Orthogonal versus covariant Lyapunov vectors for rough hard disk systems

TL;DR

This study compares covariant and orthogonal Lyapunov vectors in rough and smooth hard disk systems, revealing how disk rotation influences tangent space structure, hyperbolicity, and the Hamiltonian property.

Contribution

It provides the first detailed analysis of covariant Lyapunov vectors in rough hard disk systems and compares them to smooth systems, highlighting the effects of rotation on tangent space structure.

Findings

Both systems are hyperbolic with transverse stable and unstable manifolds.

Rotation affects the orthogonality of the central manifold in rough disks.

Rotation destroys the Hamiltonian structure in rough hard disk systems.

Abstract

The Oseledec splitting of the tangent space into covariant subspaces for a hyperbolic dynamical system is numerically accessible by computing the full set of covariant Lyapunov vectors. In this paper, the covariant Lyapunov vectors, the orthogonal Gram-Schmidt vectors, and the corresponding local (time-dependent) Lyapunov exponents, are analyzed for a planar system of rough hard disks (RHDS). These results are compared to respective results for a smooth-hard-disk system (SHDS). We find that the rotation of the disks deeply affects the Oseledec splitting and the structure of the tangent space. For both the smooth and rough hard disks, the stable, unstable and central manifolds are transverse to each other, although the minimal angle between the unstable and stable manifolds of the RHDS typically is very small. Both systems are hyperbolic. However, the central manifold is precisely orthogonal to the rest of the tangent space only for the smooth-particle case and not for the rough disks. We also demonstrate that the rotations destroy the Hamiltonian character for the rough-hard-disk system.