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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send feedback and suggestions to bombelli at olemiss.edu – modified 4
aug 2016