Deciding stability and mortality of piecewise affine dynamical systems

This paper studies problems such as: given a discrete time dynamical system of the form x(t + 1) = f(x(t)) where f : R → R is a (possibly discontinuous) piecewise affine function, decide whether all trajectories converge to 0. We show in our main theorem (Theorem 2) that this Attractivity Problem is undecidable as soon as n ≥ 2. The same is true of two related problems: Stability (is the dynamical system globally asymptotically stable?) and Mortality (decide whether all trajectories go through 0). In section 4 we show that Attractivity and Stability become decidable in dimension 1 for continuous functions, and these two notions become in fact equivalent. One can show with similar techniques that Mortality is also decidable for piecewise affine continuous functions of one variable. It is well-known that various types of dynamical systems, such as hybrid systems or piecewise affine functions, can simulate Turing machines, see, e.g., [11], [7]. In these simulations, a machine configuration is encoded by a point