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Deciding stability and mortality of piecewise affine dynamical systems. (English) Zbl 0973.68067
Summary: In this paper we study problems such as: given a discrete time dynamical system of the form \(x(t+1)=f (x(t))\) where \(f: R^{n}\rightarrow R^{n}\) is a piecewise affine function, decide whether all trajectories converge to 0. We show in our main theorem that this attractivity problem is undecidable as soon as \(n\geqslant 2\). The same is true of two related problems: Stability (is the dynamical system globally asymptotically stable?) and mortality (do all trajectories go through 0?). We then show that attractivity and stability become decidable in dimension 1 for continuous functions.

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