Abstract
We formulate two simplicity criteria for dynamical systems based on the concepts of finite automata and regular languages. Finite automata are regarded as dynamical systems on discontinuum and their factors yield the first simplicity class. A finite cover of a topological space is almost disjoint, if it consists of closed sets which have the same dimension as the space, and meet in sets whose dimension is smaller. A dynamical system is regular, if it yields a regular language when observed through any finite almost disjoint cover. Next we formulate two topological simplicity criteria. A dynamical system has finite attractors, if the Ω-limit of its state space is finite. A dynamical system has chaotic limits, if every point is included in a set whose Ω-limit is either a finite orbit or a chaotic subsystem. We show the relations between these criteria and classify according to them several classes of zero-dimensional and one-dimensional dynamical systems.
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References
B.Balcar, P.Simon: Appendix on general topology. Handbook of Boolean Algebras (J.D.Monk and R.Bonnet, eds.), Elsevier Science Publishers B.V., 1241 (1989).
J.Brooks, G.Cairns, G.Davis, P.Stacey: On Devaney's definition of chaos. The American Mathematical Monthly, 99,4, 332 (1992).
A.W. Burks: Essays on Cellular automata. University of Illinois Press. Urbana 1970.
P.Collet, J.P.Eckmann.: Iterated Maps on the Interval as Dynamical Systems. Birkhauser, Basel 1980.
J.P.Crutchfield, K.Young: Computation at the onset of chaos. in: Complexity, Entropy and the Physics of Information, SFI Studies in the Sciences of Complexity, vol VIII (W.H.Zurek, ed.), Addison Wesley 223 (1990).
K. Culik II, S.Yu: Cellular automata, ΩΩ-regular sets, and sofic systems. Discrete Applied Mathematics 32,85–101 (1991).
K.Culik II, L.P.Hurd, S.Yu: Computation theoretic aspects of cellular automata. Physica D 45, 357–378 (1990).
R.L. Devaney: An Introduction to Chaotic Dynamical Systems. Addison-Wesley, Redwood City 1989.
D.Fiebig, U.R.Fiebig: Covers for coded systems. Symbolic Dynamics and its Applications (Peter Walters, ed.) American Mathematical Society, Providence 1992.
J.G.Hocking: Topology. Addison-Wesley, Reading 1961.
J.E.Hopcroft, J.D.Ullmann: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Menlo Park 1990.
J.Guckenheimer: Sensitive dependence to initial conditions for one dimensional maps. Commun.Math.Phys. 70, 133 (1979).
J.Guckenheimer, P. Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences 42, Springer-Verlag, Berlin 1983.
S.A.Kauffman, S.Johnson: Co-evolution at the edge of chaos: coupled fitness landscapes, poised states, and co-evolutionary avalanches. in: Artificial Life II. SFI Studies in the Sciences of Complexity, vol. X (Ch.G.Langton, Ch. Taylor, J.D.Farmer, S.Rasmunsen, eds.) Addison-Wesley, Redwood City 1992.
P.Koiran,P.Cosnard,M.Garzon: Computability with low-dimensional dynamical systems. Rapport n 92-31, Ecole Normal Superieure de Lyon, 1992.
C.Kuratowski: Topologie. Polskie Towarzystwo Matematyczne, Warszawa 1952.
P. Kůrka: Ergodic languages. Theoretical Computer Science 21:351–355, (1982).
P.Kůrka: One-dimensional dynamics and factors of finite automata. to appear in Acta Universitatis Carolinae, Mathematica et Physica.
P.Kůrka: Regular unimodal systems and factors of finite automata. submitted to Theoretical Computer Science.
Ch.G.Langton: Life at the Edge of Chaos. in: Artificial Life II. SFI Studies in the Sciences of Complexity, vol. X (Ch.G.Langton, Ch.Taylor, J.D.Farmer, S.Rasmunsen, eds.) Addison-Wesley, Redwood City 1992.
J.Milnor, W.Thurston: On iterated maps of the interval. Dynamical Systems (J.C.Alexander, ed.) Lecture Notes in Mathematics 1342, pp. 465–563, Springer-Verlag, Berlin 1988.
M.Mitchell, P.T.Hraber, J.P.Crutchfield: Revisiting the edge of chaos: evolving cellular automata to perform computations. Complex Systems, in press.
Ch. Moore: Unpredictability and undecidability in dynamical systems. Physical Review Letters 64(20), 2354–2357, (1990).
M.Shub: Global Stability of Dynamical Systems. Springer-Verlag, Berlin 1987.
A.R.Smith: Simple computation-universal cellular spaces. J. ACM 18, 339–353 (1971).
S.M.Ulam, J.von Neumann: On combinations of stochastic and deterministic processes. Bull.Amer.Math.Soc. 53, 1120, (1947).
S.Wolfram: Theory and Applications of Cellular Automata. World Scientific, Singapore, 1986.
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Kůrka, P. (1995). Simplicity criteria for dynamical systems. In: Andersson, S.I. (eds) Analysis of Dynamical and Cognitive Systems. Lecture Notes in Computer Science, vol 888. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58843-4_19
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DOI: https://doi.org/10.1007/3-540-58843-4_19
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