Flows of Ordinary Differential Equations

Differential equations are used in many sciences to describe among other things such different concepts as fluid dynamics, options pricing and predator-prey-systems. Differential equations come in many flavours, but the most simple ones are ordinary differential equations (ODEs). In most cases, the solution to an ODE (which is a function) depends on the initial situation (e.g. to know how a ball will fly when you throw it, you need to know its initial direction and speed). Given the initial conditions the solution can be interpreted as a description of how the system behaves in time under the rules given by the ODE. One can visualize this behaviour using a curve. If one now changes the initial conditions slightly then one gets a different solution which means a different behaviour of the system, and therefore a different curve (e.g. a ball will fly differently if you throw it with half the force). The set of all solutions to all the different initial conditions are called the flow of the ODE.

The images below visualize parts of the flows of different ODEs by visualizing their solutions for some initial conditions using curves — one curve represents one solution. The colors of the curves depend on the speed with which the described system changes at that point in time: White parts represent slow change while red parts indicate fast changes in the system.

Source code

The calculations for the images were done in MATLAB and the rendering was done using POV-Ray. The source code is released under the GPL.