By Frederic Geurts
This self-contained monograph is an built-in research of commonly used platforms outlined via iterated kinfolk utilizing the 2 paradigms of abstraction and composition. This comprises the complexity of a few state-transition structures and improves figuring out of advanced or chaotic phenomena rising in a few dynamical structures. the most insights and result of this paintings drawback a structural kind of complexity bought by means of composition of easy interacting platforms representing antagonistic attracting behaviors. This complexity is expressed within the evolution of composed structures (their dynamics) and within the kin among their preliminary and ultimate states (the computation they realize). The theoretical effects are demonstrated via examining dynamical and computational houses of low-dimensional prototypes of chaotic platforms, high-dimensional spatiotemporally complicated platforms, and formal structures.
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Extra info for Abstract Compositional Analysis of Iterated Relations: A Structural Approach to Complex State Transition Systems
Geurts: Abstract Compositional Analysis of Iterated Relations, LNCS 1426, pp. 21-52, 1998. Springer-Verlag Berlin Heidelberg 1998 22 2. Dynamics of Relations Juxtaposition of symbols stands for concatenation; exponentiation stands for multiple concatenation. For any sequence s of length at least n, s|n represents its preﬁx of length n. For any bi-inﬁnite s ∈ X Z, we denote the subsequence s0 s1 s2 · · · by s+ and s0 s−1 s−2 · · · by s− . g. ). The dynamical system imposes a temporal ordering on the underlying space.
The iterative evolution from a state can be described from two viewpoints, emphasizing the intrinsic nondeterminism of relations at the point-level, or by means of multi-valued functions and set-transformers at the set-level. 1 Point-Level Nondeterministic Dynamics Let (X, f ) be a RDS, and x ∈ X. If there exists a y such that (x, y) ∈ f , then f x −→ y represents a possible iteration-step from x, and many other evolutions can coexist; f is regarded as a nondeterministic function from X to X. The subsequent steps of the dynamics can be deﬁned using this view, leading to the following deﬁnitions.
68. Take a function deﬁned as follows (see Fig. 5 1 Fig. 6. Graph of f (x) (and y = x, dotted line) f (x) = 1 12 x 1 12 x + + 1 3 7 11 4 on [0, 11 ) 4 on [ 11 , 1]. 4 4 The part of f deﬁned on [0, 11 ) seems to have a ﬁxed point in 11 but there it is not deﬁned. Actually, at this point, the right branch is deﬁned, for which 84 . e. the there is a true ﬁxed point in 121 1 contractivity factor is strictly smaller than 1 (here, it is equal to 12 ). 4 Thus, starting from any point in [0, 11 ), ω iterations are needed to reach the 4 “virtual” attracting ﬁxed point 11 .
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