Narrow Results

In this work, we study a system of Kuramoto oscillators with identical frequencies in a Cayley tree. Heterogeneity in the frequency distribution is introduced in the root of the tree, allowing for analytical calculations of the phase evolution. In this work, we study a system of Kuramoto oscillators with identical frequencies in a Cayley tree. Heterogeneity in the frequency distribution is introduced in the root of the tree, allowing for analytical calculations of the phase evolution. This simple case can be regarded as a starting point in order to understand how to extract topological features of the connectivity pattern from the dynamic state of the system, and vice versa, for the general situation of a set of phase oscillators located on a tree-like network.

It has been demonstrated that excitable media with a tree structure performed better than other network topologies, therefore it is natural to consider neural networks defined on Cayley trees. The investigation of a symbolic space called tree-shift of finite type is important when it comes to the discussion of the equilibrium solutions of neural networks on Cayley trees. Entropy is a frequently used invariant for measuring the complexity of a system, and constant entropy for an open set of coupling weights between neurons means that the specific network is stable. This paper gives a complete characterization of entropy spectrum of neural networks on Cayley trees and reveals whether the entropy bifurcates when the coupling weights change.

In this paper, we study linear cellular automata (CAs) on Cayley tree of order $k$ over the field ${𝔽}_p$ (the set of prime numbers modulo $p$). After revealing the rule matrix corresponding to cellular automata on Cayley tree with the null boundary condition, we analyze the reversibility problem of these cellular automata for some given values of $a,b,c,d\in {𝔽}_p$ and the levels $n$ of Cayley tree. The necessary and sufficient conditions for determining whether a CA is reversible or not are demonstrated. Furthermore, we compute the measure-theoretical entropy of the cellular automata which we define on Cayley tree. We show that for CAs on Cayley tree, the measure entropy with respect to uniform Bernoulli measure is infinity.