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Standard map

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Area-preserving chaotic map from a square with side 2π onto itself

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The phase-space of the standard map with the variation of the parameter $K {\displaystyle K}$ from 0 to 5.19 ( $p_{n}$ in y axes, $\theta _{n}$ in x axes). Notice the appearance of a "dotted" zone, a signature ofchaotic behavior.

{\displaystyle p_{n}} — The phase-space of the standard map with the variation of the parameter $K {\displaystyle K}$ from 0 to 5.19 ( $p_{n}$ in y axes, $\theta _{n}$ in x axes). Notice the appearance of a "dotted" zone, a signature ofchaotic behavior.

Orbits of the standard map forK = 0.971635.

Orbits of the standard map forK = 2.0. The large green region is the main chaotic region of the map.

A single orbit of the standard map forK=2.0. Magnified close-up centered at $\theta =0.282$ ,p = 0.666, of total width/height 0.02. Note the extremely uniform distribution of the orbit.

{\displaystyle \theta =0.282} — A single orbit of the standard map forK=2.0. Magnified close-up centered at $\theta =0.282$ ,p = 0.666, of total width/height 0.02. Note the extremely uniform distribution of the orbit.

Thestandard map (also known as theChirikov–Taylor map or as theChirikov standard map) is an area-preservingchaotic map from a square with side $2\pi$ onto itself.^[1] It is constructed by aPoincaré's surface of section of thekicked rotator, and is defined by:

p_{n+1}=p_{n}+K\sin(\theta _{n})

\theta _{n+1}=\theta _{n}+p_{n+1}

where $p_{n}$ and $\theta _{n}$ are taken modulo $2\pi$ .

The properties of chaos of the standard map were established byBoris Chirikov in 1969.

Physical model

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This map describes thePoincaré's surface of section of the motion of a simple mechanical system known as thekicked rotator. The kicked rotator consists of a stick that is free of the gravitational force, which can rotate frictionlessly in a plane around an axis located in one of its tips, and which is periodically kicked on the other tip.

The standard map is a surface of section applied by astroboscopic projection on the variables of the kicked rotator.^[1] The variables $\theta _{n}$ and $p_{n}$ respectively determine the angular position of the stick and its angular momentum after then-th kick. The constantK measures the intensity of the kicks on the kicked rotator.

Thekicked rotator approximates systems studied in the fields ofmechanics of particles,accelerator physics,plasma physics, andsolid state physics. For example, circularparticle accelerators accelerate particles by applying periodic kicks, as they circulate in the beam tube. Thus, the structure of the beam can be approximated by the kicked rotor. However, this map is interesting from a fundamental point of view in physics and mathematics because it is a very simple model of a conservative system that displaysHamiltonian chaos. It is therefore useful to study the development of chaos in this kind of system.

Main properties

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For $K=0$ the map is linear and only periodic and quasiperiodicorbits are possible. When plotted inphase space (the θ–p plane), periodic orbits appear as closed curves, and quasiperiodic orbits as necklaces of closed curves whose centers lie in another larger closed curve. Which type of orbit is observed depends on the map's initial conditions.

Nonlinearity of the map increases withK, and with it the possibility to observechaotic dynamics for appropriate initial conditions. This is illustrated in the figure, which displays a collection of different orbits allowed to the standard map for various values of $K>0$ . All the orbits shown are periodic or quasiperiodic, with the exception of the green one that is chaotic and develops in a large region of phase space as an apparently random set of points. Particularly remarkable is the extreme uniformity of the distribution in the chaotic region, although this can be deceptive: even within the chaotic regions, there are an infinite number of diminishingly small islands that are never visited during iteration, as shown in the close-up.

Circle map

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The standard map is related to thecircle map, which has a single, similar iterated equation:

\theta _{n+1}=\theta _{n}+\Omega -K\sin(\theta _{n})

as compared to

\theta _{n+1}=\theta _{n}+p_{n}+K\sin(\theta _{n})

p_{n+1}=\theta _{n+1}-\theta _{n}

for the standard map, the equations reordered to emphasize similarity. In essence, the circle map forces the momentum to a constant.

Notes

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^^a ^bOtt, Edward (2002).Chaos in Dynamical Systems. Cambridge University Press New, York.ISBN 0-521-01084-5.

References

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Chirikov, B.V.Research concerning the theory of nonlinear resonance and stochasticity. Preprint N 267, Institute of Nuclear Physics, Novosibirsk (1969) (in Russian) [Engl. Transl., CERN Trans. 71 - 40, Geneva, October (1971), Translated by A.T.Sanders].link
Chirikov, B.V.A universal instability of many-dimensional oscillator systems. Phys. Rep. v.52. p.263 (1979) Elsevier, Amsterdam.
Lichtenberg, A.J. & Lieberman, M.A. (1992).Regular and Chaotic Dynamics. Springer, Berlin.ISBN 978-0-387-97745-4.Springer link
Ott, Edward (2002).Chaos in Dynamical Systems. Cambridge University Press New, York.ISBN 0-521-01084-5.
Sprott, Julien Clinton (2003).Chaos and Time-Series Analysis. Oxford University Press.ISBN 0-19-850840-9.

External links

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Standard map atMathWorld
Chirikov standard map atScholarpedia
Website dedicated to Boris Chirikov
Interactive Java Applet visualizing orbits of the Standard Map, by Achim Luhn
Mac Application for the Standard Map, by James Meiss
Interactive Javascript Applet Standard Map on experiences.math.cnrs.fr

Chaos theory

Concepts

Core	Attractor Bifurcation Fractal Limit set Lyapunov exponent Orbit Periodic point Phase space
Anosov diffeomorphism Arnold tongue axiom A dynamical system Bifurcation diagram Box-counting dimension Correlation dimension Conservative system Ergodicity False nearest neighbors Hausdorff dimension Invariant measure Lyapunov stability Measure-preserving dynamical system Mixing Poincaré section Recurrence plot SRB measure Stable manifold Topological conjugacy
Theorems	Ergodic theorem Liouville's theorem Krylov–Bogolyubov theorem Poincaré–Bendixson theorem Poincaré recurrence theorem Stable manifold theorem Takens's theorem

Theoretical
branches

Chaotic
maps (list)

Discrete	Arnold's cat map Baker's map Complex quadratic map Coupled map lattice Duffing map Dyadic transformation Dynamical billiards outer Exponential map Gauss map Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Irrational rotation Kaplan–Yorke map Langton's ant Logistic map Standard map Tent map Tinkerbell map Zaslavskii map
Continuous	Double scroll attractor Duffing equation Lorenz system Lotka–Volterra equations Mackey–Glass equations Rabinovich–Fabrikant equations Rössler attractor Three-body problem Van der Pol oscillator