Standard map
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The standard map (also known as the Chirikov–Taylor map or as the Chirikov standard map) is an area-preserving chaotic map from a square with side 2\pi onto itself. It is constructed by a Poincaré's surface of section of the kicked rotator, and is defined by: :p_ = p_n + K \sin(\theta_n) :\theta_ = \theta_n + p_ where p_n and \theta_n are taken modulo 2\pi. The properties of chaos of the standard map were established by Boris Chirikov in 1969.


Physical model

This map describes the Poincaré's surface of section of the motion of a simple mechanical system known as the kicked 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 a stroboscopic projection on the variables of the kicked rotator. The variables \theta_n and p_n respectively determine the angular position of the stick and its angular momentum after the ''n''-th kick. The constant ''K'' measures the intensity of the kicks on the kicked rotator. The kicked rotator approximates systems studied in the fields of
mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...
of particles,
accelerator physics Accelerator physics is a branch of applied physics, concerned with designing, building and operating particle accelerators. As such, it can be described as the study of motion, manipulation and observation of relativistic charged particle beams ...
,
plasma physics Plasma () is a state of matter characterized by the presence of a significant portion of charged particles in any combination of ions or electrons. It is the most abundant form of ordinary matter in the universe, mostly in stars (including th ...
, and
solid state physics Solid-state physics is the study of rigid matter, or solids, through methods such as solid-state chemistry, quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state p ...
. For example, circular
particle accelerator A particle accelerator is a machine that uses electromagnetic fields to propel electric charge, charged particles to very high speeds and energies to contain them in well-defined particle beam, beams. Small accelerators are used for fundamental ...
s 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 displays Hamiltonian chaos. It is therefore useful to study the development of chaos in this kind of system.


Main properties

For K=0 the map is linear and only periodic and quasiperiodic
orbits In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificia ...
are possible. When plotted in
phase space The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
(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 with ''K'', and with it the possibility to observe chaotic 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

The standard map is related to the
circle map In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamic ...
, which has a single, similar iterated equation: :\theta_ = \theta_n + \Omega - K \sin(\theta_n) as compared to :\theta_ = \theta_n + p_n + K \sin(\theta_n) :p_ = \theta_ - \theta_ for the standard map, the equations reordered to emphasize similarity. In essence, the circle map forces the momentum to a constant.


See also

* Ushiki's theorem


Notes


References


link
*
Springer link
* *


External links



at
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