Global Attractors and Robustness of the Boissonade System

Global Attractors for Hindmarsh-Rose Equations in Neurodynamics

“…These topics of global dynamic patterns are also important in neural field and neural network theories [5,10]. For the autocatalytic and Boissonade reaction-diffusion systems [31,33,34,35], it is proved that global attractors exist.…”

Random attractor for stochastic Boissonade system with time-dependent deterministic forces and white noises

“…The system (1) arised in many fields of biology and chemistry, for example, Turing pattern, chlorite-iodide-malonic acid reaction-diffusion systems, cancer model involving mutating cancel, genuine 2D systems, 3D monolayers, Schnackenberg model and activator-inhibitor model, see [14,22,28,29] and the references therein.…”

“…Tu in [28] proved the existence of a global attractor for the autonomous deterministic system (1) without any external effects (there f 1 (x, t) = f 2 (x, t) = h 1 (x) = h 2 (x) = 0, ∀x ∈ Ω), and obtained upper bounds of the Hausdorff and fractal dimension of this attractor and the upper-semicontinuity of the attractors with respect to γ → 0.…”

“…Motivated by [28,30,31,40,41], we will consider the existence, the uppersemicontinuity, the fractal dimensions of random attractors for the stochastic nonautonomous Boissonade systems (1) and (2).…”

“…First we present some notations and formulation from [28]. Let H = L 2 (Ω) × L 2 (Ω) with Borel σ-algebra B(H), E = H 1 0 (Ω) × H 1 0 (Ω), and we denote the norm and inner-product of H and L 2 (Ω) by • and •, • , the norm of L p (Ω) or product space L p (Ω) × L p (Ω) by • L p if p = 2, and the Lebesgue measure or an absolute value or a vector norm in an Euclidean space by | • |.…”