scholarly journals Reduction of Dynamics with Lie Group Analysis

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
M. Iwasa
Keyword(s):  
Dynamical System ◽  
Lie Group ◽  
Perturbation Methods ◽  
Group Analysis ◽  
Lie Symmetries ◽  
Time Behavior ◽  
Lie Group Analysis ◽  
Long Time Behavior ◽  
Singular Perturbation Methods ◽  
Long Time

This paper is mainly a review concerning singular perturbation methods by means of Lie group analysis which has been presented by the author. We make use of a particular type of approximate Lie symmetries in those methods in order to construct reduced systems which describe the long-time behavior of the original dynamical system. Those methods can be used in analyzing not only ordinary differential equations but also difference equations. Although this method has been mainly used in order to derive asymptotic behavior, when we can find exact Lie symmetries, we succeed in construction of exact solutions.

scholarly journals Uniform attractors of stochastic three-component Gray-Scott system with multiplicative noise

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Junwei Feng ◽  
Hui Liu ◽  
Jie Xin
Keyword(s):  
Dynamical System ◽  
Bounded Domain ◽  
Multiplicative Noise ◽  
Random Dynamical System ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Uniform Estimates ◽  
Estimates Of Solutions ◽  
Long Time ◽  
Uniform Attractors

<p style='text-indent:20px;'>In a bounded domain, we study the long time behavior of solutions of the stochastic three-component Gray-Scott system with multiplicative noise. We first show that the stochastic three-component Gray-Scott system can generate a non-autonomous random dynamical system. Then we establish some uniform estimates of solutions for stochastic three-component Gray-Scott system with multiplicative noise. Finally, the existence of uniform and cocycle attractors is proved.</p>

scholarly journals A Stochastic Weakly Damped, Forced KdV-BO Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Guolian Wang
Keyword(s):  
Dynamical System ◽  
White Noise ◽  
Global Solution ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
Energy Type ◽  
Bo Equation

We investigate the long time behavior of the damped, forced KdV-BO equation driven by white noise. We first show that the global solution generates a random dynamical system. By energy type estimates and dispersive properties, we then prove that this system possesses a weak random attractor in the spaceH1(ℝ).

scholarly journals Parametric Resonance of Hopf Bifurcation

2005 ◽  
Author(s):  
Richard Rand ◽  
Albert Barcilon ◽  
Tina Morrison
Keyword(s):  
Hopf Bifurcation ◽  
Perturbation Methods ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Damping Parameter ◽  
Long Time ◽  
Negative Damping ◽  
Parametric Forcing

We investigate the dynamics of a system consisting of a simple, harmonic oscillator with small nonlinearity, small damping and small parametric forcing in the neighborhood of 2:1 resonance. We assume that the unforced system exhibits the birth of a stable limit cycle as the damping changes sign from positive to negative (a supercritical Hopf bifurcation). Using perturbation methods and numerical integration, we investigate the changes which occur in long-time behavior as the damping parameter is varied. We show that for large positive damping, the origin is stable, whereas for large negative damping a quasiperiodic behavior occurs. These two steady states are connected by a complicated series of bifurcations which occur as the damping is varied.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

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