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 = 
2/(6
ln2 2).
@ Overview: Hobill, Burd, ... in Hobill et al ed-94.
@ 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-in.
@ 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)-in.
@ Analogies: Pavlov gq/95 [generalized
Toda]; Graham gq/94,
Imponente & Montani
PRD(01)ap [as
billiard].
@ As geodesic flow: Szydlowski & Lapeta PLA(90); Uggla et al
PRD(90); Szydlowski & Biesiada PRD(91); Szydlowski & 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-in;
Motter & Letelier PLA(01)gq/00.
@ Related topics: Demaret & De Rop PLB(93)
[fractal power spectrum]; Cushman
& Sniatycki 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 [homoclinic];
Soares & Stuchi PRD(05).
@ In Brans-Dicke: 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].
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-in [and vector
fields].
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send feedback and suggestions to bombelli at olemiss.edu – modified 16
nov 2009