This paper presents a detailed analysis on the stability and instability of a coupled oscillator network with small world connections. This network consists of regular connections, excitatory short-cuts or inhibitory short-cuts. By using the perturbation theory of matrix, we give the upper and lower bounds of maximum and minimum eigenvalues of the coupling strength matrix, and then give the generalized sufficient conditions that guarantee the system complete stability or complete instability. In addition, we analyze the effects of the short-cut possibility, excitatory or inhibitory short-cut strength and time delay on the system stability. We also analyze the instability mechanism and bifurcation modes. In addition, the studies on the robustness stability show that the stability of this network is more robust to change of short-cut connections than the regular network. Compared to the mean-field theory, the stability conditions from the proposed method are more conservational. However, the proposed method can guarantee the complete stability even if the randomness is in the system. They are more useful and adaptive than mean-field theory especially when the excitatory and inhibitory connections exist simultaneously.
Renormalization of the Higgs Model: Minimizers, Propagators and the Stability of Mean Field Theory
