Complex behaviours,
visualised

Starting from a grid of particles of position \(x \in \mathbb{C}\), each of them is animated according to a complex function \(f(x)\) which is considered their instantaneous velocity. For each frame, all particles move and end up in a new position, which in turn will define the istantaneous velocity in the following frame. If a sufficiently small \(dx\) of displacement is taken, the animation accurately depicts the motion direction of many particles inside the vector field defined by \(f\).

The tool is made to be played around with, allowing the user to:

• Select a function \(f\) among many;
• Set numerical values appearing in the function definitions;
• Pan and zoom in/out the viewbox;
• Choose the pixel gap between the particle initial positions;
• Set the particle life (animation loop duration in frames);
• Set the number of colors (representing the speed);
• Increase or decrease the speeds-for-colors thresholds;

Currently, two version of the streamplot are available and still work in progress. If you have any suggestions or reports on issues and bugs, feel free to contact me.

1. The first version computes \(f\) at each particle each frame. In older, or less-performant, devices it can lead to a laggy animation: be aware.
2. The second one precomputes all the frames beforehand, and simply displays them afterwards. When changing some setting that requires a re-computation, some freezing can be encountered (use of multi-threading is still work in progress).

Chosen \(0 \lt n \in \mathbb{N}\), a polynomial of degree \(n\), with its complex roots, is considered. The Guassian plane is colored based on the exact root closest to the \(m\)-th iteration of the Newton's method applied at each \(z \in \mathbb{C}\) visible in the viewbox.

Still work in progress...