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Unpredictability and undecidability in dynamical systems. (English) Zbl 1050.37510

Summary: We show that motion with as few as three degrees of freedom (for instance, a particle moving in a three-dimensional potential) can be equivalent to a Turing machine, and so be capable of universal computation. Such systems possess a type of unpredictability qualitatively stronger than that which has been previously discussed in the study of low-dimensional chaos: Even if the initial conditions are known exactly, virtually any question about their long-term dynamics is undecidable.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
03D10 Turing machines and related notions
37E99 Low-dimensional dynamical systems
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[1] S. Wolfram, Commun. Math. Phys. 96 pp 15– (1984) · Zbl 0587.68050 · doi:10.1007/BF01217347
[2] S. Wolfram, Phys. Rev. Lett. 54 pp 735– (1984) · doi:10.1103/PhysRevLett.54.735
[3] S. Omohundro, Physica (Amsterdam) 10D pp 128– (1984)
[4] E. Fredkin, Int. J. Theor. Phys. 21 pp 219– (1982) · Zbl 0496.94015 · doi:10.1007/BF01857727
[5] S. Smale, in: Differential and Combinatorial Topology (1963)
[6] , in: Hamiltonian Dynamical Systems (1987)
[7] P. Cvitanović, Phys. Rev. Lett. 61 pp 2729– (1988) · doi:10.1103/PhysRevLett.61.2729
[8] H. Rogers, Jr., in: Theory of Recursive Functions and Effective Computability (1967)
[9] C. Crebogi, Phys. Lett. 99A pp 415– (1983)
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