Convergence of the Eckmann and Ruelle algorithm for the estimation of Liapunov exponents

We analyse the convergence conditions of the Eckmann and Ruelle algorithm (ERA) used to estimate the Liapunov exponents, for the tangent map, of an ergodic measure, invariant under a smooth dynamical system. We find sufficient conditions for this convergence that are related to those ensuring the convergence to the tangent map of the best linear $L^p$-fittings of the action of a mapping $f$ on small balls. Under such conditions, we show how to use ERA to obtain estimates of the Liapunov exponents, up to an arbitrary degree of accuracy. We propose an adaptation of ERA for the computation of Liapunov exponents in smooth manifolds, which allows us to avoid the problem of detecting the spurious exponents.We prove, for a Borel measurable dynamics $f$, the existence of Liapunov exponents for the function $S_r(x)$, mapping each point $x$ to the matrix of the best linear $L^p$-fitting of the action of $f$ on the closed ball of radius $r$ centred at $x$, and we show how to use ERA to get reliable estimates of the Liapunov exponents of $S_r$. We also propose a test for checking the differentiability of an empirically observed dynamics.

Chaos and Force Fluctuations in Frictional Dynamics

ABSTRACTA model is presented of a particle that interacts with two periodic potentials, representing two confining plates, one of which is externally driven. The model leads to various behaviors in the motion of the top driven plate: stick-slip, intermittent regime, characterized by force fluctuations, and two types of sliding above a critical driving velocity vc. Similar behaviors are typical of a broad range of systems including thin sheared liquids. A detailed analysis of the different regimes displays a transition between the stick-slip and the kinetic regimes, ω−2 power spectra of the force over a wide range of velocities below vc, and a decrease of the force fluctuations that follows (vc – v)½ for v < vc. The velocity dependent Liapunov exponents demonstrate that the stick-slip motion is characterized by a chaotic behavior of the top plate and the embedded particle. An extension of the model to an embedded chain is introduced and preliminary results are presented and confronted with the single particle case. The role of the internal excitations of the chain in frictional dynamics is discussed.