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

TL;DR

This paper constructs a chaotic discrete-time Hopfield network with piecewise-affine activation functions, demonstrating sensitive dependence on initial conditions and complex dynamics within a specific Cantor set in the state space.

Contribution

It introduces a novel chaotic Hopfield network model with explicit parameters, expanding understanding of chaos in neural networks with piecewise-affine activations.

Findings

Existence of a Cantor set with sensitive dependence on initial conditions

Network orbits have the Cantor set as their omega-limit set

Explicit parameters for the chaotic network are provided

Abstract

We construct a chaotic discrete-time continuous-state Hopfield network with 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.