scholarly journals Pullback and forward dynamics of nonautonomous Laplacian lattice systems on weighted spaces

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiaoying Han ◽  
Peter E. Kloeden
Keyword(s):  
Weighted Space ◽  
Lattice System ◽  
Laplacian Operator ◽  
Pullback Attractor ◽  
Forward Dynamics ◽  
Discrete Laplacian ◽  
Lattice Systems ◽  
Nonempty Compact ◽  
Omega Limit Set ◽  
Dense Inclusion

<p style='text-indent:20px;'>A nonautonomous lattice system with discrete Laplacian operator is revisited in the weighted space of infinite sequences <inline-formula><tex-math id="M1">\begin{document}$ {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula>. First the existence of a pullback attractor in <inline-formula><tex-math id="M2">\begin{document}$ {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula> is established by utilizing the dense inclusion of <inline-formula><tex-math id="M3">\begin{document}$ \ell^2 \subset {{\ell_{\rho}^2}} $\end{document}</tex-math></inline-formula>. Moreover, the pullback attractor is shown to consist of a singleton trajectory when the lattice system is uniformly strictly contracting. Then forward dynamics is investigated in terms of the existence of a nonempty compact forward omega limit set. A general class of weights for the sequence space are adopted, instead of particular types of weights often used in the literature. The analysis presented in this work is more direct compare with previous studies.</p>

Random exponential attractor for second-order nonautonomous stochastic lattice systems with multiplicative white noise

2019 ◽  
Vol 19 (06) ◽  
pp. 1950044
Author(s):  
Haijuan Su ◽  
Shengfan Zhou ◽  
Luyao Wu
Keyword(s):  
White Noise ◽  
Weighted Space ◽  
Sufficient Conditions ◽  
Second Order ◽  
Lattice System ◽  
Exponential Attractor ◽  
Lattice Systems ◽  
Random System ◽  
Multiplicative White Noise ◽  
Infinite Sequences

We studied the existence of a random exponential attractor in the weighted space of infinite sequences for second-order nonautonomous stochastic lattice system with linear multiplicative white noise. Firstly, we present some sufficient conditions for the existence of a random exponential attractor for a continuous cocycle defined on a weighted space of infinite sequences. Secondly, we transferred the second-order stochastic lattice system with multiplicative white noise into a random lattice system without noise through the Ornstein–Uhlenbeck process, whose solutions generate a continuous cocycle on a weighted space of infinite sequences. Thirdly, we estimated the bound and tail of solutions for the random system. Fourthly, we verified the Lipschitz continuity of the continuous cocycle and decomposed the difference between two solutions into a sum of two parts, and carefully estimated the bound of the norm of each part and the expectations of some random variables. Finally, we obtained the existence of a random exponential attractor for the considered system.

scholarly journals Discrete Laplacian Operator and Its Applications in Signal Processing

IEEE Access ◽  
2020 ◽  
Vol 8 ◽  
pp. 89692-89707
Author(s):  
Waseem Waheed ◽  
Guang Deng ◽  
Bo Liu
Keyword(s):  
Signal Processing ◽  
Laplacian Operator ◽  
Discrete Laplacian

scholarly journals Dataset Denoising Based on Manifold Assumption

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Zhonghua Hao ◽  
Shiwei Ma ◽  
Hui Chen ◽  
Jingjing Liu
Keyword(s):  
Point Cloud ◽  
Geometric Distribution ◽  
Lagrange Equation ◽  
Optimal Solution ◽  
Classification Problem ◽  
Machine Learning Algorithms ◽  
High Dimensional ◽  
Laplacian Operator ◽  
Discrete Laplacian ◽  
Denoising Method

Learning the knowledge hidden in the manifold-geometric distribution of the dataset is essential for many machine learning algorithms. However, geometric distribution is usually corrupted by noise, especially in the high-dimensional dataset. In this paper, we propose a denoising method to capture the “true” geometric structure of a high-dimensional nonrigid point cloud dataset by a variational approach. Firstly, we improve the Tikhonov model by adding a local structure term to make variational diffusion on the tangent space of the manifold. Then, we define the discrete Laplacian operator by graph theory and get an optimal solution by the Euler–Lagrange equation. Experiments show that our method could remove noise effectively on both synthetic scatter point cloud dataset and real image dataset. Furthermore, as a preprocessing step, our method could improve the robustness of manifold learning and increase the accuracy rate in the classification problem.

Pullback and Uniform Exponential Attractors for Nonautonomous Boussinesq Lattice System

2015 ◽  
Vol 25 (08) ◽  
pp. 1550100 ◽  
Author(s):  
Min Zhao ◽  
Shengfan Zhou
Keyword(s):  
Dynamical System ◽  
Boussinesq Equations ◽  
Lattice System ◽  
Time Dependent ◽  
Pullback Attractor ◽  
Exponential Attractor ◽  
Uniform Attractor ◽  
Exponential Attractors ◽  
Lattice Dynamical System ◽  
External Forces

We first prove the existence of a pullback attractor and a pullback exponential attractor for a nonautonomous lattice dynamical system of nonlinear Boussinesq equations affected by time-dependent coupled coefficients and forces. Then, we prove the existence of a uniform attractor and a uniform exponential attractor for the system driven by quasi-periodic external forces.

scholarly journals Existence and Upper Semicontinuity of Attractors for Nonautonomous Stochastic Sine-Gordon Lattice Systems with Random Coupled Coefficients

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Zhaojuan Wang ◽  
Shengfan Zhou
Keyword(s):  
White Noise ◽  
Weighted Space ◽  
Upper Semicontinuity ◽  
Lattice Systems ◽  
Random Attractors ◽  
Multiplicative White Noise

We study nonautonomous stochastic sine-Gordon lattice systems with random coupled coefficients and multiplicative white noise. We first consider the existence of random attractors in a weighted space for this system and then establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

scholarly journals Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Yiju Chen ◽  
Xiaohu Wang
Keyword(s):  
Asymptotic Behavior ◽  
Multiplicative Noise ◽  
Upper Semicontinuity ◽  
Random Attractor ◽  
Discrete Laplacian ◽  
Lattice Systems ◽  
Random Attractors ◽  
Infinite Range Interactions ◽  
Infinite Range

<p style='text-indent:20px;'>In this paper, we study the asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise. The considered systems are driven by the fractional discrete Laplacian, which features the infinite-range interactions. We first prove the existence of pullback random attractor in <inline-formula><tex-math id="M1">\begin{document}$ \ell^2 $\end{document}</tex-math></inline-formula> for stochastic lattice systems. The upper semicontinuity of random attractors is also established when the intensity of noise approaches zero.</p>

Upon Generating Discrete Expanding Integrable Models of the Toda Lattice Systems and Infinite Conservation Laws

2017 ◽  
Vol 72 (1) ◽  
pp. 77-86 ◽  
Author(s):  
Yufeng Zhang ◽  
Xiangzhi Zhang ◽  
Yan Wang ◽  
Jiangen Liu
Keyword(s):  
Conservation Laws ◽  
Statistical Physics ◽  
Quantum Physics ◽  
Toda Lattice ◽  
Lattice System ◽  
Integrable Models ◽  
Matrix Approach ◽  
Lattice Systems ◽  
Lax Pairs ◽  
Dimensional Lattice

AbstractWith the help ofR-matrix approach, we present the Toda lattice systems that have extensive applications in statistical physics and quantum physics. By constructing a new discrete integrable formula byR-matrix, the discrete expanding integrable models of the Toda lattice systems and their Lax pairs are generated, respectively. By following the constructing formula again, we obtain the corresponding (2+1)-dimensional Toda lattice systems and their Lax pairs, as well as their (2+1)-dimensional discrete expanding integrable models. Finally, some conservation laws of a (1+1)-dimensional generalised Toda lattice system and a new (2+1)-dimensional lattice system are generated, respectively.

Uniform Exponential Attractor for Second Order Lattice System with Quasi-Periodic External Forces in Weighted Space

2014 ◽  
Vol 24 (01) ◽  
pp. 1450006 ◽  
Author(s):  
Shengfan Zhou ◽  
Min Zhao
Keyword(s):  
Weighted Space ◽  
Second Order ◽  
Lattice System ◽  
Exponential Attractor ◽  
Uniform Attractor ◽  
External Forces ◽  
Infinite Sequences

In this paper, we study the existence of a uniform exponential attractor for second order lattice system with quasi-periodic external forces in weighted space of infinite sequences. We first prove that the system possesses a uniform attractor. Then we obtain the existence of a uniform exponential attractor for the system.

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