Stable phase-locked periodic solutions in a delay differential system.
The subject of the paper is the following system with delay \[ \dot x_i(t) = - \mu x_i(t) +a f(x_i(t-\tau)) + b[f(x_{i-1}(t-\tau)) + f(x_{i+1}(t-\tau))], \quad i(\text{mod}) \; 3, \] where \(x_i \in \mathbb{R}\), \(f: \mathbb{R}\to \mathbb{R}\), \(\mu>0\), \(b>0\), and \(a<0\). This system appears in the theory of neural networks. The nonlinearity \(f\) is assumed to be close to the step function. The author obtains existence, stability, and the limiting profile of phase-locked periodic solutions \(P(t)=(p_1(t),p_2(t),p_3(t))\) with period \(T\), which have the following symmetry properties: \(p_1(t)=p_2(t+T/3)\), \(p_2(t)=p_3(t+T/3)\), \(p_3(t)=p_1(t+T/3)\).
scientific article; zbMATH DE number 2005879
scientific article