A chaotic discrete-time continuous-state Hopfield network with piecewise-affine activation functions

Abstract

We construct a chaotic discrete-time continuous-state Hopfield networkwith piecewise-affine nonnegative activation functions and weight matrix with small positive entries. More precisely, there exists a Cantor set $ C $ in the state space such that the network has sensitive dependence on initial conditions at initial states in $ C $ and the network orbit of each initial state in $ C $ has $ C $ as its $ \omega $-limit set. The approach we use is based on tools developed and employed recently in the study of the topological dynamics of piecewise-contractions. The parameters of the chaotic network are explicitly given.