13 research outputs found
Theory and computation of covariant Lyapunov vectors
Lyapunov exponents are well-known characteristic numbers that describe growth
rates of perturbations applied to a trajectory of a dynamical system in
different state space directions. Covariant (or characteristic) Lyapunov
vectors indicate these directions. Though the concept of these vectors has been
known for a long time, they became practically computable only recently due to
algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and
by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in
covariant Lyapunov vectors and their wide range of potential applications, in
this article we summarize the available information related to Lyapunov vectors
and provide a detailed explanation of both the theoretical basics and numerical
algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The
angles between these vectors and the original covariant vectors are
norm-independent and can be considered as characteristic numbers. Moreover, we
present and study in detail an improved approach for computing covariant
Lyapunov vectors. Also we describe, how one can test for hyperbolicity of
chaotic dynamics without explicitly computing covariant vectors.Comment: 21 pages, 5 figure
Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: Testing absence of tangencies of stable and unstable manifolds for phase trajectories
Dynamical equations are formulated and a numerical study is provided for
self-oscillatory model systems based on the triple linkage hinge mechanism of
Thurston -- Weeks -- Hunt -- MacKay. We consider systems with holonomic
mechanical constraint of three rotators as well as systems, where three
rotators interact by potential forces. We present and discuss some quantitative
characteristics of the chaotic regimes (Lyapunov exponents, power spectrum).
Chaotic dynamics of the models we consider are associated with hyperbolic
attractors, at least, at relatively small supercriticality of the
self-oscillating modes; that follows from numerical analysis of the
distribution for angles of intersection of stable and unstable manifolds of
phase trajectories on the attractors. In systems based on rotators with
interacting potential the hyperbolicity is violated starting from a certain
level of excitation.Comment: 30 pages, 18 figure
Determination of scattering lengths from measurement of atom lifetime
The DIRAC experiment at CERN has achieved a sizeable production of
atoms and has significantly improved the precision on its lifetime
determination. From a sample of 21227 atomic pairs, a 4% measurement of the
S-wave scattering length difference
has been attained, providing an important test of Chiral Perturbation Theory.Comment: 6 pages, 6 figure