Wei, Junjie; Yuan, Yuan Synchronized Hopf bifurcation analysis in a neural network model with delays. (English) Zbl 1085.34058 J. Math. Anal. Appl. 312, No. 1, 205-229 (2005). The subject of the paper is the following network model with two constant delays \[ \dot x_i(t) = -x_i(t) +\alpha f(x_i(t-\tau_s))+ \beta [g(x_{i-1}(t-\tau))+g(x_{i+1}(t-\tau))], \quad (i \;\mathrm{mod} \;n), \] where \(x_i\in \mathbb{R}\). Here, \(\beta\) and time delay \(\tau\) are considered as the main bifurcation parameters. The authors focus their analysis on the bifurcations of synchronous solutions, i.e., solutions that satisfy \(x_i(t)=x_j(t)\) for all \(i\) and \(j\). First, they analyse the stability of the trivial solution. Then, using the center manifold theory and normal forms, they give the direction and stability of the Hopf bifurcation. Finally, they study global existence and stability of periodic solutions. 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