Chaotic Systems  

In General > s.a. phase transitions; yang-mills gauge theories.
@ And (quantum) field theory: Matinyan & Müller FP(97)ht/96, PRL(97) [quantum fluctuations]; Cvitanovic PhyA(00)n.CD.

Theoretical Models > s.a. dissipation; Jerk; network.
* Remark: Almost all conservative dynamical systems are at least partly chaotic.
* Discrete models: For example, the baker map, the Farey map and the Hénon map below.
* Continuous models: Particle motion (Billiard – chaotic with convex boundary; Motion in a potential – chaotic if the Gaussian curvature of the potential surface is negative); Gas (Hard spheres – see Sinai's theorem); Simple example

d3x/dt3 = –(dx/dt)3 + (d2x/dt2) (x + d2x/dt2) / (dx/dt) .

* Hénon map: An unstable map R2 → R2, given by (x, y) (1+y–ax2, bx); Its behavior depends crucially on the values of a and b; For an interesting case, look at a = 1.4, b = 0.3.
* Hénon-Heiles system: The potential can be written as

V(x, y) = m 2 [ (x2 + y2) + (x2y – y3)/a)] ;

For a = = m = 1, E [0, 1/10] the behavior is regular, E [1/10, 1/6] is chaotic, E > 1/6 is unbounded.
@ Particle motion: Bogomolny et al PRP(97) [geodesics on R < 0 manifold]; Cho & Kao PLA(03)n.CD/02 [spinning]; Müller a0802 [geodesics on genus g = 0 manifolds].
@ Discrete: Benedicks & Carleson AM(91) [Hénon]; Abraham et al 97; Waelbroeck & Zertuche JPA(99); > s.a. Baker Map, entropy [dynamical], Farey Map, Standard Map.
@ Hénon-Heiles: Hénon & Heiles AJ(64); Fordy PhyD(91); Vernov TMP(03)mp/02 [solutions]; > s.a. toda lattice.
@ Field theory: Latora & Bazeia IJMPA(99) [2 scalar fields]; Salasnich JMP(99) [homogeneous]; > s.a. yang-mills gauge theory.
@ Other models: Kawabe & Ohta PRA(90) [3 particles, Yukawa interaction]; Ginelli et al JPA(02) [linearly stable]; Hasegawa et al PLA(03) [Arnold cat]; Grammaticos et al PLA(05)mp/04 [solvable]; Chetrite et al JSP(07) [integrable, Kraichnan flow].

Computer Models and Calculations > s.a. computation; quantum computation.
* History: 1963, First model introduced by E Lorenz for weather prediction.
* Ubiquity: Chaos is (theoretically) exhibited not only by systems with many degrees of freedom or quantum systems, but also by macroscopic ones with few degrees of freedom, such as hydrodynamic flows near turbulence, mechanical oscillators, plasmas, etc.
@ Chaos and numerical methods: Corless et al PLA(91); Sprott AJP(08) [simple models].
@ Chaos introduced by approximations: Ge & Leng PLA(94).

Real Systems > s.a. Billiard; brownian motion; fractals; matter; oscillator; quantum chaos; types of measurement.
* Ubiquity: Found in the onset of fluid turbulence, chemical reactions, electrochemical and other special systems; For less well-controlled systems (full turbulent flow, biological systems, climate, ...) one can only infer chaotic behavior, although quantitative studies have been attempted; We believe chaos to be a universal feature.
* Specific examples: Dendritic growth, group decisions (& Meyer & Brown), snowflakes.
@ Dripping faucet: D'Innocenzo & Renna IJTP(96) [model]; Tufaile et al PLA(99) [simulations]; Reyes et al PLA(02) [heteroclinic]; Kiyono et al PLA(03); [> s.a. fluid].
@ Related topics: Levien & Tan AJP(93) [double pendulum undergraduate demo]; Kantz & Huggard AJP(94) [amusement park]; Sanders & Jensen AJP(96), AJP(96) [ionization of atoms]; Sprott PLA(00) [circuits]; DeSerio AJP(03) [chaotic pendulum]; Téi & Lai PRP(08) [spatiotemporal chaotic transients]; > s.a. turbulence.

In Astronomy, Gravitation and Cosmology > s.a. chaos in gravitation; newtonian orbits [three-body]; solar system [asteroids]; strings.
* History: This problem started the study of chaos in a way, with questions about the stability of the solar system.
* Results: Results of simulations show that the solar system, while chaotic, is not seriously unstable over time scales of billions of years.
@ Reviews: Contopoulos in(79); Gurzadyan ap/04-in [astrophysics/cosmology]; Regev a0705 [astrophysics].
@ Solar system objects: Sussman & Wisdom Sci(88)jul [Pluto]; Peterson 93; Lissauer RMP(99); Murray & Holman ap/99/Sci; Haghighipour JMP(02)ap/01 [partial averaging]; Lecar et al ARAA(01)ap [rev]; Murray & Holman Nat(01)ap; Quillen ap/02/ApJ [solar neighborhood]; Hayes ap/07 [outer solar system]; Batygin & Laughlin a0804-ApJ [stability]; news S&T(08)apr [Mercury instability].
@ Galaxies: Merritt ap/95-in, ap/96/Sci [elliptical]; Merritt & Valluri ap/97-in; Kandrup ap/00-in, ap/02-in, et al MNRAS(03)ap/02.

In Other Disciplines, Applications > s.a. computation.
* Meteorology: It is not just due to the winds, but to the double effect of clouds, both cooling and warming.
@ Medicine: BWest 90; Witkowski et al PRL(95) [heart attacks].
@ Ecology and evolution: May BAMS(95).

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 23 jun 2008