Stability and synchronization of a ring of identical cells with delayed coupling (Q1768370)

The subject of the paper is a ring of coupled identical systems with time delayed, nearest neighbor coupling \[ x_i' = -x_i(t) + \alpha f(x_i(t-\tau_s)) + \beta [g(x_{i-1}(t-\tau))+ g (x_{i+1}(t-\tau))], \quad (i \mod n), \tag{1} \] where \(f,g\in C^3\) are bounded functions. In addition, it assumed that \(f(0)=g(0)=0\), \(f'(0)=g'(0)=1\); \(f'(x)>0\), \(g'(x)>0\) for all \(x\in \mathbb{R}\); \(xf''(x)<0\) and \(xg''(x)<0\) for all \(x\neq 0\); and \(f'''(0)<0\), \(g'''(0)<0\). The authors study the local stability of the trivial solution \(x_i=0\) as well as the existence and stability of the synchronized solutions of (1). Lyapunov functionals are used to establish global stability of the synchronized solutions.