Hydrodynamic Lyapunov modes and effective degrees of freedom of extended systems
Hydrodynamic Lyapunov modes and effective degrees of freedom of extended systems
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DOI：
10.1088/1751-8113/46/25/254015
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发表时间：
2013-06
期刊：
影响因子：
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通讯作者：
Hong-liu Yang;G. Radons
中科院分区：
文献类型：
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作者：
Hong-liu Yang;G. Radons
本文综述了作者对扩展动力系统的李雅普诺夫分析的贡献。流体动力学李雅普诺夫模（HydrodynamicLyapunov Modes，HLM）是一类特殊的李雅普诺夫向量，它具有长波长结构和慢振荡特性，在硬球系统、动力学XY模型、Lennard-Jones流体、耦合映象格子和偏微分方程等具有连续对称性的扩展系统中被观测到.它们对于非线性动力学及其粗粒度描述之间的联系具有潜在的重要性。本文的第一部分回顾了我们最近关于扩展系统中李雅普诺夫模的一些结果，包括哈密顿系统和耗散系统中HLM的普适性，双原子哈密顿系统的李雅普诺夫谱中出现有意义或“模糊”模的条件以及出现分支分裂的条件。特别是，大多数通过正交LV获得的HLM的结果已检查使用协变LV。从这些研究中出现的一幅图景是，哈密顿系统中的HLM可以被看作是谐波系统的正常模式到非线性混沌系统的推广。第二部分讨论了耗散偏微分方程的双曲性和有效自由度。利用协变的LV，在这些无限维系统中发现了两组LV之间的双曲分离。被称为“物理模”的相互纠缠的LV的有限集合被构造为跨越惯性流形的局部线性近似，惯性流形是包含原始PDE系统的物理相关动力学的有限维光滑流形。超越李雅普诺夫分析，提出了一种投影方法来探测惯性流形的几何结构，这使得我们能够提供耗散偏微分方程的物理模式和惯性流形之间的直接关系.本文是《物理学杂志A：数学与理论》特刊的一部分，专门讨论“李雅普诺夫分析：从动力系统理论到应用”。
This work reviews the authors’ contributions to the Lyapunov analysis of extended dynamical systems. Hydrodynamic Lyapunov modes (HLMs), the special Lyapunov vectors (LVs) associated with near-zero Lyapunov exponents exhibiting long wavelength structures and slow oscillations, have recently been observed in many extended systems with continuous symmetry, such as hard sphere systems, dynamic XY models, Lennard-Jones fluids, coupled map lattices and partial differential equations. They are of potential importance for the connection between nonlinear dynamics and its coarse-grained description. In the first part of this paper, we review our recent results on Lyapunov modes in an extended system, which includes the universality of HLMs in Hamiltonian and dissipative systems, the condition for the appearance of significant or ‘vague’ modes and the appearance of branch splitting in the Lyapunov spectra of diatomic Hamiltonian systems. In particular, most results on HLMs obtained via orthogonal LVs have been checked by using covariant LVs. An emerging picture from these studies is that HLMs in Hamiltonian systems can be viewed as the generalization of normal modes of harmonic systems to nonlinear, chaotic systems. The second part is devoted to the hyperbolicity and the effective degrees of freedom of dissipative partial differential equations. By using covariant LVs, a hyperbolic separation between two sets of LVs was found in these infinite-dimensional systems. The finite set of mutually entangled LVs, named ‘physical modes’, was conjectured to span a local linear approximation of the inertial manifold, a finite-dimensional smooth manifold containing the physically relevant dynamics of the original PDE system. Going beyond the Lyapunov analysis, a projection method was proposed to probe the geometric structures of the inertial manifold, which enables us to provide a direct relation between the physical modes and the inertial manifold of dissipative PDEs. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’.