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.