School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China.
Chaos. 2011 Jun;21(2):023130. doi: 10.1063/1.3602226.
Oscillation death (also called amplitude death), a phenomenon of coupling induced stabilization of an unstable equilibrium, is studied for an arbitrary symmetric complex network with delay-coupled oscillators, and the critical conditions for its linear stability are explicitly obtained. All cases including one oscillator, a pair of oscillators, regular oscillator networks, and complex oscillator networks with delay feedback coupling, can be treated in a unified form. For an arbitrary symmetric network, we find that the corresponding smallest eigenvalue of the Laplacian λ(N) (0 >λ(N) ≥ -1) completely determines the death island, and as λ(N) is located within the insensitive parameter region for nearly all complex networks, the death island keeps nearly the largest and does not sensitively depend on the complex network structures. This insensitivity effect has been tested for many typical complex networks including Watts-Strogatz (WS) and Newman-Watts (NW) small world networks, general scale-free (SF) networks, Erdos-Renyi (ER) random networks, geographical networks, and networks with community structures and is expected to be helpful for our understanding of dynamics on complex networks.
震荡死亡（也称为振幅死亡），是一种耦合诱导不稳定平衡点稳定的现象，针对具有时滞耦合振荡器的任意对称复网络进行了研究，并明确得到了其线性稳定性的临界条件。所有情况，包括一个振荡器、一对振荡器、规则振荡器网络和具有时滞反馈耦合的复杂振荡器网络，都可以用统一的形式进行处理。对于任意对称网络，我们发现拉普拉斯算子的最小特征值λ（N）（0＞λ（N）≥-1）完全决定了死亡岛，并且由于λ（N）位于几乎所有复杂网络的不敏感参数区域内，因此死亡岛保持几乎最大，并且不会对复杂网络结构敏感。这种不敏感性效应已经在许多典型的复杂网络中进行了测试，包括 Watts-Strogatz（WS）和 Newman-Watts（NW）小世界网络、广义无标度（SF）网络、Erdos-Renyi（ER）随机网络、地理网络以及具有社区结构的网络，并有望有助于我们理解复杂网络上的动力学。
