An Introduction to Sequential Dynamical Systems by Henning Mortveit, Christian Reidys

By Henning Mortveit, Christian Reidys

This introductory textual content to the category of Sequential Dynamical platforms (SDS) is the 1st textbook in this well timed topic. pushed by means of a number of examples and thought-provoking difficulties all through, the presentation deals solid foundational fabric on finite discrete dynamical structures, which then leads systematically to an advent of SDS. From a huge diversity of issues on constitution thought - equivalence, mounted issues, invertibility and different section area houses - thereafter SDS kinfolk to graph conception, classical dynamical platforms in addition to SDS functions in laptop technological know-how are explored. this can be a flexible interdisciplinary textbook.

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The automorphisms of Y form a group under function composition. This is the automorphism group of Y , and it is denoted Aut(Y ). Let Y and Z be undirected graphs and let ϕ : Y −→ Z be a graph morphism. 5) are all surjective or all injective, respectively. A graph morphism that is both locally surjective and locally injective is called a local isomorphism or a covering. 1. 1 is surjective but not locally surjective. −→ Y = =Z Fig. 1. 1. 1 Simple Graphs and Combinatorial Graphs An undirected graph Y is a simple graph if the mapping {e, e} → {ω(e), τ (e)} is injective.

Closely related to this is the #predecessor problem, which asks for the number of predecessors of a system state x. 1) in, for example, [17]. Exactly as for the previous problems the predecessor existence problem is NP-complete in the general case, but can be solved efficiently for restricted classes of vertex functions and/or graphs. Examples include SDS where the vertex functions are given by logical And 4 The problem is PSPACE-complete; see, for example, [15]. 20 1 What is a Sequential Dynamical System?

7. Let Y be a graph with adjacency matrix A. The number of walks of length k in Y that start at vertex vi and end at vertex vj is [Ak ]i,j , the (i, j) entry of the kth power of A. The result is proved by induction. Obviously, the assertion holds for k = 1. Assume it is true for k = m. We can show that it holds for k = m + 1 by decomposing a walk of length m + 1 from vertex vi to vertex vj into a walk of length m from the initial vertex vi to an intermediate vertex vk followed by a walk of length 1 from the intermediate vertex vk to the final vertex vj .

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