The dynamics of network coupled phase oscillators: An ensemble approach
The dynamics of network coupled phase oscillators: An ensemble approach
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DOI：
10.1063/1.3596711
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发表时间：
2011-06-01
期刊：
影响因子：
2.9
通讯作者：
Ott, Edward
中科院分区：
文献类型：
--
作者：
Barlev, Gilad;Antonsen, Thomas M.;Ott, Edward
我们考虑了许多相位振荡器，通过耦合网络相互作用的动力学。对于一个给定的网络连接，我们进一步考虑这样的系统的合奏，对于每个合奏成员，一组振荡器的自然频率是独立和随机选择根据一个给定的分布函数。然后，我们寻求一个统计描述这个合奏的动态。使用这种方法，我们可以将最近开发的Ott和Antonsen [Chaos 18，037113（2008）]的分析应用于每个节点处状态系综的边缘分布。这反过来又会导致一组减少的常微分方程确定这些边际分布函数。新的方程组在几个方面有利于网络动力学的分析：（i）简化的系综方程组的时间演化更加平滑，因此可以通过使用更长的时间步长更快地获得数值解;（ii）新的方程组可以用作获得分析结果的基础;以及（iii）对于某种类型的网络，可以简化为整个网络动态的低维描述。我们说明了我们的方法与网络版本的经典仓本问题的数值实验，首先与单峰频率分布，然后与双峰分布。在后一种情况下，网络动力学的特点是分叉和滞后涉及各种稳定和周期性吸引子。(C)2011年美国物理学会。[doi：10.1063/1.3596711]
We consider the dynamics of many phase oscillators that interact through a coupling network. For a given network connectivity we further consider an ensemble of such systems where, for each ensemble member, the set of oscillator natural frequencies is independently and randomly chosen according to a given distribution function. We then seek a statistical description of the dynamics of this ensemble. Use of this approach allows us to apply the recently developed ansatz of Ott and Antonsen [Chaos 18, 037113 (2008)] to the marginal distribution of the ensemble of states at each node. This, in turn, results in a reduced set of ordinary differential equations determining these marginal distribution functions. The new set facilitates the analysis of network dynamics in several ways: (i) the time evolution of the reduced system of ensemble equations is much smoother, and thus numerical solutions can be obtained much faster by use of longer time steps; (ii) the new set of equations can be used as a basis for obtaining analytical results; and (iii) for a certain type of network, a reduction to a low dimensional description of the entire network dynamics is possible. We illustrate our approach with numerical experiments on a network version of the classical Kuramoto problem, first with a unimodal frequency distribution, and then with a bimodal distribution. In the latter case, the network dynamics is characterized by bifurcations and hysteresis involving a variety of steady and periodic attractors. (C) 2011 American Institute of Physics. [doi:10.1063/1.3596711]