非线性振荡器网络中局部同步的必要和充分条件
A Necessary and Sufficient Condition for Local Synchronization in Nonlinear Oscillator Networks
AI总结：
研究通过Lyapunov-Floquet理论和主稳定性函数框架，证明了正耦合强度是线性全状态耦合振荡器网络局部同步的必要和充分条件，并扩展到非相同耦合的定向图网络。
摘要：
确定互联振荡器网络中同步所需的耦合强度条件是非线性动力学中的挑战性问题。尽管已使用复杂数学方法推导条件，但这些条件通常仅是充分条件或基于数值方法。我们利用Lyapunov-Floquet理论和主稳定性函数框架，填补了充分耦合强度与数值观测之间的差距。我们证明，正耦合强度是线性全状态耦合相同振荡器网络局部同步的必要和充分条件。对于部分状态耦合，正耦合常数导致状态空间轨迹的渐近收缩，从而实现二维振荡器的同步。我们进一步扩展到非相同耦合的定向图网络，证明正耦合常数是同步的充分条件。这些理论结果通过数值模拟和实验验证。我们的结果有助于弥合理论推导的充分耦合强度与数值观测之间的差距。
英文摘要：
Determining conditions on the coupling strength for the synchronization in networks of interconnected oscillators is a challenging problem in nonlinear dynamics. While sophisticated mathematical methods have been used to derive conditions, these conditions are usually only sufficient and/ or based on numerical methods. We addressed the gap between the sufficient coupling strength and numerically observations using the Lyapunov-Floquet Theory and the Master Stability Function framework. We showed that a positive coupling strength is a necessary and sufficient condition for local synchronization in a network of identical oscillators coupled linearly and in full state fashion. For partial state coupling, we showed that a positive coupling constant results in an asymptotic contraction of the trajectories in the state space, which results in synchronisation for two-dimensional oscillators. We extended the results to networks with non-identical coupling over directed graphs and showed that positive coupling constants is a sufficient condition for synchronisation. These theoretical results are validated using numerical simulations and experimental implementations. Our results contribute to bridging the gap between the theoretically derived sufficient coupling strengths and the numerically observed ones.
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