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Abstract: In this paper, we construct a two-layer feedback neural network to theoretically investigate the influence of symmetry and time delays on patterned rhythmic behaviors. Firstly, linear stability of the model is investigated by analyzing the associated transcendental characteristic equation. Next, by means of the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups, we not only investigate the effect of synaptic delays of signal transmission on the pattern formation, but also obtain some important results about the spontaneous bifurcation of multiple branches of periodic solutions and their spatio-temporal patterns. Thirdly, based on the normal form approach and the center manifold theory, we derive the formula to determine the bifurcation direction and stability of Hopf bifurcating periodic solutions. Finally, some numerical examples and the corresponding numerical simulations are used to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a simple discrete two-neuron network model with three delays. The characteristic equation of the linearized system at the zero solution is a polynomial equation involving very high order terms. We derive some sufficient and necessary conditions on the asymptotic stability of the zero solution. Regarding the eigenvalues of connection matrix as the bifurcation parameters, we also consider the existence of three types of bifurcations: Fold bifurcations, Flip bifurcations, and Neimark–Sacker bifurcations. The stability and direction of these three kinds of bifurcations are studied by applying the normal form theory and the center manifold theorem. Our results are a very important generalization to the previous works in this field. [Copyright &y& Elsevier]

The emergence of rumors is subjective and aggressive which will bring disturbance and misleading to people's lives. This paper is dedicated to comprehensively investigating the influence of forced silence function, time delay and network topology on rumor propagation in online social networks. Considering the network supervision such as forced silence and network topology, a novel delayed SIR(susceptible-propagating-recovery) epidemic-like rumor propagation model in both homogeneous and heterogeneous networks is proposed. By using the mean-field theory, we analyze the propagation dynamics of the model in detail. Firstly, we demonstrate the boundedness of the solutions, then we obtain the basic reproduction number and the existence of the rumor-prevailing equilibrium points. Secondly, the forward and backward bifurcation as well as the local and global stabilities of the equilibrium points in both homogeneous and heterogeneous networks are theoretically analyzed and revealed. Finally, the possible impact is analyzed through numerical simulations, and the factors affecting rumor propagation are theoretically analyzed, which prove the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]