Sequential Dynamical System¶

A finite graph dynamical system whose one-step evolution is the ordered composition of neighborhood-local vertex updates, so later updates in the schedule read changes made earlier in the same sweep.

Version: v2 · 2026-09-06 · History
Domain-specific #: 2758
Origin domain: discrete dynamical systems
Subdomain: asynchronous graph dynamics

Core Idea¶

A sequential dynamical system (SDS) is a finite graph dynamical system whose global evolution rule is built by applying vertex-local update maps one at a time in a specified order. Each vertex carries a state. Its local rule reads the current states in a graph-defined neighborhood and writes a new state only at that vertex. Because an update is committed immediately, a vertex appearing later in the schedule can read values written earlier in the same sweep. The schedule is therefore part of the mathematical model: changing it can change the global map, its transient trajectories, and its periodic behavior even when the graph and local functions stay fixed.

Scope of Application¶

SDS belongs to discrete mathematics, finite dynamical systems, graph dynamics, theoretical computer science, and simulation theory. It was developed as a mathematical framework for systems whose global behavior emerges from local, interacting updates implemented in a sequence. The formalism supports both structural questions—when do different orders induce the same map or cycle structure?—and computational questions—does a configuration have a predecessor, can a target be reached, or does some update permutation realize a requested transition?

Clarity¶

The fastest diagnostic is to compare two formulas. In a synchronous system,

\[ F_{\mathrm{sync}}(x)=\bigl(f_1(x[1]),\ldots,f_n(x[n])\bigr), \]

so every component reads the same old \(x\). In an SDS,

\[ F_{\mathrm{seq}}(x)=F_{w_m}(\cdots F_{w_2}(F_{w_1}(x))\cdots), \]

Manages Complexity¶

An arbitrary map on \(K^n\) can require a table with \(|K|^n\) entries. SDS factorizes the map into \(m\) local operations, each constrained by a graph neighborhood. This exposes a compact generative description: network structure says who can influence whom, local functions say how influence changes one vertex, and the update word says when that change becomes visible.

Abstract Reasoning¶

Several deductions follow directly from the composition structure.

First, schedule sensitivity is a noncommutativity question. If two consecutive local maps satisfy \(F_iF_j=F_jF_i\), swapping them leaves the global map unchanged. For standard neighborhood-local rules on an undirected graph, updates at nonadjacent vertices do not read one another's coordinates and therefore commute. Schedule differences generated solely by such swaps are functionally neutral. Edges mark where order dependence may enter, though special functions on adjacent vertices can commute as well.

Knowledge Transfer¶

SDS carries a useful modeling pattern across finite networked domains:

\[ \text{interaction graph} + \text{local state rules} + \text{visibility schedule} \longrightarrow \text{global map and phase space}. \]

The transfer is literal when a target system supplies all four parts. A Boolean regulatory model can use genes or logical variables as vertices, local regulatory logic as \(f_i\), and an explicit sequential schedule as \(w\). A communicating-state model can use process states at vertices and neighbor messages in the local rules if the finite-state and fixed-schedule assumptions are maintained.

Relationships to Other Abstractions¶

Current abstraction Sequential Dynamical System Domain-specific

Parents (1) — more general patterns this builds on

Sequential Dynamical System is a kind of Network Prime

Network is the most literal parent.

Hierarchy path (1) — routes to 1 parentless root

Sequential Dynamical System → Network → Reservoir-Flux Network → Conservation Laws → Invariance

Neighborhood in Abstraction Space¶

Sequential Dynamical System sits in a sparse region of the domain-specific corpus (85^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08