Department of Computer Science, Katholieke Universiteit Leuven, Heverlee, Belgium.
Chaos. 2009 Sep;19(3):033110. doi: 10.1063/1.3187792.
We consider the synchronization problem of an arbitrary number of coupled nonlinear oscillators with delays in the interconnections. The network topology is described by a directed graph. Unlike the conventional approach of deriving directly sufficient synchronization conditions, the approach of the paper starts from an exact stability analysis in a (gain, delay) parameter space of a synchronized equilibrium and extracts insights from an analysis of its bifurcations and from the corresponding emerging behavior. Instrumental to this analysis a factorization of the characteristic equation is employed that not only facilitates the analysis and reduces computational cost but also allows to determine the precise role of the individual agents and the topology of the network in the (in)stability mechanisms. The study provides an algorithm to perform a stability and bifurcation analysis of synchronized equilibria. Furthermore, it reveals fundamental limitations to synchronization and it explains under which conditions on the topology of the network and on the characteristics of the coupling the systems are expected to synchronize. In the second part of the paper the results are applied to coupled Lorenz systems. The main results show that for sufficiently large coupling gains, delay-coupled Lorenz systems exhibit a generic behavior that does not depend on the number of systems and the topology of the network, as long as some basic assumptions are satisfied, including the strong connectivity of the graph. Here the linearized stability analysis is strengthened by a nonlinear stability analysis which confirms the predictions based on the linearized stability and bifurcation analysis. This illustrates the usefulness of the exact linearized analysis in a situation where a direct nonlinear stability analysis is not possible or where it yields conservative conditions from which it is hard to get qualitative insights in the synchronization mechanisms and their scaling properties. In the examples several network topologies are considered.
我们研究了具有连接时滞的任意数量的耦合非线性振荡器的同步问题。网络拓扑结构由有向图描述。与传统的直接推导同步条件的方法不同，本文的方法从同步平衡点的（增益、延迟）参数空间的精确稳定性分析开始，并从其分叉分析和相应的出现行为中提取见解。这种分析的一个重要工具是特征方程的因式分解，它不仅简化了分析并降低了计算成本，而且还可以确定各个代理和网络拓扑在（不）稳定性机制中的精确作用。该研究提供了一种用于执行同步平衡点稳定性和分叉分析的算法。此外，它揭示了同步的基本限制，并解释了在网络拓扑和耦合特性的条件下，系统预期会同步的条件。在本文的第二部分，将这些结果应用于耦合 Lorenz 系统。主要结果表明，对于足够大的耦合增益，延迟耦合 Lorenz 系统表现出一种不依赖于系统数量和网络拓扑的通用行为，只要满足一些基本假设，包括图的强连通性。这里通过非线性稳定性分析加强了线性稳定性分析，该分析证实了基于线性稳定性和分叉分析的预测。这说明了在直接进行非线性稳定性分析不可行或从其中很难获得有关同步机制及其缩放性质的定性见解的情况下，精确线性分析的有用性。在示例中考虑了几种网络拓扑。
