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]; > s.a. non-extensive statistics [Tsallis entropy].


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