Mathematical Description of Chaos| 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];
Gidea & de la Llave a1710 [general theory].
Other Concepts and Techniques > s.a. computational physics.
@ 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].
> Related topics:
see Catastrophe Theory; correlations;
Poincaré Recurrence; Poincaré
Section; Quasiperiodic Functions.
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.
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