lattice dynamical systems
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Author(s):  
Panpan Zhang ◽  
Anhui Gu

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.


2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Radosław Czaja

<p style='text-indent:20px;'>In this paper we study long-time behavior of first-order non-autono-mous lattice dynamical systems in square summable space of double-sided sequences using the cooperation between the discretized diffusion operator and the discretized reaction term. We obtain existence of a pullback global attractor and construct pullback exponential attractor applying the introduced notion of quasi-stability of the corresponding evolution process.</p>


2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Peng Chen ◽  
Die Hu ◽  
Yuanyuan Zhang

Abstract Sun and Ma (J. Differ. Equ. 255:2534–2563, 2013) proved the existence of a nonzero T-periodic solution for a class of one-dimensional lattice dynamical systems, $$\begin{aligned} \ddot{q_{i}}=\varPhi _{i-1}'(q_{i-1}-q_{i})- \varPhi _{i}'(q_{i}-q_{i+1}),\quad i\in \mathbb{Z}, \end{aligned}$$ q i ¨ = Φ i − 1 ′ ( q i − 1 − q i ) − Φ i ′ ( q i − q i + 1 ) , i ∈ Z , where $q_{i}$ q i denotes the co-ordinate of the ith particle and $\varPhi _{i}$ Φ i denotes the potential of the interaction between the ith and the $(i+1)$ ( i + 1 ) th particle. We extend their results to the case of the least energy of nonzero T-periodic solution under general conditions. Of particular interest is a new and quite general approach. To the best of our knowledge, there is no result for the ground states for one-dimensional lattice dynamical systems.


Author(s):  
Jason J. Bramburger

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.


2019 ◽  
Vol 20 (2) ◽  
pp. 485-515
Author(s):  
Jan W. Cholewa ◽  
Radosław Czaja

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).


2019 ◽  
Vol 30 (2) ◽  
pp. 603-644 ◽  
Author(s):  
Jason J. Bramburger ◽  
Björn Sandstede

2019 ◽  
Vol 32 (3) ◽  
pp. 1457-1474 ◽  
Author(s):  
Xiaoying Han ◽  
Peter E. Kloeden ◽  
Stefanie Sonner

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