Belousov–Zhabotinsky (Excitable)
A discrete Greenberg–Hastings cellular automaton on a toroidal grid stands in for the Belousov–Zhabotinsky reaction. Each cell is in one of three states — quiescent (0), excited (1), or refractory (2…N−1). A quiescent cell becomes excited when enough of its eight Moore neighbours are excited; an excited cell then runs through a fixed refractory countdown before becoming sensitive again. The model is qualitative, not chemically calibrated, but it reproduces the hallmark behaviour of an excitable medium: target waves expanding from local pacemakers and counter-rotating spiral waves around free wave-tips.
Who it's for: Intro nonlinear chemistry, complex systems, and computational physics; an accessible bridge between reaction–diffusion PDEs and cellular automata.
Key terms
- Belousov–Zhabotinsky
- excitable medium
- Greenberg–Hastings
- spiral waves
- target patterns
- refractory period
- cellular automaton
How it works
Belousov–Zhabotinsky-style **target waves and rotating spirals** from a tiny excitable cellular automaton — a qualitative, lattice version of the bromate–malonic acid clock.
Key equations
Frequently asked questions
- Why do spirals appear instead of just expanding rings?
- Spirals are seeded wherever a wavefront is broken — an open tip cannot annihilate against a partner, so it curls around itself indefinitely. Random initial conditions give the wavefronts plenty of free ends, so spirals usually win over the slower target patterns.
- What does the refractory period correspond to chemically?
- In the real BZ reaction, an oxidised patch of catalyst (e.g., ferroin → ferriin) cannot be re-excited until reductant is replenished. The integer countdown in the automaton plays the same role: a cell that has just fired is locked out for several frames before it can respond again.
- Is this a true reaction–diffusion model?
- No. The real chemistry is captured by Oregonator-style PDEs with diffusion. The cellular automaton is a coarse cartoon: it keeps the three essential ingredients — local excitation threshold, refractoriness, and propagation — and is fast enough to render full-screen patterns in real time.
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