chirikov 
Oliver Knill 
Overview 
chirikov allows to compute the entropy and phase space pictures of toral maps. For possible updates, check out the the home and downlowad page of "chirikov". 
chirikov is invoked on the command line. Currently implemented is a threeparameter family of toral maps T: (x,y) > (2xy + f(x),x) called 'twist maps' or 'standard maps'. If f(x)=g sin(x) the map T is called the ChirikovTaylor Standard map. Implemented is the three parameter family f(x)=a sin(x) + b sin(2 x) + c sin(3 x). The user can easily change the definition of the map f in the header file chirikov.h and recompile the code to investigate different maps. 
The program chirikov was used over years for research purposes. We had written earlier versions in Pascal, Fortran, Mathematica, Java. While this C program version grew while more features were added, this release is a reduction to the basics. It can be used as a building block in shellscripts using image processing, image viewing and movie rendering programs. 
Installation 
Installation is generic: type  


to produce the binary. Type (eventually as root)


chirikov has currently been tested for  
You might to want to modify chirikov.h to change default features or flags or the map in consideration. 
Try it ! 
In order to try out chirikov in a X environment, just type


Next, try

Why no graphics front end? 
This program was developed while doing research on toral maps.
It was used to compute numerically the entropy, for producing
slides for talks or web illustrations. Terminalbased
programming under Unix has many advantages [KerPik99].

Who is Chirikov? 
Boris V. Chirikov's is a Russian physisict at the Budker Institute of Nuclear Physics. He was one of the first people studing seriously toral areapreserving maps. Such maps appear in plasma dynamics, celestial mechanics and other dynamical systems. 
What is the Chirikov map? 


In the case f(x) = sin(x), the map T is called the Chirikov map, the Standard map or the ChirikovTaylor map.  
The map appeared first in 1960 in the context of electron dynamics in microtrons. It was first numerically studied by Taylor in 1968 and Chirikov in 1969. In the physical literature it is called the "kicked rotator" and describs ground states of the "FrenkelKontorova model". The map is often used to illustrate or motivate various mathematical theorems in the theory of dynamical systems.  
chirikov allows to study maps T with

Entropy calculation 
The Jacobean of the map T is a linear map given by the 2x2 matrix
The limit of (log dT ^{ n } (x,y))/n for n > infinity measures the exponential growth rate of the norm dT ^{ n } . It is defined for almost all points (x,y) and called the Lyapunov exponent of T at the point (x,y). If we integrate the Lyapunov exponent over the torus, we obtain a number called the entropy of the map. The program chirikov estimates this number by taking a large n, say n = 5'000'000 and computes the integral by approximating the integral through a Riemann sum. Example: the entropy of (x,y) > (2xy + 3.0 sin(x),x) is computed with
Example: The entropy of (x,y) > (2xy + 1.0 sin(x)+2.4 sin(2x),x) is computed with the following command:

Phase space pictures 
If we take a point (x,y) and look at its iterates T(x,y), T (T(x,y))
= T ^{ 2 } etc., we obtain an orbit.
By taking a grid of initial starting points and
coloring each orbit according to the Lyapunov exponent measured along
this orbit, chirikov obtains the phase space picture. Examples:
You can look closer at the 'shore' of the island by taking a picture of radius 0.2 (taxi metric) centered at (0.73,0.73)

Example1: Making background patterns 
An example on how to use chirikov to produce a background in X
try out

Example2: Zoom into the phase space 
With a small shellscript the zooming can be done automatically.
The gif files are connected to a gif animation using 'gifmerge'
(which is also included here). The shellscript
(located in the Demos directory)

Example3: Parameter Movies 
As an other demonstration, we move along a paths in the
ab parameter space.

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