On a Simple Hysteresis Network
Kenya Jin’no, Toshimichi Saito · 1993
In order to approach the explication of rich dynamics from artificial neural networks (ab. ANN), this article analyses a simple continuous-time hysteresis network including only two parameters. The objective system is described by the following equation: where \(x\equiv (x_1,\cdots,x_m)^t, x_i\epsilon \ R\) , is a state variable vector, \(y\equiv (y_1,\cdots,y_m)^t,y_i \; \epsilon\;\left \{ 0,1 \right \}\) , is an output vector and m is the number of the cell. T is a self feedback parameter and h ( xi ) is a binary hysteresis function as shown in Fig.l. That is, y i is switched from 0 to 1 if x i - hits the left threshold a and vice versa. This network is a simplified version of our original hysteresis neural network [1] and an implementation example is shown in [2]. The net includes only two parameters ( a , T ), but the system behavior is very interesting. Then we give the following results. If the parameters satisfy 0 < K — a < 1 and 0 < ( T + l ) + ( K — a ) < 1, where K and l are non-negative intergers, then all attractors of the net are stable equilibrium points such that the number of ”1” in corresponding output vector y is K + n , n = 0 ~ l . For any K and l , we can completely clarify the number of attractors and the their domain of attraction. If the parameters satisfy 0 < K — a < 1 and 1 < T + ( K — a ) < 2, where K is a non-negative integer, then all attractors are stable periodic orbits such that the number of ”1” in corresponding output vector y vibrates between K and K — 1. For any K , we can clarify the number of attractors and the their local domain of attraction. We can rigorously calculate all periodic orbits. Hysteresis Function