Sequential Dynamical System - Definition

Definition

An SDS is constructed from the following components:

• A finite graph Y with vertex set v = {1,2, ..., n}. Depending on the context the graph can be directed or undirected.

• A state x_v for each vertex i of Y taken from a finite set K. The system state is the n-tuple x = (x₁, x₂, ..., x_n), and x is the tuple consisting of the states associated to the vertices in the 1-neighborhood of i in Y (in some fixed order).

• A vertex function f_i for each vertex i. The vertex function maps the state of vertex i at time t to the vertex state at time t + 1 based on the states associated to the 1-neighborhood of i in Y.

• A word w = (w₁, w₂, ..., w_m) over v.

It is convenient to introduce the Y-local maps F_i constructed from the vertex functions by

The word w specifies the sequence in which the Y-local maps are composed to derive the sequential dynamical system map F: Kn → Kn as

If the update sequence is a permutation one frequently speaks of a permutation SDS to emphasize this point. The phase space associated to a sequential dynamical system with map F: Kn → Kn is the finite directed graph with vertex set Kn and directed edges (x, F(x)). The structure of the phase space is governed by the properties of the graph Y, the vertex functions (f_i)_i, and the update sequence w. A large part of SDS research seeks to infer phase space properties based on the structure of the system constituents.