Multi-valued random dynamics of stochastic wave equations with infinite delays

<p style='text-indent:20px;'>This paper is devoted to the asymptotic behavior of solutions to a non-autonomous stochastic wave equation with infinite delays and additive white noise. The nonlinear terms of the equation are not expected to be Lipschitz continuous, but only satisfy continuity assumptions along with growth conditions, under which the uniqueness of the solutions may not hold. Using the theory of multi-valued non-autonomous random dynamical systems, we prove the existence and measurability of a compact global pullback attractor.</p>

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

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.

Robustness of a dynamical systems model with a plastic self-organising vector field to noisy input signals

We investigate the robustness with respect to random stimuli of a dynamical system with a plastic self-organising vector field, previously proposed as a conceptual model of a cognitive system and inspired by the self-organised plasticity of the brain. This model of a novel type consists of an ordinary differential equation subjected to the time-dependent “sensory” input, whose time-evolving solution is the vector field of another ordinary differential equation governing the observed behaviour of the system, which in the brain would be neural firings. It is shown that the individual solutions of both these differential equations depend continuously over finite time intervals on the input signals. In addition, under suitable uniformity assumptions, it is shown that the non-autonomous pullback attractor and forward omega limit set of the given two-tier system depend upper semi-continuously on the input signal. The analysis holds for both deterministic and noisy input signals, in the latter case in a pathwise sense.

Upper semicontinuity of pullback attractors for a nonautonomous damped wave equation

In this paper, we study the local uniformly upper semicontinuity of pullback attractors for a strongly damped wave equation. In particular, under some proper assumptions, we prove that the pullback attractor $\{A_{\varepsilon }(t)\}_{t\in \mathbb{R}}$ { A ε ( t ) } t ∈ R of Eq. (1.1) with $\varepsilon \in [0,1]$ ε ∈ [ 0 , 1 ] satisfies $\lim_{\varepsilon \to \varepsilon _{0}}\sup_{t\in [a,b]} \operatorname{dist}_{H_{0}^{1}\times L^{2}}(A_{\varepsilon }(t),A_{ \varepsilon _{0}}(t))=0$ lim ε → ε 0 sup t ∈ [ a , b ] dist H 0 1 × L 2 ( A ε ( t ) , A ε 0 ( t ) ) = 0 for any $[a,b]\subset \mathbb{R}$ [ a , b ] ⊂ R and $\varepsilon _{0}\in [0,1]$ ε 0 ∈ [ 0 , 1 ] .

Equivalence between invariant measures and statistical solutions for the 2D non-autonomous magneto-micropolar fluid equations

In this article, we aim to investigate the regularity of statistical solution for the 2D non-autonomous magneto-micropolar fluid equations as well as the relationship between invariant measures and statistical solutions. Firstly, to get the regularity of the statistical solution, we prove the existence and regularity of the pullback attractor for the equations. Then we prove the statistical solution possesses some regularity properties by using regularity of the pullback attractor. Finally, we prove the statistical solution is actual an invariant measure for the equations.

Tipping points induced by parameter drift in an excitable ocean model

Numerous systems in the climate sciences and elsewhere are excitable, exhibiting coexistence of and transitions between a basic and an excited state. We examine the role of tipping between two such states in an excitable low-order ocean model. Ensemble simulations are used to obtain the model’s pullback attractor (PBA) and its properties, as a function of a forcing parameter $$\gamma $$ γ and of the steepness $$\delta $$ δ of a climatological drift in the forcing. The tipping time $$t_{\mathrm{{tp}}}$$ t tp is defined as the time at which the transition to relaxation oscillations (ROs) arises: at constant forcing this occurs at $$\gamma =\gamma _{\mathrm{c}}$$ γ = γ c . As the steepness $$\delta $$ δ decreases, $$t_{\mathrm{{tp}}}$$ t tp is delayed and the corresponding forcing amplitude decreases, while remaining always above $$\gamma _{\mathrm{c}}$$ γ c . With periodic perturbations, that amplitude depends solely on $$\delta $$ δ over a significant range of parameters: this provides an example of rate-induced tipping in an excitable system. Nonlinear resonance occurs for periods comparable to the RO time scale. Coexisting PBAs and total independence from initial states are found for subsets of parameter space. In the broader context of climate dynamics, the parameter drift herein stands for the role of anthropogenic forcing.

Statistical Solutions for a Nonautonomous Modified Swift-Hohenberg Equation

We consider the nonautonomous modified Swift-Hohenberg equation $$u_t+\Delta^2u+2\Delta u+au+b|\nabla u|^2+u^3=g(t,x)$$ on a bounded smooth domain $\Omega\subset\R^n$ with $n\leqslant 3$. We show that, if $|b|<4$ and the external force $g$ satisfies some appropriate assumption, then the associated process has a unique pullback attractor in the Sobolev space $H_0^2(\Omega)$. Based on this existence, we further prove the existence of a family of invariant Borel probability measures and a statistical solution for this equation.

Attractor Analysis of Cohen–Grossberg Neural Networks with Multiple Time-Varying Delays

This paper investigates the pullback attractor of Cohen–Grossberg neural networks with multiple time-varying delays. Compared with the existing references, the networks considered here are more general and cannot be expressed in the vector-matrix form due to multiple time-varying delays. After constructing a proper Lyapunov–Krasovskii functional and eliminating the terms involving multiple time-varying delays, two sets of new sufficient criteria on the existence of the pullback attractor are derived based on the theory of pullback attractors. In the end, two examples are given to demonstrate the effectiveness of our theoretical results.