Irving Daniel, Sorrentino Francesco
University of New Mexico, Albuquerque, New Mexico 87131, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2012 Nov;86(5 Pt 2):056102. doi: 10.1103/PhysRevE.86.056102. Epub 2012 Nov 5.
We present a general framework to study stability of the synchronous solution for a hypernetwork of coupled dynamical systems. We are able to reduce the dimensionality of the problem by using simultaneous block diagonalization of matrices. We obtain necessary and sufficient conditions for stability of the synchronous solution in terms of a set of lower-dimensional problems and test the predictions of our low-dimensional analysis through numerical simulations. Under certain conditions, this technique may yield a substantial reduction of the dimensionality of the problem. For example, for a class of dynamical hypernetworks analyzed in the paper, we discover that arbitrarily large networks can be reduced to a collection of subsystems of dimensionality no more than 2. We apply our reduction technique to a number of different examples, including the class of undirected unweighted hypermotifs with 3 nodes.
我们提出了一个用于研究耦合动力系统超网络同步解稳定性的通用框架。通过使用矩阵的同时块对角化，我们能够降低问题的维度。我们根据一组低维问题获得了同步解稳定性的充要条件，并通过数值模拟检验了我们低维分析的预测结果。在某些条件下，该技术可能会大幅降低问题的维度。例如，对于本文分析的一类动力超网络，我们发现任意大的网络都可以简化为维度不超过2的子系统集合。我们将降维技术应用于许多不同的例子，包括具有3个节点的无向无权超基序类。
