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' 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].
@ Curvature and geodesic deviation: Szydlowski PLA(95) [N-body]; Di Bari & Cipriani PSS(98)cd/97; Szczesny & Dobrowolski AP(99).
@ 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-in [Tsallis].
@ And observation: Kossakowski et al OSID(03)qp/04 [information theory].

Types of Chaos > s.a. Attractor; 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 ito Riemann surfaces]; Horwitz et al phy/07 [ito 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]; phase space [stochastic layer]; 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.
@ General references: Zaslavsky et al 91; Reichl 92; Buric et al JPA(94); Haller 99 [near resonance]; Chandre & Jauslin PRP(02) [and renormalization].
@ Homoclinic orbits: Glendinning & Laing PLA(96) [types, examples]; Grotta Ragazzo PLA(97) [and diffusion]; Yagasaki PLA(01) [infinite-dimensional systems].
@ 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 infty]; Bricmont et al mp/01 [for field theory]; Roy mp/05 [geometrical].

Other Concepts and Techniques > s.a. correlations; Poincaré Recurrence; Poincaré Section.
@ 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]; Mrozek & Wójcik T&A(05) [geometric, discrete systems]; Contopoulos & Harsoula a0802 [stickiness].

General References > s.a. classical mechanics; irreversibility; statistical mechanics.
@ II, texts: Bergé et al 86; Cuerno et al AJP(92); Shinbrot et al AJP(92); Devaney 92; Field & Golubitsky 92; Moon 92; Abarbanel et al 93; Hilborn 94; Nagashima & Baba 98.
@ III, reviews: Eckmann & Ruelle RMP(85); Amann et al ed-88; McCauley PS(88); Cvitanovic 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 page – Abbreviations – Journals – Comments – Other sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified 25 may 2008