About This Page

The dynamical systems visualization behind the landing page

What You're Seeing

The animated background on the landing page displays the trajectories of a dynamical system, integrated forwards in time from a collection of random initial conditions. Points that diverge to infinity are resampled to initial conditions. The previous points slowly fade over time, leaving a temporary imprint as you adjust the parameters that govern the dynamical system. The gentle 'motion' of the system is caused by small oscillating perturbations to the system parameters.

The Hénon Map

The default system is the Hénon map, which is defined by the iterative equations:

$$\begin{align} x_{n+1} &= 1 - ax_n^2 + y_n \\ y_{n+1} &= bx_n, \end{align}$$

where $a$ and $b$ are system parameters (typically $a = 1.4$ and $b = 0.3$). This system displays a mix of regular and chaotic trajectories depending on the initial conditions $(x_0, y_0)$. Regular trajectories trace out one-dimensional curves (bright lines) on the two-dimensional plane, whereas chaotic trajectories create space-filling curves (broad, fuzzy areas). Playing around with $a$ and $b$, you'll likely build some intuition for how these parameters change the shape of where trajectories collect -- maybe you could even start to guess the values of $a$ and $b$ just by looking at the size and shapes of the trajectories that the system traces out. This might give you the idea that a computer-vision model could be trained to do something like this (and you would be correct).

For a little pizazz I also make $a$ and $b$ dynamic:

$$\begin{align} a_{n+1} &= a_0 + 0.1 \cdot \sin(0.01 \cdot n) \\ b_{n+1} &= b_0 + 0.05 \cdot \cos(0.01 \cdot n). \end{align}$$

As long as we change the parameters $a$ and $b$ smoothly and slowly, we essentially get an adiabatic process where we observe a slowly shifting steady state set of points.

Interactive Controls

Click "Show Controls" to reveal parameter sliders that let you modify the mathematical constants in real-time. Each system has different parameters that control aspects like:

Shape and size of the attractor
Density and distribution of points
Symmetry and orientation of patterns
Chaotic vs. regular behavior

The "Reset Parameters" button returns each system to its defaults, selected to represent a visually interesting starting point.

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