Diffusively coupled Lorenz model
The Lorenz system is a well-known system of three ordinary differential equations that exhibit chaotic dynamics. If we put a copy of this system at every spatial point and couple these via diffusion (that is, adding a Laplacian to each equation), we arrive at the following system:
\[\begin{aligned} \pd{X}{t} &= D \nabla^2 X + \sigma \left(Y-X\right),\\ \pd{Y}{t} &= D \nabla^2 Y + X \left(\rho-Z\right)-Y,\\ \pd{Z}{t} &= D \nabla^2 Z + X Y-\beta Z, \end{aligned}\]which can exhibit a variety of spatiotemporal behaviours.
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Load the interactive simulation to see a random initial condition evolve into several complex oscillating structures. These change over long timescales, so you may want to watch the simulation for a while to see how they coalesce and interact.
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The initial condition uses a random perturbation across the spatial domain. If you change $X(t=0)=0$, the system will be at an equilibrium state and not move. If you then click, you can initiate localised travelling oscillations, with multiple clicks interacting with one another in interesting ways.
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Modifying the value of $D$, which is effectively the coupling strength between local chaotic systems, can be one fun way to explore the parameter space. For large values (e.g. $D=5$) the system tends toward a uniform state with large-wavelength oscillations, whereas for small values (e.g. $D=0.2$) it behaves more erratically, breaking up into local patches of oscillating and chaotic regions.