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];
Goodson 16 [mathematical].

@ __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];
Nakagawa et al JHEP(18)-a1805 [and relative entropy];
Xu et al PRL(20)-a1912 [scrambling does not imply chaos].

> __And fundamental physics__:
see information; QED; topological field theories.

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