Attractors for a Neural Network Equation
Under consideration is the neural network equation $$\begin{aligned} \frac{\partial v(w,t)}{\partial t} = -v(w,t) + J*(f\circ v)(w,t) + h \end{aligned}$$posed on the bounded domain \(\mathbb {S}^1\) in \(L^2\). Recently it has been shown that the semigroup of solution operators corresponding to the initial value problem admits a family of global attractors that is upper-semicontinuous with respect to the kernel \(J\) (Horcia da Silva, Differ. Eqn. Dyn. Syst. 19:8796, 2011). In this paper we provide further information about the upper-semicontinuity of the global attractors; namely, we obtain explicit control over the semidistances in terms of the perturbation parameter. In addition, we show that under an additional assumption, the semigroup admits a family of exponential attractors in \(L^2(\mathbb {S}^1)\) that is also upper-semicontinuous with respect to \(J\). The result also means that the corresponding global attractors are finite dimensional.
neural network equation bounded domain global attractor exponential attractor continuous families of sets 35b41 35k55 92b20