Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2011-11-25
Nonlinear Sciences
Chaotic Dynamics
31 pages, 10 figures
Scientific paper
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 Gram-Schmidt vectors, as well as 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 systems (SHDS). We establish that the rotation of the disks deeply affects the Oseledec splitting and the structure of the tangent space. For the rough hard disks, the localization of conjugate covariant vectors in physical space agrees, whereas it does not for conjugate Gram-Schmidt vectors. 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 unstable 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.
Bosetti Hadrien
Posch Harald A.
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