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Undecidable event detection problems for ODEs of dimension one and two. (English) Zbl 0878.68062

Summary: The ability of dynamical systems of various kinds to simulate Turing machines and thus manifest a universal computation power (and beyond) has gathered a lot of interest lately. A similar line of investigation for ordinary differential equations was started by the author [J. Inf. Process. Cybern. 29, No. 2, 101-113 (1993; Zbl 0771.65035)] and continued in [Event detection and nonrecursive hierarchies, Lect. Notes Comput. Sci. 812, 358-371 (1994), Decidability and complexity of event detection problems for ODEs (to appear in complexity)]. In this context the minimum dimension required for universal computation is of interest. The dynamical systems in C. Moore [Unpredictability and undecidability in dynamical systems, Phys. Rev. Lett. 64, 2354-2357 (1990) and Nonlinearity 4, No. 2, 199-230 (1991; Zbl 0725.58013)] are of small dimension and the topic of P. Koiran, M. Cosnard and M. Garzon [Theor. Comput. Sci. 132, No. 1-2, 113-128 (1994; Zbl 0821.68053)] is to find the smallest dimension for certain types of dynamical systems. The results in this paper show that for ODEs dimension two can be reached and, allowing somewhat complicated events, even dimension one.

MSC:

68Q05 Models of computation (Turing machines, etc.) (MSC2010)

Keywords:

Turing machines
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References:

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