An analytic System with a Computable Hyperbolic Sink Whose Basin of Attraction is Non-Computable

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.

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Acknowledgments

D. Graça was partially supported via R&D Unit 50008, financed by the applicable financial framework (FCT/MEC through national funds and when applicable co-funded by FEDER - PT2020 partnership agreement).

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Correspondence to Daniel S. Graça.

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Graça, D.S., Zhong, N. An analytic System with a Computable Hyperbolic Sink Whose Basin of Attraction is Non-Computable. Theory Comput Syst 57, 478–520 (2015). https://doi.org/10.1007/s00224-015-9609-5

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Keywords

  • Computability with real numbers
  • Basins of attractions
  • Asymptotically stable equilibrium points
  • Reachability problem