Oscillator death in coupled functional differential equations near Hopf bifurcation
Stability of the equilibrium solution is analyzed for coupled systems of retarded functional-differential equations \[ \dot x_i(t) = L_i x_i^{t} + \varepsilon f_i(x_i^{t},\varepsilon)+\varepsilon \kappa g_i(x_1^{t},\dots,x_N^{t},\varepsilon), \quad i=1,\dots,N, \] near a supercritical Hopf bifurcation. Here, \(x^{t}\) denotes the function defined on \([-\tau,0]\) by \(x^{t}(\theta)=x(t+\theta)\) for \(x: \mathbb{R}\to \mathbb{R}^{n}\) and \(\tau\geq 0\). For each \(i\), the operator \(L_i: C([-\tau,0],\mathbb{R}^{n})\to \mathbb{R}^{n}\) is linear and continuous. Moreover, the equation \(\dot x_i(t)=L_i x_i^{t}\) has a pair of characteristic values \(\pm i \omega_i\). \(f_i\) and \(g_i\) vanish at \(x^{t}=0\). The author gives conditions in terms of the coupling rate \(\kappa\) for the stability of the zero equilibrium solution. In particular, it is shown that the largest eigenvalue of the graph Laplacian completely characterizes the effect of the connection topology on the stability of the diffusively and symmetrically coupled identical systems.
scientific article; zbMATH DE number 5013577
scientific article