In this paper, some sufficient conditions for the local and global exponential stability of the discrete-time Hopfield neural networks with general activation functions are derived, which generalize those existing results. By means of Mmatrix theory and some inequality analysis techniques, the exponential convergence rate of the neural networks to the equilibrium is estimated, and for the local exponential stability, the basin of attraction of the stable equilibrium is also characterized. © 2004 Elsevier Ltd. All rights reserved.

In this paper, we study a class of neural networks, which includes bidirectional associative memory networks and cellular neural networks as its special cases. By Brouwer's fixed point theorem, a continuation theorem based on Gains and Mawhin's coincidence degree, matrix theory, and inequality analysis, we not only obtain some different sufficient conditions ensuring the existence, uniqueness, and global exponential stability of the equilibrium but also estimate the exponentially convergent rate. Our results are less restrictive than previously known criteria and can be applied to neural networks with a broad range of activation functions assuming neither differentiability nor strict monotonicity.

We study the existence and branching patterns of wave trains in the one-dimensional in nite Fermi-Pasta-Ulam (FPU) lattice. A wave train Ansatz in this Hamiltonian lattice leads to an advance-delay di erential equation on a space of periodic functions, which carries a natural Hamiltonian structure. The existence of wave trains is then studied by means of a Lyapunov Schmidt reduction, leading to a nite-dimensional bifurcation equation with an inherited Hamiltonian structure. While exploring some of the additional symmetries of the FPU lattice, we use invariant theory to nd the bifurcation equations describing the branching patterns of wave trains near p∶q resonant waves. We show that at such branching points, a generic nonlinearity selects exactly two two-parameter families of mixed-mode wave trains.

We consider a network of two coupled neurons with delayed feedback. We show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension 1 bifurcations (including a fold bifurcation and a Hopf bifurcation) and codimension 2 bifurcations (including fold-Hopf bifurcations and Hopf-Hopf bifurcations). We also give concrete formulae for the normal form coefficients derived via the center manifold reduction that give detailed information about the bifurcation and stability of various bifurcated solutions. In particular, we obtain stable or unstable equilibria, periodic solutions, quasi-periodic solutions, and sphere-like surfaces of solutions.We also show how to evaluate critical normal form coefficients from the original system of delay-differential equations without computing the corresponding center manifolds.

In this paper, we consider a simple discrete-time single-directional network of four neurons. The characteristics equation of the linearized system at the zero solution is a polynomial equation involving very high-order terms. We first derive some sufficient and necessary conditions ensuring that all the characteristic roots have modulus less than 1. Hence, the zero solution of the model is asymptotically stable. Then, we study the existence of three types of bifurcations, such as fold bifurcations, flip bifurcations, and Neimark-Sacker (NS) bifurcations. Based on the normal form theory and the center manifold theorem, we discuss their bifurcation directions and the stability of bifurcated solutions. In addition, several codimension two bifurcations can be met in the system when curves of codimension one bifurcations intersect or meet tangentially. We proceed through listing smooth normal forms for all the possible codimension 2 bifurcations. © 2007 Elsevier B.V. All rights reserved.