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arXiv 1105.4674nlin.CD
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耦合动力系统超网络的同步

Synchronization of hypernetworks of coupled dynamical systems

Francesco Sorrentino

备注：
Accepted for publication in New Journal of Physics
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摘要：

我们考虑耦合动力系统的同步时，同时存在不同类型的相互作用。我们假设一组动力系统通过两个或多个不同的网络（每个网络对应于不同类型的相互作用）的连接而耦合，我们将这样的系统称为超网络。应用包括由电间隙连接和化学突触形成的神经网络，通过视觉和流量传感进行通信的鱼群的协调运动，以及耦合混沌振荡器的超网络。我们首先分析由$m=2$网络形成的超网络的情况。我们寻找同步的充分必要条件。我们试图以主稳定性函数的形式减少线性稳定性问题，即，从网络的结构中解耦耦合函数的影响。不幸的是，我们无法获得一个主稳定性函数形式的一般情况下减少。然而，我们表明，这样的减少是可能的，在三种情况下的利益：（i）与两个网络的拉普拉斯矩阵交换;（ii）两个网络之一是未加权和完全连接;（iii）两个网络之一是这样的，从节点$i$到节点$j$的耦合强度是一个函数的$j$，但不是$i$。此外，我们定义了一类网络，使得如果两个耦合网络中的任何一个属于这一类，则可以独立于另一个网络获得约简。作为感兴趣的一个例子，我们研究了神经超网络的同步，其连接可以是化学突触或电间隙连接。我们提出了一个推广我们的稳定性结果的情况下形成的超网络的$m\geq 2$网络。

英文摘要：

We consider synchronization of coupled dynamical systems when different types of interactions are simultaneously present. We assume that a set of dynamical systems are coupled through the connections of two or more distinct networks (each of which corresponds to a distinct type of interaction), and we refer to such a system as a hypernetwork. Applications include neural networks formed of both electrical gap junctions and chemical synapses, the coordinated motion of shoals of fishes communicating through both vision and flow sensing, and hypernetworks of coupled chaotic oscillators. We first analyze the case of a hypernetwork formed of $m=2$ networks. We look for necessary and sufficient conditions for synchronization. We attempt at reducing the linear stability problem in a master stability function form, i.e., at decoupling the effects of the coupling functions from the structure of the networks. Unfortunately, we are unable to obtain a reduction in a master stability function form for the general case. However, we show that such a reduction is possible in three cases of interest: (i) the Laplacian matrices associated with the two networks commute; (ii) one of the two networks is unweighted and fully connected; (iii) one of the two networks is such that the coupling strength from node $i$ to node $j$ is a function of $j$ but not of $i$. Furthermore, we define a class of networks such that if either one of the two coupling networks belongs to this class, the reduction can be obtained independently of the other network. As an example of interest, we study synchronization of a neural hypernetwork for which the connections can be either chemical synapses or electrical gap junctions. We propose a generalization of our stability results to the case of hypernetworks formed of $m\geq 2$ networks.

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