An analytic system with a computable hyperbolic sink whose basin of attraction is non-computable

TL;DR

This paper demonstrates that the basins of attraction for certain stable systems are non-computable, introducing a novel embedding method to prove this in analytic hyperbolic systems.

Contribution

It presents a new method for embedding discrete systems into continuous ones and proves non-computability of basins of attraction in analytic hyperbolic systems.

Findings

Basins of attraction can be non-computable in analytic systems.

A new embedding method for discrete into continuous systems is developed.

Non-computability holds even for hyperbolic, stable equilibrium points.

Abstract

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.