(a) ProveTheorem 3 regardingLyapunov functions of continuous Hopfield models.

416 17. Neural Networks

(b) Consider the recurrent Hopfield network modeled using the differential equations

˙ x = −x + 7 2

π

tan−1

γπx 2

+ 6 2

π

tan−1

γπy 2

,

˙ y = −y + 6 2

π

tan−1

γπx 2

− 2 2

π

tan−1

γπy 2

.

Plot a vector field portrait and derive a suitable Lyapunov function.

(c) Plot surface plots for the Lyapunov functions for Examples 3 and 4 and Exercise 5(b). Plot the surfaces for |ai| ≤ 1, i = 1, 2.