Fourier Series and Animations

UNDER DEVELOPMENT

This small Python library was inspired by the 3Brown1Blue video linked here. After watching this video I took some time to remind myself of the structure of a Fourier series (which I, at one point, taught in ungrad.) before building the classes in the package FOUR.

The idea of a 2-D Fourier series is relatively simple. The general theory states that any function, $f$, which maps between two domains such that $f : \mathbb{X} \rightarrow \mathbb{Y}$ can be represented by an infinite sum of $\cos(x)$ and $\sin(x)$ terms.

$$ f(x) = \sum_{k=0}^\infty \left[ A_k \sin(\frac{\pi k x}{L}) + B_k \cos(\frac{\pi k x}{L}) \right] $$

Where $x \in \mathbb{X}$ and $f(x) \in \mathbb{Y}$.

References:
    A. Gilat and V. Subramaniam, Numerical methods for engineers and
        scientists: an introduction with applications using matlab, 3. ed.
        Hoboken, NJ: Wiley, 2014.

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Imprinting Temporal-dynamics onto a Grid

In this quick example I have imprinted the motion of the duffing oscillator (as begun from a single set point) onto a grid. Each space can be made visible or invisible dependent on whether the point is present in that cell. That is,

$$ c_{i,j} = \begin{cases} 1 & \text{if $x(t)$ is present} \\ 0 & \text{otherwise} \end{cases} $$

Where $c_{i,j}$ is the binary value for the $j$-th cell in the $i$-th row. The binary grid was saved at each step of the simulation and used to calculate the Fourier transform. This resulted in a playback that looks like...