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 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 Random Attractors for Stochastic Three-Component Reversible Gray-Scott System on Infinite Lattices

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Anhui Gu ◽  
Zhaojuan Wang ◽  
Shengfan Zhou
Keyword(s):  
Dynamical System ◽  
White Noise ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Random Attractors ◽  
Multiplicative White Noise ◽  
Infinite Lattices

We prove the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with a multiplicative white noise on infinite lattices.

QUALITATIVE ANALYSIS OF A STOCHASTIC PREDATOR–PREY SYSTEM WITH DISEASE IN THE PREDATOR

2013 ◽  
Vol 06 (01) ◽  
pp. 1250068 ◽  
Author(s):  
SHUANG LI ◽  
XINAN ZHANG
Keyword(s):  
White Noise ◽  
Predator Population ◽  
Deterministic System ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Predator Prey ◽  
Prey System ◽  
Long Time ◽  
Global Positive Solution ◽  
Disease Free

A stochastic predator–prey system with disease in the predator population is proposed, the existence of global positive solution is derived. When the white noise is small, there is a stationary distribution. In addition, conditions of global stability for the deterministic system are also established from the above result. By Lyapunov function, the long time behavior of solution around the disease-free equilibrium of deterministic system is derived. These results mean that stochastic system has the similar property with the corresponding deterministic system. When the white noise is small, however, large environmental noise makes the result different. Finally, numerical simulations are carried out to support our findings.

scholarly journals Random Attractors for Stochastic Three-Component Reversible Gray-Scott System with Multiplicative White Noise

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Anhui Gu
Keyword(s):  
Dynamical System ◽  
White Noise ◽  
Random Attractor ◽  
Random Dynamical System ◽  
Random Attractors ◽  
Multiplicative White Noise

The paper is devoted to proving the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with multiplicative white noise.

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 Long-time behavior of 3D stochastic planetary geostrophic viscous model

2018 ◽  
Vol 18 (05) ◽  
pp. 1850038
Author(s):  
Zhao Dong ◽  
Rangrang Zhang
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Random Noise ◽  
Random Attractor ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Exponential Mixing ◽  
Model Driven ◽  
Long Time ◽  
Mixing Property

In this paper, we consider the 3D planetary geostrophic viscous model driven by an additive random noise. It is proved that this model has exponential mixing property and global random attractor at the same time. Further, we deduce that the support of the integration of the invariant measure for the dynamic generated by this model is exactly a minimal global random attractor. Moreover, we show that the global random attractor has finite Hausdorff dimension.

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