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Almost Periodic Dynamics of a Class of Delayed Neural Networks with Discontinuous Activations

wenlian{at}fudan.edu.cn Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University, 200433, Shanghai, P.R.C., and Max Planck Institute for Mathematics in the Sciences, 04103 Leipzig, Germany

Tianping Chen

tchen{at}fudan.edu.cn Laboratory of Mathematics forNonlinear Sciences, School of Mathematical Sciences, Fudan University, 200433, Shanghai, P.R.C.

Abstract

We use the concept of the Filippov solution to study the dynamics of a class of delayed dynamical systems with discontinuous right-hand side, which contains the widely studied delayed neural network models with almost periodic self-inhibitions, interconnection weights, and external inputs. We prove that diagonal-dominant conditions can guarantee the existence and uniqueness of an almost periodic solution, as well as its global exponential stability. As special cases, we derive a series of results on the dynamics of delayed dynamical systems with discontinuous activations and periodic coefficients or constant coefficients, respectively. From the proof of the existence and uniqueness of the solution, we prove that the solution of a delayed dynamical system with high-slope activations approximates to the Filippov solution of the dynamical system with discontinuous activations.