An analytic system with a computable hyperbolic sink whose basin of   attraction is non-computable
  In many applications one is interested in finding the stability regions (basins of attraction) of some stationary states (attractors). In this paper we show that one cannot compute, in general, the basins of attraction of even very regular systems, namely analytic systems with hyperbolic asymptotically stable equilibrium points. To prove the main theorems, a new method for embedding a discrete-time system into a continuous-time system is developed. 
Daniel S. Gra\c{c}a and Ning Zhong
math.LO