Lyapunov
Exponents| Lyapunov
Exponents |
In General > s.a. Cellular
Automaton; chaos; entropy;
Mixing System.
* Idea: A measure of
how fast nearby orbits in phase space starting from a given point converge
or diverge from each other; A dynamical system has as many
Lyapunov exponents
as the dimensionality of its phase space.
* Applications: Widely
used in celestial mechanics as chaos indicator to characterize the dynamical
behavior
of bodies.
$ Def: Given two orbits
initially separated by d(0) in phase space, the corresponding
Lyapunov exponent is
:=
limt to infty (1/t) ln(d(t)/d(0))
, meaning d(t)
d(0) exp{t} as t → 
.
* Time scale: The time tL:= 1/ over which nearby trajectories
separate by a factor of e.
* And attractors: Fixed-point
attractors yield all negative Lyapunov exponents, periodic orbits a zero one,
and strange attractors at least one positive one.
References
@ General: Ginelli et al PRL(07)
[covariant Lyapunov vectors]; Motter & Saa PRL(09)-a0905 [relativistic invariance].
@ Calculation: Wolf et al PhyD(85)
[from time series]; Habib & Ryne
cd/94 [symplectic];
Kandrup et al PRE(02)ap/01 [and
microcanonical
distribution]; Terzic & Kandrup
PLA(03)ap/02 [estimate];
Stachowiak a0810-PhD [algorithm].
@ Test bodies in curved spacetime: Wu & Huang PLA(03)gq;
Wu et al PRD(06).
@
And general relativity dynamics:
Motter
PRL(03)gq [cosmological
models]; > s.a. chaotic motion, chaos
in the gravitational field.
@ Other applications: Gerlach a0901-in
[asteroids, numerical].
@ In quantum mechanics: Man'ko & Vilela Mendes PhyD(00)qp [phase
space
approach]; Ballentine
PRA(01)
[for
classical-quantum
differences];
Falsaperla et al FP(02)qp/06;
Kondratieva & Osborn qp/05-in
[based on Moyal phase space quantization]; Majewski & Marciniak JPA(06)qp/05.
@ Finite-time: Aurell et al JPA(97); Szezech et al PLA(05) [and dynamical
traps of chaotic orbits].
@ Related topics: Ziehmann et al PLA(00) [local, and predictability]; Tanase-Nicola
& Kurchan JPA(03)
[statistical mechanics formulation].
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send feedback and suggestions to bombelli at olemiss.edu – modified 5 jul
2009