相互作用非线性随机微分方程的亚稳定性I：从弱耦合到同步
Metastability in Interacting Nonlinear Stochastic Differential Equations I: From Weak Coupling to Synchronisation
摘要：
研究了噪声作用下N个耦合过阻尼粒子周期链的动力学行为。每一个粒子都受到一个局域势的作用，受到与其最近邻粒子的线性耦合，以及受到一个白噪声的独立信源的作用。我们发现，随着耦合强度的增加，系统的平衡点的数目从3 ^N变为3。对于弱耦合，系统的行为类似于具有自旋翻转动力学的伊辛模型，而对于强耦合（阶数为N^2），它是同步的，因为所有振子在大部分时间里在各自的局域势中占据几乎相同的位置。我们推导出指数渐近的过渡时间，并描述同步状态之间的最可能的过渡路径，特别是耦合强度低于同步阈值。我们的技术涉及一个中心流形分析的deskenisation分叉，精确控制的稳定性分叉的解决方案，使我们能够给出一个详细的描述系统的潜在景观，其中亚稳态行为编码。
英文摘要：
We consider the dynamics of a periodic chain of N coupled overdamped particles under the influence of noise. Each particle is subjected to a bistable local potential, to a linear coupling with its nearest neighbours, and to an independent source of white noise. We show that as the coupling strength increases, the number of equilibrium points of the system changes from 3^N to 3. While for weak coupling, the system behaves like an Ising model with spin-flip dynamics, for strong coupling (of the order N^2), it synchronises, in the sense that all oscillators assume almost the same position in their respective local potential most of the time. We derive the exponential asymptotics for the transition times, and describe the most probable transition paths between synchronised states, in particular for coupling intensities below the synchronisation threshold. Our techniques involve a centre-manifold analysis of the desynchronisation bifurcation, with a precise control of the stability of bifurcating solutions, allowing us to give a detailed description of the system's potential landscape, in which the metastable behaviour is encoded.
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