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节律行为, 即系统行为呈现随时间的周期变化, 在我们的周围随处可见. 不同节律之间可以通过相互影响、相互作用产生自组织, 其中同步是最典型、最直接的有序行为, 它也是非线性波、斑图、集群行为等的物理内在机制. 不同的节律可以用具有不同频率的振子(极限环)来刻画, 它们之间的同步可以用耦合极限环系统的动力学来加以研究. 微观动力学表明, 随着耦合强度增强, 振子同步伴随着动力学状态空间降维到一个低维子空间, 该空间由序参量来描述. 序参量的涌现及其所描述的宏观动力学行为可借助于协同学与流形理论等降维思想来进行. 本文从统计物理学的角度讨论了耦合振子系统序参量涌现的几种降维方案, 并对它们进行了对比分析. 序参量理论可有效应用于耦合振子系统的同步自组织与相变现象的分析, 通过进一步研究序参量的动力学及其分岔行为, 可以对复杂系统的涌现动力学有更为深刻的理解.Rhythmic behaviors, i.e. temporally periodic oscillations in a system, can be ubiquitously found in nature. Interactions among various rhythms can lead to self-organized behaviors and synchronizations. This mechanism is also responsible for many phenomena such as nonlinear waves, spatiotemporal patterns, and collective behaviors in populations emerging in complex systems. Mathematically different oscillations are described by limit-cycle oscillators (pacemakers) with different intrinsic frequencies, and the synchrony of these units can be described by the dynamics of coupled oscillators. Studies of microscopic dynamics reveal that the emergence of synchronization manifests itself as the dimension reduction of phase space, indicating that synchrony can be considered as no-equilibrium phase transition and can be described in terms of order parameters. The emergence of order parameters can be theoretically explored based on the synergetic theory, central manifold theorem and statistical physics. In this paper, we discuss the order-parameter theory of synchronization in terms of statistical physics and set up the dynamical equations of order parameters. We also apply this theory to studying the nonlinear dynamics and bifurcation of order parameters in several typical coupled oscillator systems.
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Keywords:
- synchronization/
- order parameter/
- emergence/
- bifurcation
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