Chaos  

In General > s.a. complexity; physics paradigms; Predictability; quantum chaos; randomness; thermodynamics.
* Idea: Chaos is the study of (non-linear) dynamical systems with unstable behavior; Deterministic, but unpredictable and irregular for almost all initial conditions and almost all variations thereof – a small variation results in a totally different trajectory; The mechanism by which this happens is that the paths locally diverge exponentially, then fold back and mix; The motion resembles a random process, because its description requires a maximally long sequence of symbols.
* History: First observed by Hadamard in 1898 for geodesic flow in a constant negative curvature manifold; The consequences were understood by Duhem [@1906] and Poincaré [@1908]; It has now become the second holistic XX century innovation (the first one was quantum mechanics), questioning our mechanistic view.
* Remark: Many non-chaotic systems exhibit sensitive dependence on initial conditions, but only for unstable fixed points or unstable periodic orbits; Also, "chaos" sometimes stands for Boltzmann's molecular disorder (> see statistical mechanics).
* Characteristics: Universality, low-dimensionality, period doubling.
* Description: For few degrees of freedom, the transition from order to chaos is well described and understood, theoretically and experimentally; Separatrices are seeds of chaos when disturbances are added; Full turbulent behavior is not understood, but described phenomenologically in terms of fractals and strange attractors (dissipative systems).
@ I: Gleick 87; Von Baeyer ThSc(91)jul; Gutzwiller SA(92)jan [quantum]; Ruelle 92; Kellert 93 [conceptual]; Lorenz 93; Smith 07 [r JPA(07)].
@ II: Kadanoff PT(83)dec; Zabusky PT(84)jul; Chernikov et al PT(88)nov; Ornstein Sci(89)jan; Stewart NS(89)nov; Gaponov-Grekhov & Rabinovich PT(90)jul; PW(90)apr; NS(90)sep29, p49-52, NS(90)oct10; Hall 92; Tél & Gruiz 06; Kautz 10; Gulick 12; Letellier 13; Stetz 16 [including numerical].
@ Historical: Sinai JSP(10) [overview]; Shepelyansky PT(14)-a1306 [Chirikov's 1959 pioneering results]; Motter & Campbell PT(13)may [chaos at 50].
@ And randomness: Svozil PLA(89); Winnie PhSc(92)jun; Amigó et al PLA(06) [non-statistical test]; Caprara & Vulpiani in(16)-a1605 [and stochastic models].
@ Related topics: Iooss et al ed-83 [lectures]; Sprott PLA(93) [genericity]; Crisanti et al JPA(94) [changing parameters]; Antoniou & Suchanecki FP(94) [and logic]; Koperski BJPS(01) [conceptual]; Yahalom et al IJGMP-a1112 [necessary conditions]; Li a1305 [some open problems]; > s.a. Fermat's Last Theorem.
> Online resources: see The Chaos Hypertextbook.

Control
* Stochastic resonance: The amplification and optimization of feeble input with the assistance of noise.
@ References: Ditto & Pecora SA(93)aug; Peak & Frame 94; Ott & Spano PT(95)may; Gammaitoni et al RMP(98) [stochastic resonance]; Boccaletti et al PRP(00), PRP(02) [synchronization]; Bowong & Kakmeni PS(03) [stability and suration of synchronization]; Chacón 05 [of homoclinic chaos]; Gauthier AJP(03)aug [RL]; Vargas et al AJP(09)sep [bouncing ball].

Obtaining Information / Analysis of Chaotic Data
* Experimentally: Fluctuations in the evolution of the system are not always easy to distinguish from noise.
* Criteria: They include the measurement of the correlation dimension.
@ Data analysis: news Nat(90)oct; Ruelle PRS(90), PT(94)jul [criteria and criticism]; Abarbanel et al RMP(93); Ott et al 94; Olbrich & Kantz PLA(97), Xiaofeng & Lai JPA(00) [time series]; Abarbanel 97.
@ Related topics: Steeb et al JPA(94) [maximum entropy formalism]; in Kaplan & Glass 95 [phase space reconstruction, II]; Buchler cd/97-conf [global flow reconstruction method].

Other Effects and Topics > s.a. chaotic systems; mathematical description; statistical mechanics.
* Routes to chaos: The three universal routes to chaos displayed by the prototypical logistic and circle maps are period doubling, intermittency, and quasiperiodicity routes; In these situations the dynamical behavior is exactly describable through infinite families of Tsallis’ q-exponential functions.
* Universality: Behavior that is quantitatively identical for a broad class of systems; The first and most famous example is the period-doubling route to chaos.
@ Period doubling: Feigenbaum JSP(78), JSP(79); Coppersmith AJP(99)jan [Feigenbaum's renormalization group equation].
@ And transport: Zaslavsky PRP(02) [anomalous]; Vollmer PRP(02) [and non-equilibrium thermodynamics].
@ Related topics: Kandrup et al MNRAS(00)ap/99 [low-amplitude noise in Hamiltonian systems]; Firpo & Ruffo JPA(01) [suppression in large-size limit]; Zheng et al IJTP(03) [observer dependence]; Baldovin PhyA(06) [incipient chaos, routes, and glass formation].
> And fundamental physics: see information; QED; topological field theories.


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