Estimation of delay on synchronization stability in a class of complex systems with coupling delays
Consider the network \[ \dot x_i= f(x_i)+ d\sum^N_{j=1} c_{ij}\Gamma x_j(t-\tau),\quad i= 1,\dots, N\tag{1} \] with \(x_i\in \mathbb{R}^n\), \(d\in\mathbb{R}_+\), \(\Gamma= \text{diag}(r_1,\dots, r_N)\) with \(r_i\in \{0,1\}\), \(C= \{c_{ij}\}\in \mathbb{R}^{N\times N}\), \(C\) symmetric, \(c_{ij}\in \{0,1\}\) for \(i\neq j\), \(c_{ii}=- \sum^N_{j=1} c_{ij}\) for \(i= 1,\dots, N\), \(f= (f_1,\dots, f_n)^T\) is continuously differentiable. Synchronization of (1) is defined by \[ x_1(t)\to x_2(t)\to\cdots\to x_N(t)\to x^*\in \mathbb{R}^n, \] where \(x^*\) is a stable equilibrium. The stability of the synchronized states is determined by the stability of the origin of the systems \[ \dot w(t)= J(x^*) w(t)+ d\mu_i\Gamma w(t-\tau),\quad i= 2,\dots, N,\tag{2} \] where \(\mu_i\) are the eigenvalues of the matrix \(C\) which are different from zero. The authors describe a criterion for the asymptotic stability of (2) which does not depend on the delay \(\tau\), and a stability criterion based on some estimate of the delay. The paper contains several misprints.
scientific article; zbMATH DE number 5548656