Joel Midgley-Volpato | Searcharxiv

arXiv2017

An example of a deterministic cellular automaton exhibiting linear-exponential convergence to the steady state

In a recent paper [arXiv:1506.06649 [nlin.CG]], we presented an example of a 3-state cellular automaton which exhibits behaviour analogous to degenerate hyperbolicity often observed in finite-dimensional dynamical systems. We also calculated densities of 0, 1 and 2 after n iterations of this rule, using finite state machines to conjecture patterns present in preimage sets. Here, we re-derive the same formulae in a rigorous way, without resorting to any semi-empirical methods. This is done by analysing the behaviour of continuous clusters of symbols and by considering their interactions.

nlin.CG↗

arXiv2015

An example of degenerate hyperbolicity in a cellular automaton with 3 states

We show that a behaviour analogous to degenerate hyperbolicity can occur in nearest-neighbour cellular automata (CA) with three states. We construct a 3-state rule by "lifting" elementary CA rule 140. Such "lifted" rule is equivalent to rule 140 when arguments are restricted to two symbols, otherwise it behaves as identity. We analyze the structure of multi-step preimages of 0, 1 and 2 under this rule by using minimal finite state machines (FSM), and exploit regularities found in these FSM. This allows to construct explicit expressions for densities of 0s and 1s after $n$ iterations of the rule starting from Bernoulli distribution. When the initial Bernoulli distribution is symmetric, the densities of all three symbols converge to their stationary values in linearly-exponential fashion, similarly as in finite-dimensional dynamical systems with hyperbolic fixed point with degenerate eigenvalues.

nlin.CG↗