Chaos in Bianchi Models| Chaos in Bianchi Models |
Vacuum Bianchi IX > s.a. chaos in gravitation;
minisuperspace quantum cosmology [quantum chaos].
* Qualitatively: From the
relation between the discrete approximation and the continuum dynamics,
the behavior is like a "punctuated integrability", with chaos
produced in short kicks at the bounces between Kasner epochs.
* Quantitatively: Lyapunov
exponents are problematic, because of the time-reparametrization problem (also,
numerical studies led to different conclusions, due in part to errors in the
simulations), so try different approach; Fractal methods concluded that the
system is chaotic; The Gauss map has Kolmogorov entropy h
= π2/(6 ln2 2).
@ Overview: Hobill, Burd, ... in Hobill et al ed-94;
Kamenshchik PU(10)-a1006.
@ Discrete dynamics vs continuum time:
Rugh & Jones PLA(90) [not chaotic];
Berger CQG(90),
GRG(91) [ADM, chaotic],
in(94);
Creighton & Hobill in(94);
Imponente & Montani PhyA(04)gq,
gq/04-MGX.
@ Discrete dynamics, chaotic:
Barrow PRL(81),
PRP(82);
Chernoff & Barrow PRL(83);
Barrow in(85);
Khalatnikov et al JSP(85);
Mayer PLA(87) [relaxation time];
Berger PRD(93).
@ Numerical, positive Lyapunov exponents:
Zardecki PRL(83) [BKL, Hamiltonian constraint violated];
Ferraz et al PLA(91);
Ferraz & Francisco PRD(92) [different def of time].
@ Numerical, zero Lyapunov exponents:
Francisco & Matsas GRG(88);
Burd et al GRG(90).
@ Analytical, zero Lyapunov exponents:
Hobill et al CQG(91).
@ Problem with the Lyapunov exponents:
Burd et al CQG(91),
in(91) [local Lyapunov exponents];
Pullin in(91).
@ Geometrical methods:
Di Bari & Cipriani in(00)gq/98 [Finsler geometry];
Imponente & Montani IJMPD(03)gq/01;
Montani & Benini PRD(04)gq;
Benini & Montani IJMPA(08)-proc.
@ Analogies: Pavlov gq/95 [generalized Toda];
Graham gq/94,
Imponente & Montani PRD(01)ap [as billiard].
@ As geodesic flow:
Szydłowski & Łapeta PLA(90);
Uggla et al PRD(90);
Szydłowski & Biesiada PRD(91);
Szydłowski & Szczesny PRD(94).
@ Painlevé, non-integrability:
Contopoulos et al JPA(93),
JPA(94),
JPA(95);
Cotsakis & Leach JPA(94);
Latifi et al PLA(94)gq;
Christiansen et al JPA(95);
Scheen & Demaret CQG(96).
@ Fractal basins of attraction: Cornish & Levin PRL(97)gq/96,
PRD(97)gq/96,
gq/97-MG8;
Motter & Letelier PLA(01)gq/00.
@ Related topics: Demaret & De Rop PLB(93) [fractal power spectrum];
Cushman & Śniatycki RPMP(95) [local integrability];
Imponente & Montani NPPS(02)gq/01,
IJMPD(03)gq/04 [and quantum gravity];
Andrianopoulos & Leach JPA(08);
Battisti & Montani PLB(09)-a0808 [and generalized uncertainty principle].
Bianchi IX with Matter and in Other Theories
> s.a. bianchi IX models.
@ With matter: Bruni & Sopuerta CQG(03)gq
[fluid, role of Hab];
Fay & Lehner GRG(05) [massive scalar].
@ With matter and cosmological constant: de Oliveira et al PRD(97)gq,
gq/97-MG8,
PRD(02)gq,
Soares & Stuchi PRD(05);
Corrêa et al PRD(10)-a1005 [homoclinic].
@ In Brans-Dicke theory: Carretero-González et al PLA(94);
Scheen & Demaret CQG(96).
@ In higher dimensions:
Barrow & Stein-Schabes PRD(85);
Demaret et al PLB(86),
PLB(88);
Helmi & Vucetich PLA(95) [Kaluza-Klein];
Halpern GRG(03)gq/02 [5D, no chaos].
@ Other theories: Spindel & Zinque IJMPD(93) [higher-derivative];
Erickson et al PRD(04)ht/03 [stringy, w > 1];
Di Menza & Lehner GRG(04) [scalar-tensor, suppression of chaos];
Kim & Kawai PRD(13)-a1301 [Gauss-Bonnet gravity];
Moriconi et al a1411
[R + qR 2, chaos removal].
Other Bianchi Models > s.a. bianchi
models; born-infeld theory [I, Einstein-Yang-Mills].
@ General references:
Jantzen PRD(86) [Einstein-Maxwell-scalar];
de Buyl et al CQG(03) [Einstein billiards];
Jin & Maeda PRD(05)gq/04 [with Yang-Mills field];
Larena & Perez CQG(07)-a0706 [integrability in scalar-tensor gravity, based on Kovalewski exponents].
@ V: in Rebouças et al GRG(98)gq.
@ VI: Berger CQG(96)gq/95 [magnetic VI0];
LeBlanc et al CQG(95) [magnetic VI0].
@ VIII: Halpern GRG(87);
Graham gq/94 [including quantum];
Barrow & Gaspar CQG(01) [far future];
Maciejewski et al JMP(01) [non-integrability];
Gaspar GRG(04)gq.
@ Higher-dimensional homogeneous cosmologies: Benini et al gq/07-MGXI [and vector fields].
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