1,260 research outputs found
How to test for partially predictable chaos
For a chaotic system pairs of initially close-by trajectories become
eventually fully uncorrelated on the attracting set. This process of
decorrelation may split into an initial exponential decrease, characterized by
the maximal Lyapunov exponent, and a subsequent diffusive process on the
chaotic attractor causing the final loss of predictability. The time scales of
both processes can be either of the same or of very different orders of
magnitude. In the latter case the two trajectories linger within a finite but
small distance (with respect to the overall extent of the attractor) for
exceedingly long times and therefore remain partially predictable.
Tests for distinguishing chaos from laminar flow widely use the time
evolution of inter-orbital correlations as an indicator. Standard tests however
yield mostly ambiguous results when it comes to distinguish partially
predictable chaos and laminar flow, which are characterized respectively by
attractors of fractally broadened braids and limit cycles. For a resolution we
introduce a novel 0-1 indicator for chaos based on the cross-distance scaling
of pairs of initially close trajectories, showing that this test robustly
discriminates chaos, including partially predictable chaos, from laminar flow.
One can use furthermore the finite time cross-correlation of pairs of initially
close trajectories to distinguish, for a complete classification, also between
strong and partially predictable chaos. We are thus able to identify laminar
flow as well as strong and partially predictable chaos in a 0-1 manner solely
from the properties of pairs of trajectories.Comment: 14 pages, 9 figure
Chaotic Scattering Theory, Thermodynamic Formalism, and Transport Coefficients
The foundations of the chaotic scattering theory for transport and
reaction-rate coefficients for classical many-body systems are considered here
in some detail. The thermodynamic formalism of Sinai, Bowen, and Ruelle is
employed to obtain an expression for the escape-rate for a phase space
trajectory to leave a finite open region of phase space for the first time.
This expression relates the escape rate to the difference between the sum of
the positive Lyapunov exponents and the K-S entropy for the fractal set of
trajectories which are trapped forever in the open region. This result is well
known for systems of a few degrees of freedom and is here extended to systems
of many degrees of freedom. The formalism is applied to smooth hyperbolic
systems, to cellular-automata lattice gases, and to hard sphere sytems. In the
latter case, the goemetric constructions of Sinai {\it et al} for billiard
systems are used to describe the relevant chaotic scattering phenomena. Some
applications of this formalism to non-hyperbolic systems are also discussed.Comment: 35 pages, compressed file, follow directions in header for ps file.
Figures are available on request from [email protected]
On the generalized spectral subradius
AbstractIn this note we study the concepts of generalized spectral subradius and joint spectral subradius of a family of matrices. We show that both these numbers are equal and we present two different formulas for them. We also explain the relation between them and maximal Lyapunov exponent of discrete linear time varying system. Finally we show that the spectral subradius is less than one if and only if a discrete linear inclusion is stable in a certain sense
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