A damped electrostatic pendulum is modeled using a system of differential equations and solved using numerical integration methods. A two dimensional map of initial conditions is integrated to construct a colored fractal image formed by the discrete points of convergence. Multiple programmatic solutions were implemented using Mathematica, Python and C++ to compare and improve computational performance. A decrease of two orders of magnitude in computation time was obtained by writing an adaptive step Runge Kutta integration method in C++. With significantly improved computation times high resolution images are constructed and stitched together to form videos that demonstrate how the system progressively changes with varying physical parameters: gravity, air drag, attractor strength, etc. The optimized code enables further statistical analysis, and may be easily modified for other systems of differential equations, initial condition maps, and integration methods.
CMakeLists.txt included. External libraries used: lodepng (https://github.com/lvandeve/lodepng) and jsoncpp (https://github.com/open-source-parsers/jsoncpp). Source for those libraries is included directly in this repository for simplicity but could be better maintained using submodules.
No external libraries are required outside of this repository, but C++11/14 features and libraries are used (greatest performance found using GCC 4.9 using -ffastmath, -O3, etc.). I have only tested on two machines: an AMD Phenom II 720BE with unlocked 4th core and overclocked to 3.3Ghz with Windows 7, and an Intel i7-3630QM Ubuntu laptop. I found GCC to be the fastest, next Clang, and finally MSVC. GCC is very effective at optimizing the tight loops with floating point operations, and I was unable to speed up the integration by writing inline sse intrinsic instructions. However MSVC is significantly slower and may be sped up by writing some sse intrinsic instructions.
The program takes in a json map configuration file and png image file name. An example configuration file is provided (map.json). Example use: $staticpendulum.exe map.json mapImage.png
Check the Presentation.pdf for example map images and general overview of the project.