Identical Phase Oscillator Networks: Bifurcations, Symmetry and Reversibility for Generalized Coupling
Identical Phase Oscillator Networks: Bifurcations, Symmetry and Reversibility for Generalized Coupling
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同相振荡器网络：广义耦合的分叉、对称性和可逆性
DOI：
10.3389/fams.2016.00007
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
O. Burylko
中科院分区：
文献类型：
--
作者：
P. Ashwin;C. Bick;O. Burylko
对于一个具有全排列对称性的耦合同相振子系统，任何动力学行为中的对称性破缺都必须来自于自发对称性破缺，即来自于系统的非线性动力学。这样一个系统的相位差的动力学只取决于耦合（相位相互作用）函数g（\varphi）$和振荡器的数量N$。本文首先简要回顾了这类系统在一般耦合g$情况下的一些结果，然后详细讨论了两种情况：（a）一般的双调和形式：g（\varphi）=q\sin（\varphi-\alpha）+r\sin（2\varphi-\beta）$和N$ small（B）耦合g$是奇数或偶数。我们扩展以前发表的分歧分析一般两个谐波的情况下，甚至$g$的相位差的动态有一些时间反转对称性。对于偶数g的情况，已知系统有N-2个运动常数。这是真实的$N=4$和任何$g$，而$N=4$和两个以上的谐波在$g$，我们表明系统必须有较少的独立常数的运动。
For a system of coupled identical phase oscillators with full permutation symmetry, any broken symmetries in dynamical behaviour must come from spontaneous symmetry breaking, i.e. from the nonlinear dynamics of the system. The dynamics of phase differences for such a system depends only on the coupling (phase interaction) function $g(\varphi)$ and the number of oscillators $N$. This paper briefly reviews some results for such systems in the case of general coupling $g$ before exploring two cases in detail: (a) general two harmonic form: $g(\varphi)=q\sin(\varphi-\alpha)+r\sin(2\varphi-\beta)$ and $N$ small (b) the coupling $g$ is odd or even. We extend previously published bifurcation analyses to the general two harmonic case, and show for even $g$ that the dynamics of phase differences has a number of time-reversal symmetries. For the case of even $g$ with one harmonic it is known the system has $N-2$ constants of the motion. This is true for $N=4$ and any $g$, while for $N=4$ and more than two harmonics in $g$, we show the system must have fewer independent constants of the motion.