Standard map
Encyclopedia
The standard map is an area-preserving chaotic map from a square with side 
onto itself. It is defined by:
where
and 
are taken modulo 
.
This map is also known as the Chirikov–Taylor map or as the Chirikov standard map,
The properties of chaos of the standard map were established by Boris Chirikov
in 1969. See more details at Scholarpedia entry.
The quantized map is also known as the kicked rotator in the quantum chaos
community.
and 
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.
The kicked rotator approximates systems studied in the fields of mechanics
of particles, accelerator physics
, plasma physics, and solid state physics. For example, circular particle accelerator
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.
the map is linear and only periodic and quasiperiodic orbits
are possible. When plotted in phase 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 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
. 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.
, which has a single, similar iterated equation:
as compared to
for the standard map, the equations reordered to emphasize similarity. In essence, the circle map forces the momentum to a constant.
onto itself. It is defined by:where
and are taken modulo .This map is also known as the Chirikov–Taylor map or as the Chirikov standard map,
The properties of chaos of the standard map were established by Boris Chirikov
Boris Chirikov
Boris Valerianovich Chirikov was an outstanding Soviet and Russian physicist....
in 1969. See more details at Scholarpedia entry.
The quantized map is also known as the kicked rotator in the quantum chaos
Quantum chaos
Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is, "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle...
community.
Physical model
This map describes the motion of a simple mechanical system known as the kicked rotator. It 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 variables and 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.The kicked rotator approximates systems studied in the fields of mechanics
Mechanics
Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....
of particles, accelerator physics
Accelerator physics
Accelerator physics deals with the problems of building and operating particle accelerators.The experiments conducted with particle accelerators are not regarded as part of accelerator physics. These belong to particle physics, nuclear physics, condensed matter physics, materials physics, etc...
, plasma physics, and solid state physics. For example, circular particle accelerator
Particle accelerator
A particle accelerator is a device that uses electromagnetic fields to propel charged particles to high speeds and to contain them in well-defined beams. An ordinary CRT television set is a simple form of accelerator. There are two basic types: electrostatic and oscillating field accelerators.In...
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 the map is linear and only periodic and quasiperiodic orbitsIterated function
In mathematics, an iterated function is a function which is composed with itself, possibly ad infinitum, in a process called iteration. In this process, starting from some initial value, the result of applying a given function is fed again in the function as input, and this process is repeated...
are possible. When plotted in phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase 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 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
. 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 mapCircle map
In mathematics, a circle map is a member of a family of dynamical systems on the circle first defined by Andrey Kolmogorov. Kolmogorov proposed this family as a simplified model for driven mechanical rotors . The circle map equations also describe a simplified model of the phase-locked loop in...
, which has a single, similar iterated equation:
as compared to
for the standard map, the equations reordered to emphasize similarity. In essence, the circle map forces the momentum to a constant.
External links
- Standard map at MathWorldMathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...
- Chirikov standard map at Scholarpedia
- Website dedicated to Boris Chirikov
- Interactive Java Applet visualizing orbits of the Standard Map, by Achim Luhn