Topological Structure of Chaos in Discrete-Time Neural Networks with Generalized Input–Output Function.

By analogue of [Chen & Aihara, 1995, 1997, 1999], we theoretically investigate the topologically chaotic structure, attracting set and global searching ability of discrete-time recurrent neural networks with the form of u[sub i](t+1)=ku[sub i](t)+Δt(∑[sup n][sub j=1]a[sub ij]v[sub j](t)+a[sub i]) i = 1, 2, …, n where the input-output function is defined as a generalized sigmoid function, such as v[sub i] = tanh(µ[sub i]u[sub i]), v[sub i] = 2/π arctan (π/2 µ[sub i]u[sub i]) and v[sub i] = 1/1+e[sup -u[sub i]/ε] etc. We first derive sufficient conditions of existence for a fixed point, and then prove that this fixed point eventually evolves into a snap-back repeller which generates chaotic structure when certain conditions are satisfied. Furthermore we prove that there exists an attracting set which includes not only the homoclinic orbit but also the globally unstable set of fixed points, thereby ensuring the neural networks to have global searching ability. Numerical simulations are also provided to demonstrate the theoretical results. The results indicated in this paper can be viewed as an extension of the works of [Chen & Aihara, 1997, 1999] and others [Gopalsamy & He, 1994; Wang & Smith, 1998]. [ABSTRACT FROM AUTHOR]