Hyperbolic decoupling of tangent space and effective dimension of dissipative systems

TL;DR

This paper demonstrates that the tangent space of spatially-extended dissipative systems splits into two hyperbolically decoupled subspaces, distinguishing physically relevant modes from decaying spurious modes, with implications for understanding system dynamics.

Contribution

It introduces the concept of hyperbolic decoupling of tangent space into physical and spurious modes using covariant Lyapunov vectors, applicable across various spatially-extended systems.

Findings

Physical modes carry relevant information of the trajectory.

Decoupling is characterized by absence of tangencies between subspaces.

Evidence found in PDEs, lattices, and coupled oscillators.

Abstract

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.