The Standard Chirikov Map Visualization

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General Information

The standard Chirikov map is a symplectic transformation of the phase plane given by formula $$(x,y)\mapsto(x+y+\varepsilon\sin x,y+\varepsilon\sin x)$$

where ε is a small parameter. This mappping is correctly restricted to a torus when $x,y\in[-\pi;\pi)$ are taken modulo $2\pi$.

For $\varepsilon=0$ the map is linear and only periodic and quasiperiodic orbits are possible and the map is integrable. With a non-zero value of $\varepsilon$, some of the orbits exhibit chaotic behavior.

As $\varepsilon$ increases, the measure of the set of points with chaotic orbits increases.

The video visualizes this process. Each frame corresponds to the next increasing value of $\varepsilon$. Each pixel of the frame is shaded according to the chaoticity of the corresponding point’s orbit.

The chaoticity of the orbit is calculated as follows. Let $z$ be the starting point of the orbit, $T$ be the Chirikov map, $n$ be the smallest natural number such that $$\mathrm{distance}(T^{2n}(z),T^n(z))<\rho$$ for a small positive $\rho$. The larger $n$, the darker the pixel. This method is inspired by the Floyd’s cycle detection algorithm.

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Assembling Video

The assembling of the video is a lengthy process. It took us several months intermittently on CPU Intel Core i9-9900K (video dimensions 1200, 6000 frames).

The UNIX-like operating system is required. Prerequisites:

• java ≥ 11
• make
• ffmpeg

Simply run

make

The build can be interrupted and restarted at any time.

If the build is overheating your CPU, you can limit the number of threads used:

make THREADS=4

Full list of variables you may want to tweak:

Description	Variable	Default value
The number of threads	THREADS	0 (thread pool uses all available processors)
The video frame rate	FRAMERATE	30
Initial value of $\varepsilon$	INITIAL	0
Final value of $\varepsilon$	FINAL	6
Increment of $\varepsilon$	STEP	.001
The value of $\rho$	RHO	.001
The density (video width and height)	DENSITY	1200

Copyright and License

© Anton Shvetz, 2022—2023

This project is licensed under the CC-BY-SA-4.0 License.