Colliding spirals in the Rock-Paper-Scissors equation 5: norm of the change of phase
Channel: Nils Berglund
Date: 06/05/2022

This new episode of our series on colliding spirals in the Rock-Paper-Scissors reaction-diffusion equation with increasing viscosity probably shows the cleanest representation I obtained so far. The color indicates here the norm of the gradient of the phase associated with the equation: darker colors represent a fast rate of change of the chemical concentrations, while lighter colors indicate almost constant concentrations. After a first transient phase, one sees that the dark regions organize in three-armed rotating spirals, which delimit the regions in which one of the chemicals predominates. As the viscosity increases, some centers of counter-rotating spirals can be seen to annihilate each other.
The reaction-diffusion equation is defined as follows. At each point in space and time, there are three concentrations u, v, and w of chemicals, that appear as red, blue and green in the video https://youtu.be/TORwMc2AaRE . Denoting by rho = u + v + w the total concentration, the system of equations is given by 
d_t u = D(t)*Delta(u) + u*(1 - rho - a*v)
d_t v = D(t)*Delta(v) + v*(1 - rho - a*w)
d_t w = D(t)*Delta(w) + w*(1 - rho - a*u)
where Delta denotes the Laplace operator, which performs a local average, and the parameter a is equal here to 0.75 while D(t) increases from 0.05 to 5 in the course of the simulation. The terms proportional to a*v, a*w and a*u denote reaction terms, in which Red is beaten by Blue, Blue is beaten be Green, and Green is beaten by Red. The situation is thus similar to the Rock-Paper-Scissors game (see https://en.wikipedia.org/wiki/Rock_paper_scissors ), and there exist simpler cellular automata with similar properties, see for instance https://softologyblog.wordpress.com/2018/03/23/rock-paper-scissors-cellular-automata

Render time: 37 minutes
Color schemes: Twilight by Bastian Bechtold
https://github.com/bastibe/twilight

Music: "Can't Sleep" by Eveningland@EveninglandMusic

See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on  a few previous simulations of wave equations. 

The simulation solves a reaction-diffusion equation by discretization. 
C code: https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html