35,420 research outputs found
Binary Codes and Period-2 Orbits of Sequential Dynamical Systems
Let be the (global) SDS map of a sequential dynamical system
(SDS) defined over the complete graph using the update order
in which all vertex functions are equal to the same function . Let denote the maximum number of periodic
orbits of period that an SDS map of the form can have. We
show that is equal to the maximum number of codewords in a binary code
of length with minimum distance at least . This result is significant
because it represents the first interpretation of this fascinating
coding-theoretic sequence other than its original definition.Comment: 12 pages, 2 figure
Cycle Equivalence of Graph Dynamical Systems
Graph dynamical systems (GDSs) can be used to describe a wide range of
distributed, nonlinear phenomena. In this paper we characterize cycle
equivalence of a class of finite GDSs called sequential dynamical systems SDSs.
In general, two finite GDSs are cycle equivalent if their periodic orbits are
isomorphic as directed graphs. Sequential dynamical systems may be thought of
as generalized cellular automata, and use an update order to construct the
dynamical system map.
The main result of this paper is a characterization of cycle equivalence in
terms of shifts and reflections of the SDS update order. We construct two
graphs C(Y) and D(Y) whose components describe update orders that give rise to
cycle equivalent SDSs. The number of components in C(Y) and D(Y) is an upper
bound for the number of cycle equivalence classes one can obtain, and we
enumerate these quantities through a recursion relation for several graph
classes. The components of these graphs encode dynamical neutrality, the
component sizes represent periodic orbit structural stability, and the number
of components can be viewed as a system complexity measure
Adaptive Complex Contagions and Threshold Dynamical Systems
A broad range of nonlinear processes over networks are governed by threshold
dynamics. So far, existing mathematical theory characterizing the behavior of
such systems has largely been concerned with the case where the thresholds are
static. In this paper we extend current theory of finite dynamical systems to
cover dynamic thresholds. Three classes of parallel and sequential dynamic
threshold systems are introduced and analyzed. Our main result, which is a
complete characterization of their attractor structures, show that sequential
systems may only have fixed points as limit sets whereas parallel systems may
only have period orbits of size at most two as limit sets. The attractor states
are characterized for general graphs and enumerated in the special case of
paths and cycle graphs; a computational algorithm is outlined for determining
the number of fixed points over a tree. We expect our results to be relevant
for modeling a broad class of biological, behavioral and socio-technical
systems where adaptive behavior is central.Comment: Submitted for publicatio
General Iteration graphs and Boolean automata circuits
This article is set in the field of regulation networks modeled by discrete
dynamical systems. It focuses on Boolean automata networks. In such networks,
there are many ways to update the states of every element. When this is done
deterministically, at each time step of a discretised time flow and according
to a predefined order, we say that the network is updated according to
block-sequential update schedule (blocks of elements are updated sequentially
while, within each block, the elements are updated synchronously). Many
studies, for the sake of simplicity and with some biologically motivated
reasons, have concentrated on networks updated with one particular
block-sequential update schedule (more often the synchronous/parallel update
schedule or the sequential update schedules). The aim of this paper is to give
an argument formally proven and inspired by biological considerations in favour
of the fact that the choice of a particular update schedule does not matter so
much in terms of the possible and likely dynamical behaviours that networks may
display
Order Independence in Asynchronous Cellular Automata
A sequential dynamical system, or SDS, consists of an undirected graph Y, a
vertex-indexed list of local functions F_Y, and a permutation pi of the vertex
set (or more generally, a word w over the vertex set) that describes the order
in which these local functions are to be applied. In this article we
investigate the special case where Y is a circular graph with n vertices and
all of the local functions are identical. The 256 possible local functions are
known as Wolfram rules and the resulting sequential dynamical systems are
called finite asynchronous elementary cellular automata, or ACAs, since they
resemble classical elementary cellular automata, but with the important
distinction that the vertex functions are applied sequentially rather than in
parallel. An ACA is said to be pi-independent if the set of periodic states
does not depend on the choice of pi, and our main result is that for all n>3
exactly 104 of the 256 Wolfram rules give rise to a pi-independent ACA. In 2005
Hansson, Mortveit and Reidys classified the 11 symmetric Wolfram rules with
this property. In addition to reproving and extending this earlier result, our
proofs of pi-independence also provide significant insight into the dynamics of
these systems.Comment: 18 pages. New version distinguishes between functions that are
pi-independent but not w-independen
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