切空间的双曲解耦与耗散系统的有效维数
Hyperbolic decoupling of tangent space and effective dimension of dissipative systems
摘要：
我们表明，使用协变李雅普诺夫向量，空间扩展耗散系统的切空间被分裂成两个双曲解耦的子空间：一个包括有限数量的频繁纠缠的“物理”模式，其中携带的物理相关信息的轨迹，和一组残留的强烈衰减的“假”模式。物理子空间和伪子空间的解耦是由它们之间没有相切来定义的，并且通常会发生;我们在一维和二维空间的偏微分方程中找到了证据，甚至在耦合映射或振荡器的晶格中也找到了证据。我们推测，物理模式可以构成一个局部的线性描述的惯性流形在任何一点的整体吸引。
英文摘要：
We show, using covariant Lyapunov vectors, that the tangent space of spatially-extended dissipative systems is split into two hyperbolically decoupled subspaces: one comprising a finite number of frequently entangled "physical" modes, which carry the physically relevant information of the trajectory, and a residual set of strongly decaying "spurious" modes. The decoupling of the physical and spurious subspaces is defined by the absence of tangencies between them and found to take place generally; we find evidence in partial differential equations in one and two spatial dimensions and even in lattices of coupled maps or oscillators. We conjecture that the physical modes may constitute a local linear description of the inertial manifold at any point in the global attractor.
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