In this paper, we present a new training algorithm for a fuzzy perceptron. In the case where the dimension of the input vectors is two and the training examples are separable, we can prove a-nite convergence, i.e., the training procedure for the network weights will stop after-nite steps. When the dimension is greater than two, stronger conditions are needed to guarantee the-nite convergence.

Online gradient methods are widely used for training feedforward neural networks. We prove in this paper a convergence theorem for an online gradient method with variable step size for backward propagation (BP) neural networks with a hidden layer. Unlike most of the convergence results that are of probabilistic and nonmonotone nature, the convergence result that we establish here has a deterministic and monotone nature.

An online gradient method for BP neural networks is presented and discussed. The input training examples are permuted stochastically in each cycle of iteration. A monotonicity and a weak convergence of deterministic nature for the method are proved.

A survey is given for some recent developments on bifurcation from local steady-states to global dynamics governed by nonlinear ODE's with Z2 or O(2) symmetries. In particular, we are mainly concerned with a double singular point, which is a Z2 symmetric steady-state possessing a Jacobian with two zero eigenvalues. There exist, under suitable con-ditions, Hopf points and heteroclinic cycles bifurcating from the double singular point. We also considered a triple zero point with an O(2) sym-metry, from which bifurcate standing waves, rotating waves and modu-lated rotating waves.

In this paper, we study the convergence of an online gradient method for feed-forward neural networks. The input training examples are permuted stochastically in each cycle of iteration. A monotonicity and a weak convergence of deterministic nature are proved.

MRI (Madaline Rule I) neural network has wide applications. A finite convergence for the training of MRI neural network is proved for linearly separable training patterns.

The online gradient method has been widely used as a learning algorithm for neural networks. We establish a deterministic convergence of online gradient methods for the training of a class of nonlinear feedforward neural networks when the training examples are linearly independent. We choose the learning rate to be a constant during the training procedure. The monotonicity of the error function in the iteration is proved. A criterion for choosing the learning rate is also provided to guarantee the convergence. Under certain conditions similar to those for the classical gradient methods, an optimal convergence rate for our online gradient methods is proved.

This paper deals with nonlinear equations f (x,λ,α)=0 and the corresponding ODEs xt=f (x,λ,α) satisfying f (0,λ,α)=0 and a Z2-symmetry. In particular, we are interested in Hopf points, which indicate the bifurcation of periodic solutions of xt=f (x,λ,α) from (steady-state) solutions of f (x,λ,α)=0. It is shown that under suitable nondegeneracy conditions, there bifurcate two paths of Hopf points from a double singular point, where x=0 and fx (0,λ,α) has a double zero eigenvalue with one eigenvector symmetric and one anti-symmetric. This result gives a new example of -nding Hopf points through local singular points. Our main tools for analysis are some extended systems, which also provide easily implemented algorithms for the numerical computation of the bifurcating Hopf points. A supporting numerical example for a Brusselator model is also presented.

In this paper we prove that the online gradine method for continuous perceptrons coverges in finite steps when the training pat-terns are linearly separable.

Motivated by the problem of training multilayer perceptrons in neural networks, we consider the problem of minimizing E(x)=n=1 fi (ξi·x), where ξi ∈ Rs, 1≤i≤n, and each fi (ξi·x) is a ridge function. We show that when n is small the problem of minimizing E can be treated as one of minimizing univariate functions, and we use the gradient algorithms for minimizing E when n is moderately large. For largen, we present the online gradient algorithms and especially show the monotonicity and weak convergence of the algorithms.