Stability switches, bifurcation, and multi-stability of coupled networks with time delays (Q422856)

The author presents a stability analysis of a trivial steady state solution in a network with delayed coupling which consists of two identical bidirectional rings with elementwise, symmetric crosscoupling. The crosscoupling delay \(\tau\) is assumed different from the ring-coupling delay \(\sigma\). The model equations are NEWLINE\[NEWLINE \dot{x}_i (t) = -x_i(t) + af(x_{i-1}(t-\sigma)) + af(x_{i+1}(t-\sigma)) + bg(y_i(t-\tau)),NEWLINE\]NEWLINE NEWLINE\[NEWLINE \dot{y}_i (t) = -y_i(t) + af(y_{i-1}(t-\sigma)) + af(y_{i+1}(t-\sigma)) + bg(x_i(t-\tau)),NEWLINE\]NEWLINE for \(i=1,\dots,n\). First, the linearized system is studied. Some sufficient conditions for the asymptotic stability of the steady state are given and the occurence of purely imaginary eigenvalues, i.e., Hopf bifurcations, is investigated. Under some assumptions on the coupling functions, a sufficient condition for the global asymptotic stability is derived by construction of a Lyapunov functional. Numerical illustration is shown in a final chapter.