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Without assuming boundedness and differentiability of the activation functions and any symmetry of interconnections, we employ Lyapunov functions to establish some sufficient conditions ensuring existence, uniqueness, global asymptotic stability, and even global exponential stability of equilibria for the Cohen-Grossberg neural networks with and without delays. Our results are not only presented in terms of system parameters and can be easily verified and also less restrictive than previously known criteria and can be applied to neural networks, including Hopfield neural networks, bidirectional association memory neural networks, and cellular neural networks.

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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, the existence, stability, and multiplicity of spatially nonhomogeneous steady-state solution and periodic solutions for a reaction-diffusion model with nonlocal delay effect and Dirichlet boundary condition are investigated by using Lyapunov-Schmidt reduction. Moreover, we illustrate our general results by applications to models with a single delay and one-dimensional spatial domain.

In this paper, we consider the following semilinear Kirchhoff type equationwhere > 0 is a small parameter, p ∈ [3, 5), a, b are positive constants, μ > 0 is a parameter, and the nonlinear growth of |u| 4 u reaches the Sobolev critical exponent since 2 * = 6 for three spatial dimensions. We prove the existence of a positive ground state solution u with exponential decay at infinity for μ > 0 and sufficiently small under some suitable conditions on the nonnegative functions V , K and Q. Moreover, u concentrates around a global minimum point of V as → 0 + . The methods used here are based on the concentration-compactness principle of Lions.Mathematics Subject Classification (2010). 35J60 · 35J65 · 53C35.

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.