Mathematical Description of Chaos  

In General > s.a. chaos; classical mechanics formalism [dynamical systems]; dimension.
* Statistical methods: Usually statistical methods and entropy are considered applicable only to fully-chaotic systems, but they may be applicable also at the edge of chaos by using Tsallis' non-extensive generalizations.
@ Methods: Sussman & Wisdom Sci(88)jul [power spectra]; Schack & Caves in(97)qp [sensitivity to perturbations]; Gilmore RMP(98) [topological, dissipative systems]; Korsch & Leyes NJP(02) [delocalization and measure on phase space]; Barrow & Levin n.CD/03 [test]; Gottwald & Melbourne PRS(04) [test]; Li & Zhang PLA(11) [for 2D Hamiltonian systems].
@ Curvature and geodesic deviation: Szydłowski PLA(95) [N-body]; Di Bari & Cipriani PSS(98)cd/97; Szczesny & Dobrowolski AP(99); Wu JGP(09).
@ Statistical properties: Ornstein & Weiss BAMS(91).
@ And entropy: Gu & Wang PLA(97); Latora et al PLA(00)cm/99 [rate of increase]; Lissia et al cm/05-proc [Tsallis].
@ And information theory: Kossakowski et al OSID(03)qp/04; Cafaro AIP(07)-a0810, CSF(09)-a0810, Cafaro & Ali EJTP(08)-a0810, PhyA(08)-a0810 [based on entropic dynamics]; Cafaro PhD(08)-a1601.

Types of Chaos > s.a. Attractor [strange attractors]; fractals [Cantor set].
* Degree of chaoticity: From weaker to stronger, ergodic – mixing – Kolmogorov.
* Origin of chaoticity: Local sensitive dependence on initial conditions, together with global folding of orbits; Can be continuously produced or in bursts (as in the billiard or Bianchi IX models).
* Indicators: In low dimensions, Poincaré sections are a good qualitative indicator; In general, features of the power spectrum (~ square amplitude of Fourier transform in time) of one dynamical variable or fractal basins of attraction can be used, but Lyapunov exponents are quantitatively better.
@ Hamiltonian systems: von Kempis & Lustfeld JPA(93); Zaslavsky Chaos(95) [and Maxwell's demon]; Tang & Boozer PLA(97); Kandrup PRE(97)ap [2D, geometrical]; Zaslavsky 04; Calogero et al JPA(05) [transition to irregular motion in terms of Riemann surfaces]; Horwitz et al PRL(07)phy [in terms of curvature of metric].
@ Indicators: Lukes-Gerakopoulos et al PhyA(08) [and Tsallis entropy, Average Power Law Exponent].

Soft / Perturbative Chaos > s.a. KAM Theorem; Mixing [including time scale]; Separatrix.
* Idea: When perturbing an integrable system, if chaos sets in, it starts from local instabilities:
- The KAM theorem says that for small perturbations most tori are undisturbed.
-
The Melnikov method finds some unstable places (homoclinic/heteroclinic orbits) where chaos might arise.
- The Lyapunv exponents say how fast trajectories diverge locally.
- Arnold diffusion is the stronger form of soft chaos.
* Stochastic web: A thread-like region of chaotic dynamics in phase space generated by weak perturbations, discovered by Arnold.
@ General references: Zaslavsky et al 91; Reichl 92; Burić et al JPA(94); Haller 99 [near resonance]; Chandre & Jauslin PRP(02) [and renormalization]; Nguyen Thu Lam & Kurchan JPA(14)-a1305 [integrable systems perturbed by stochastic noise].
@ Stochastic web: Soskin et al CP(10) [introductory review].
@ Stochastic layer, width: Tsiganis et al JPA(99) [driven pendulum]; Shevchenko PLA(08) [new estimation method].
@ Homoclinic orbits: Glendinning & Laing PLA(96) [types, examples]; Grotta Ragazzo PLA(97) [and diffusion]; Yagasaki PLA(01) [infinite-dimensional systems]; Dong & Lan PLA(14) [variational method]; > s.a. kerr spacetime.
@ Melnikov method: Bruhn PS(91) [higher dimensions]; Soto-Treviño & Kaper JMP(96) [higher-order]; Cicogna & Santoprete PLA(99) [non-hyperbolic points], JMP(00) [critical point at infinity]; Bricmont et al CMP(01)mp [for field theory]; Roy JGP(06)mp/05 [geometrical]; Castilho & Marchesin JMP(09) [practical use].

Other Concepts and Techniques > s.a. computational physics; correlations; Poincaré Recurrence; Poincaré Section; Quasiperiodic Functions.
@ General references: Skokos JPA(01) [alignment indices]; Pingel et al PRP(04) [stability transformation]; Saa AP(04)gq [limitations of local criteria]; in Goldfain CSF(04) [fractional derivatives and diffusion]; Contopoulos & Harsoula IJBC(08)-a0802 [stickiness].
@ Geometric criteria: Mrozek & Wójcik T&A(05) [discrete systems]; Li & Zhang JPA(10) [extension of HBLSL criterion and Dicke model]; Li & Zhang PLA(11) [using the potential energy surface].

General References > s.a. classical mechanics; irreversibility; statistical mechanics.
@ II, texts: Bergé et al 86; Cuerno et al AJP(92)jan; Shinbrot et al AJP(92)jun; Devaney 92; Moon 92; Abarbanel et al 93; Hilborn 94; Nagashima & Baba 98; Field & Golubitsky 09.
@ III, reviews: Eckmann & Ruelle RMP(85); Amann et al ed-88; McCauley PS(88); Cvitanović ed-89; Holmes PRP(90); Ruelle PRS(90); Roberts & Quispel PRP(92).
@ III, texts: Guckenheimer & Holmes 83; Hao 84; Zaslavsky 84; Sagdeev et al 88; Temam 88; Wiggins 88; Devaney 89; Rasband 89; Ruelle 89; Stewart 89; Arrowsmith & Place 90; Gutzwiller 90; Jackson 91; Froyland 92 [short]; Tufillaro et al 92 [including knots]; Wiggins 92 [chaotic transport]; Mullin ed-93; Sklar 94 [conceptual]; Nicolis 95; Schuster 95; Baker & Gollub 96; Martelli 99 [discrete systems]; Ott 02.


main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 3 feb 2016