Chaotic Systems| Chaotic Systems |
In General, Theoretical Models
> s.a. chaos / Jerk;
network; phase transitions.
* Remark: Almost all
conservative dynamical systems are at least partly chaotic.
* Discrete models: For
example, the baker map, the Farey map and the Hénon map.
* Continuous models: Particle motion
(Billiard – chaotic with convex boundary; Motion in a potential – chaotic
if the Gaussian curvature of the potential surface is negative); Gas (Hard spheres –
see Sinai's theorem); Simple example
d3x/dt3 = −(dx/dt)3 + (d2x/dt2) (x + d2x/dt2) / (dx/dt) .
@ Discrete: Abraham et al 97;
Waelbroeck & Zertuche JPA(99);
Sogo & Masumizu PLA(11) [2D exactly solvable chaotic maps];
> s.a. Baker Map; entropy [dynamical];
Farey Map; Hénon Map;
Standard Map.
@ Particle motion:
Bogomolny et al PRP(97) [geodesics on R < 0 manifold];
Cho & Kao PLA(03)n.CD/02 [spinning];
Müller PLA(12)-a0802 [geodesics on genus g = 0 manifolds];
> s.a. Hénon-Heiles System.
@ Spin models:
Parisi & Rizzo JPA(10) [temperature chaos in random-lattice mean-field spin-glass models];
Santos et al PRE(12)-a1201 [isolated systems of interacting spins, chaos and relaxation];
Bodneva & Lundin TMP(14).
@ Other models: Kawabe & Ohta PRA(90) [3 particles, Yukawa interaction];
Ginelli et al JPA(02) [linearly stable];
Hasegawa et al PLA(03) [Arnold cat];
Grammaticos et al PLA(05)mp/04 [solvable];
Chetrite et al JSP(07) [integrable, Kraichnan flow];
Miritello et al PhyA(09)
[Kuramoto model, at the "edge of chaos"];
Sprott 10 [simple systems];
Yao in(10)-a1104 [non-linear differential equations];
Munmuangsaen et al PLA(11);
Groff AJP(13)oct
[logistic map activities as a model in introductory physics].
> Other systems:
see chaos in field theories and gravitational systems [including the solar system];
dissipative systems [doubly transient chaos].
Computer Models and Calculations
> s.a. computational physics; quantum computation.
* History:
1963, First model introduced by E Lorenz for weather prediction.
* Ubiquity: Chaos is (theoretically)
exhibited not only by systems with many degrees of freedom or quantum systems, but
also by macroscopic ones with few degrees of freedom, such as hydrodynamic flows
near turbulence, mechanical oscillators, plasmas, etc.
@ Chaos and numerical methods:
Corless et al PLA(91);
Sprott AJP(08)apr [simple models].
@ Chaos introduced by approximations:
Ge & Leng PLA(94).
Real Systems > s.a. Billiard; brownian motion;
fractals; matter; oscillator;
quantum chaos; types of measurement.
* Ubiquity: Found in the onset of
fluid turbulence, chemical reactions, electrochemical and other special systems; For
less well-controlled systems (full turbulent flow, biological systems, climate, ...)
one can only infer chaotic behavior, although quantitative studies have been attempted;
We believe chaos to be a universal feature.
* Specific examples:
Dendritic growth, group decisions (& Meyer & Brown), snowflakes.
@ Dripping faucet:
D'Innocenzo & Renna IJTP(96);
Tufaile et al PLA(99) [simulations]
Reyes et al PLA(02) [heteroclinic];
Kiyono et al PLA(03);
> s.a. fluids.
@ Related topics: Levien & Tan AJP(93)nov [double pendulum undergraduate demo];
Kantz & Huggard AJP(94)jan [amusement park];
Sanders & Jensen AJP(96)jan,
AJP(96)aug [ionization of atoms];
Sprott PLA(00) [circuits];
DeSerio AJP(03)mar [chaotic pendulum];
Téi & Lai PRP(08) [spatiotemporal chaotic transients];
Strzalko et al PRP(08) [coin tossing is not chaotic];
> s.a. turbulence.
Chaotic Quantum Systems
> s.a. foundations of quantum mechanics; quantum systems;
semiclassical effects; spectral geometry.
@ General references: Roberts & Muzykantskii JPA(00) [mixing, spectral decompositions];
Dabaghian & Jensen EJP(05) [in elementary quantum mechanics];
Rozenbaum et al a1902 [classically non-chaotic systems];
Gharibyan et al PRE-a1902
[characterization by 2-point correlation functions];
Ali et al a1905 [and complexity time scale];
Lando & Ozorio de Almeida a1907 [semiclassical evolution].
@ Anharmonic oscillator: Adamyan et al PRE(01)qp;
Caron et al PLA(01)qp [at finite T],
PLA(04),
JPA(04)qp [and classical].
@ Quantum billiard:
Liboff PLA(00);
Primack & Smilansky PRP(00) [semiclassical];
Bies et al JPA(03)qp/02 [2-point correlations];
Salazar & Tellez EJP(12)-a1202 [non-planar billiards].
@ In atomic physics:
Monteiro CP(94).
@ In discrete systems: Smilansky JPA(07)-a0704 [on graphs];
Gubin & Santos AJP(12)mar [chains of interacting spin-1/2 particles];
Mourik et al PRE(18)-a1703 [single nuclear spin];
> s.a. graphs [quantum].
@ And quantum computers: Kim & Mahler PLA(99)qp;
Berman et al qp/01,
PRE(01)qp.
@ Many-body systems / materials: Jona-Lasinio & Presilla PRL(96)cm [thermodynamic limit];
Gómez et al PRP(11) [and applications to nuclei];
Graefe Phy(13) [spin-orbit-coupled atomic gases];
Xu et al PT(21)feb [graphene, relativistic Dirac equation];
> s.a. many-body systems [information scrambling].
@ Other types of systems:
Spiller & Ralph PLA(94) [open system];
Dabaghian et al JETPL(01)qp [solvable spectra in 1D];
Dabaghian qp/04 [familiar example];
Ullmo RPP(08) [mesoscopic and nanoscopic systems];
Lemos et al nComm(12)-a1207 [in a beam of light];
> s.a. Baker Map; bianchi models;
chaos in field theory and astronomy; spin models.
In Other Disciplines, Applications > s.a. computation.
* Meteorology: It is not just due to the winds,
but to the double effect of clouds, both cooling and warming.
@ References: May BAMS(95) [ecology and evolution];
Witkowski et al PRL(95) [heart attacks].
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