Blabla
This repository contains code regarding the Hénon attractor; it has a number of different files. The full_attractor.py contains functions that define how the Hénon map is built up. This includes the three different steps that Hénon used in his original paper; starting from an ellipse and subsequently applying the three different transformations that define the map. Furthermore, a function that defines the full map is also defined. I have attempted to optimize the function as much as possible such that a large amount of points can be generated. As of right now it can generate 1 million points in about 7 seconds and 10 million points in a minute or two.
The lyapunov.py file contains functions associated with the calculation of the Lyapunov exponents of the Hénon map. As of right now it is not fully optimized and the computation of the exponents is not super accurate but it does give the approximate values. Moreover, the computation time for a relatively small number of Lyapunov exponents is quite long. To deal with this the code has to be further optimized by for example including a part that calculates the exponents for point attractors according to a different program which results in a smaller computation time. For the creation of a larger grid of these exponents a text file will be created to save all exponents to such that they can be reused later. This file also contains function to find the exponents for a range of 'a' and 'b' values and a function that determines the type of attractor based on its Lyapunov exponents.
The trapping_region.py file contains functions considering the trapping region of the Hénon attractor. Next to this the image of the trapping region can also be obtained with functions that are defined here. A property of the Hénon map is that the image of the trapping region will remain inside the trapping region itself. However, it is of course possible to find the image of the image; it can be shown that this also remains wihtin the trapping region. One can, in theory, continue this indefinitely; this file contains a function which makes it possible to obtain and plot the nth order image of the trapping region. This gives a more visual representation of the property that these always lay within the original trapping region.
The boxes.py file contains functions to make a grid of plots which zoom in on a specific part of the Hénon attractor. These functions are generally a bit awkward to work with; later on I will add a jupyter notebook which includes an example of how one is able to utilize these functions. One important function in this file is called 'cut_interval'; this function cuts the points of the Hénon attractor for plots where the limits are small such that not the whole attractor is displayed. The reason that this is necessary is due to the fact that matplotlib plots the points even though the limits make these points invisible. So by cutting the list of points, the plotting process becomes faster and when saving the plot the filesize is reduced.
The general.py file contains some more general functions that could potentially be used for other purposes than the Hénon attractor or even attractors in general. Some of the functions here are a bit redundant and are not used elsewhere in this project. Nevertheless, I decided to keep them in here for now, maybe for later purposes.
A rough list of what has been done and what is left to do:
- ✅ Added basic functions regarding the Hénon attractor;
- ✅ Add option for 'zooming' in on a specific part on the Hénon map;
- ✅ Added the trapping region;
- ✅ Added the image of the trapping region;
- ✅ Added code to compute Lyapunov exponents;
- Improve code for the computation of Lyapunov exponents;
- Add notebook with plots;
- Add functions to determine whether attractor is a point attractor or a limit cycle, if so what is the period?
- ✅ Find Lyapunov exponents for a point attractor;
- Add code for creating a text file to save the generated Lyapunov exponents for later use;
- More...