Transition from 3-branched to 5-branched spirals in the Rock-Paper-Scissors-Lizard-Spock equation
Channel: Nils Berglund
Date: 2022-05-21

In this variant of the Rock-Paper-Scissors-Lizard-Spock reaction-diffusion equation, the interactions between chemicals fall into two different families, with different intensities. By decreasing one of the intensities, one can induce a transition from 3-armed to 5-armed spirals. 
The game Rock-Paper-Scissors-Lizard-Spock was originally created by Sam Kass with Karen Bryla for the sitcom Big Bang theory (see https://youtu.be/x5Q6-wMx-K8 ). It is a variant of the Rock-Paper-Scissors game, in which each of the five words beats two others, and is beaten by the two other ones:
- Rock (red) beats Scissors and Lizard, and is beaten by Paper and Spock
- Paper (yellow) beats Rock and Spock, and is beaten by Scissors and Lizard
- Scissors (green) beat Paper and Lizard, and are beaten by Rock and Spock
- Spock (blue) beats Scissors and Rock, and is beaten by Lizard and Paper
- Lizard (purple) beats Spock and Paper, and is beaten by Rock and Scissors
This simulation shows a solution of a reaction-diffusion equation that imitates the game. There are five chemicals, which diffuse, and transform into each other. Each chemical can convert two other ones into its own type, while it is converted by the other two. 
Previous simulations on this channel of the Rock-Papers-Scissors equation, involving three instead of five chemicals, showed the appearance of rotating spirals. The question arose in the comments whether a version with five chemicals would lead to five-armed spirals, or to a more complex version where only three or four colors meet at the center of every spiral. The simulation https://youtu.be/-Qap_7skcIA showed that the case of a perfectly symmetric interaction, in which the latter scenario occurs. However, in this simulation, half of the interactions are gradually decrease over time, creating a transition between a regime with 3-branched spirals and 5-branched spirals.
This video has two parts, showing the same solution of the equation with two different color gradients:
Dominant type: 0:00
Norm of gradient of phase: 1:42
In part 1, the color indicates the chemical which is the most abundant at each point. In part 2, the state is first converted into an angular variable, or phase, interpolating between the different chemicals. The color hue shows the norm of the gradient of the phase, meaning that in dark areas, the concentrations change quickly, while in light areas, they are nearly constant.
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 and as orange, purple and blue in this one. Denoting by rho = u + v + w + x + y the total concentration, the system of equations is given by 
d_t u = D*Delta(u) + u*(1 - rho - a*v - b(t)*x)
d_t v = D*Delta(v) + v*(1 - rho - a*w - b(t)*y)
d_t w = D*Delta(w) + w*(1 - rho - a*x - b(t)*u)
d_t x = D*Delta(x) + x*(1 - rho - a*y - b(t)*v)
d_t y = D*Delta(y) + y*(1 - rho - a*u - b(t)*w)
where Delta denotes the Laplace operator, which performs a local average, and the parameter a is equal here to 0.75, D is equal to 1, while b(t) decreases from 0.75 to 0 in the course of the simulation. At the beginning, when b is equal to a, the dynamics features 3-armed spirals. The picture then becomes less symmetric as b decreases, and 5-armed spirals appear when b is close to 0. 

Render time: 1 hour 20 minutes
Color schemes: Part 1 - Jet 
Part 2 - Twilight by Bastian Bechtold
https://github.com/bastibe/twilight

Music: "Back To The Future Jellyfish" by NoMBe@nombe

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