scholarly journals Stationary states of weakly coupled lattice dynamical systems arising in strong competition models

2012 ◽  
Vol 25 (9) ◽  
pp. 1197-1202 ◽  
Author(s):  
Zunxian Li ◽  
Peixuan Weng
Keyword(s):  
Dynamical Systems ◽  
Stationary States ◽  
Competition Models ◽  
Weakly Coupled ◽  
Lattice Dynamical Systems ◽  
Strong Competition

Attractors for multi-valued lattice dynamical systems with nonlinear diffusion terms

2021 ◽  
Author(s):  
Panpan Zhang ◽  
Anhui Gu
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Diffusion ◽  
Upper Semicontinuity ◽  
Growth Conditions ◽  
Pullback Attractor ◽  
Random Lattice ◽  
Lipschitz Continuous ◽  
Lattice Dynamical Systems ◽  
Long Term Behavior

This paper is devoted to the long-term behavior of nonautonomous random lattice dynamical systems with nonlinear diffusion terms. The nonlinear drift and diffusion terms are not expected to be Lipschitz continuous but satisfy the continuity and growth conditions. We first prove the existence of solutions, and establish the existence of a multi-valued nonautonomous cocycle. We then show the existence and uniqueness of pullback attractors parameterized by sample parameters. Finally, we establish the measurability of this pullback attractor by the method based on the weak upper semicontinuity of the solutions.

Random attractors for second-order stochastic lattice dynamical systems

2010 ◽  
Vol 72 (1) ◽  
pp. 483-494 ◽  
Author(s):  
Xiaohu Wang ◽  
Shuyong Li ◽  
Daoyi Xu
Keyword(s):  
Dynamical Systems ◽  
Second Order ◽  
Random Attractors ◽  
Lattice Dynamical Systems

scholarly journals Lattice dynamical systems: dissipative mechanism and fractal dimension of global and exponential attractors

2019 ◽  
Vol 20 (2) ◽  
pp. 485-515
Author(s):  
Jan W. Cholewa ◽  
Radosław Czaja
Keyword(s):  
Fractal Dimension ◽  
Dynamical Systems ◽  
Global Attractor ◽  
American Mathematical Society ◽  
Reaction Diffusion Equations ◽  
Dissipative Systems ◽  
Exponential Attractor ◽  
Lattice Systems ◽  
Exponential Attractors ◽  
Lattice Dynamical Systems

Abstract In this work, we examine first-order lattice dynamical systems, which are discretized versions of reaction–diffusion equations on the real line. We prove the existence of a global attractor in $$\ell ^2$$ℓ2, and using the method by Chueshov and Lasiecka (Dynamics of quasi-stable dissipative systems, Springer, Berlin, 2015; Memoirs of the American Mathematical Society, vol 195(912), AMS, 2008), we estimate its fractal dimension. We also show that the global attractor is contained in a finite-dimensional exponential attractor. The approach relies on the interplay between the discretized diffusion and reaction, which has not been exploited as yet for the lattice systems. Of separate interest is a characterization of positive definiteness of the discretized Schrödinger operator, which refers to the well-known Arendt and Batty’s result (Differ Int Equ 6:1009–1024, 1993).

On simplicity of intermediate -algebras

2019 ◽  
Vol 40 (12) ◽  
pp. 3181-3187
Author(s):  
TATTWAMASI AMRUTAM ◽  
MEHRDAD KALANTAR
Keyword(s):  
Dynamical Systems ◽  
Simple Group ◽  
Crossed Product ◽  
Simple Groups ◽  
Stationary States ◽  
Compact Spaces ◽  
Link Type ◽  
One To One

We prove simplicity of all intermediate $C^{\ast }$-algebras $C_{r}^{\ast }(\unicode[STIX]{x1D6E4})\subseteq {\mathcal{B}}\subseteq \unicode[STIX]{x1D6E4}\ltimes _{r}C(X)$ in the case of minimal actions of $C^{\ast }$-simple groups $\unicode[STIX]{x1D6E4}$ on compact spaces $X$. For this, we use the notion of stationary states, recently introduced by Hartman and Kalantar [Stationary $C^{\ast }$-dynamical systems. Preprint, 2017, arXiv:1712.10133]. We show that the Powers’ averaging property holds for the reduced crossed product $\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$ for any action $\unicode[STIX]{x1D6E4}\curvearrowright {\mathcal{A}}$ of a $C^{\ast }$-simple group $\unicode[STIX]{x1D6E4}$ on a unital $C^{\ast }$-algebra ${\mathcal{A}}$, and use it to prove a one-to-one correspondence between stationary states on ${\mathcal{A}}$ and those on $\unicode[STIX]{x1D6E4}\ltimes _{r}{\mathcal{A}}$.

Lattice Dynamical Systems Associated with a Fractional Laplacian

2019 ◽  
Vol 40 (11) ◽  
pp. 1315-1343 ◽  
Author(s):  
Valentin Keyantuo ◽  
Carlos Lizama ◽  
Mahamadi Warma
Keyword(s):  
Dynamical Systems ◽  
Fractional Laplacian ◽  
Lattice Dynamical Systems

scholarly journals Isolas of multi-pulse solutions to lattice dynamical systems

2020 ◽  
pp. 1-36
Author(s):  
Jason J. Bramburger
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Continuum Limit ◽  
Single Pulse ◽  
Compact Sets ◽  
Bifurcation Structure ◽  
Numerical Investigations ◽  
Lattice Dynamical Systems ◽  
Bistable State ◽  
Steady State Solutions

This work investigates the existence and bifurcation structure of multi-pulse steady-state solutions to bistable lattice dynamical systems. Such solutions are characterized by multiple compact disconnected regions where the solution resembles one of the bistable states and resembles another trivial bistable state outside of these compact sets. It is shown that the bifurcation curves of these multi-pulse solutions lie along closed and bounded curves (isolas), even when single-pulse solutions lie along unbounded curves. These results are applied to a discrete Nagumo differential equation and we show that the hypotheses of this work can be confirmed analytically near the anti-continuum limit. Results are demonstrated with a number of numerical investigations.

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