Dynamics of discrete random Burgers-Huxley systems: attractor convergence and finite-dimensional approximations
Fang Su, Xue Wang
TL;DR
This paper develops a rigorous framework for the long-time dynamics of discrete and random Burgers-Huxley lattice systems under implicit Euler time discretization. It establishes the existence and upper semicontinuity of numerical attractors $\mathcal{A}^{\varepsilon}$ that converge to the continuous global attractor $\mathcal{A}$ as $\varepsilon\to0$, and constructs finite-dimensional truncations with corresponding truncated attractors $\mathcal{A}^{\varepsilon}_{m}$ that converge to $\mathcal{A}^{\varepsilon}$. For the stochastic version, it proves the existence of a random attractor $\mathcal{A}(\omega)$ via an Ornstein-Uhlenbeck transform and demonstrates upper semicontinuity to the deterministic attractor as the noise vanishes, including convergence of truncated random attractors to their deterministic analogs. Collectively, the results connect discrete-time dynamics, finite-dimensional approximations, and random perturbations in Burgers-Huxley lattices, providing a comprehensive attractor-theoretic understanding of these systems.
Abstract
In this paper, we apply the implicit Euler scheme to discretize the (random) Burgers-Huxley equation and prove the existence of a numerical attractor for the discrete Burgers-Huxley system. We establish upper semi-convergence of the numerical attractor to the global attractor as the step size tends to zero. We also provide finite-dimensional approximations for the three attractors (global, numerical and random) and prove the existence of truncated attractors as the state space dimension goes to infinity. Finally, we prove the existence of a random attractor and establish that the truncated random attractor upper semi-converges to the truncated global attractor as the noise intensity tends to zero.