Lyapunov Exponents| Lyapunov Exponents |
In General > s.a. chaos; entropy;
Mixing System; non-extensive entropy.
* 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
λ:= limt → ∞ (1/t) ln(d(t)/d(0)) , meaning d(t) ~ d(0) exp{λt} as t → ∞ .
* Time scale: The time
tL:= 1/λ 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:
Ginelli et al PRL(07) [covariant Lyapunov vectors];
Motter & Saa PRL(09)-a0905 [relativistic invariance];
Slipantschuk et al JPA(13) [and exponential decay of correlations];
Cencini & Ginelli ed-JPA(13)#25;
Viana 14,
Pikovsky & Politi 16 [textbook];
Wilkinson BAMS-a1608-conf [expository, why they are important].
> Online resources:
see MathWorld page;
Wikipedia page.
Applications and Special Topics > s.a. cellular
automaton; Kolmogorov-Sinai Entropy.
@ Calculation:
Wolf et al PhyD(85) [from time series];
Habib & Ryne PRL(95)cd/94 [symplectic];
Kandrup et al PRE(02)ap/01 [and microcanonical distribution];
Terzić & Kandrup PLA(03)ap/02 [estimate];
Stachowiak PhD(08)-a0810 [algorithm].
@ Test bodies in curved spacetime: Wu & Huang PLA(03)gq;
Wu et al PRD(06)-a1006;
> s.a. geodesics.
@ And general relativity dynamics: Motter PRL(03)gq [cosmological models];
> s.a. chaotic motion;
chaos in the gravitational field.
@ Other applications:
Gerlach a0901-proc [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-proc [based on Moyal phase space quantization];
Majewski & Marciniak JPA(06)qp/05;
Berenstein & García-García a1510 [upper bound on the Lyapunov exponents, and entanglement entropy growth rate].
@ 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];
Baptista et al PLA(11)
[information production and bound on the sum of positive exponents];
Akemann et al a1809 [universal local statistics];
Sutter et al a1905 [analytical upper and lower bounds];
> s.a. non-extensive statistics [Tsallis entropy].
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