The 4-color FCA on square lattice Ζ^2

The κ-color Firefly Cellular Automaton (FCA) is a discrete model for κ-state pulse-coupled oscillators.  While the FCA on lattices for any κ≠4 exhibit local sprials, the 4-color instance on square lattice shows clustering via interacting incomplete spirals.



The 64-state adaptive 4-coupling on uniform spanning tree of Moore lattice

The 4-coupling is a continuum extension of the 4-color FCA. The adaptive 4-coupling is a clock synchronization algorithm based on the 4-coupling, which is able to synchronizes arbitrary initial joint configuration on any trees. If we take a uniform spanning tree of the underlying lattice, the system tries to form spirals but none are completed by topological restriction.


Probability, combinatorics, and complex systems

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