 ◽  
Vol 103 (3-4) ◽  
pp. 265-274 ◽  
Author(s):  
C. Miguel Blázquez
Keyword(s):  
Periodic Orbit ◽  
Differential Equations ◽  
Equilibrium Point ◽  
Homoclinic Orbit ◽  
Parabolic Equations ◽  
Parabolic Differential Equations ◽  
Stable Periodic Orbit ◽  
Codimension One ◽  
Stable Periodic

SynopsisThis paper considers autonomous parabolic equations which have a homoclinic orbit to an isolated equilibrium point. We study these systems under autonomous perturbations. Firstly we prove that the perturbation under which the homoclinicorbit persists forms a submanifold of codimension one. Then, if a perturbation of this manifold is considered, we prove that a unique stable periodic orbit arises from the homoclinic orbit under certain conditions for the eigenvalues of thesaddle point.

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