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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