Natural Computing

, Volume 10, Issue 4, pp 1295–1312 | Cite as

Computability in planar dynamical systems

Article

Abstract

In this paper we explore the problem of computing attractors and their respective basins of attraction for continuous-time planar dynamical systems. We consider C1 systems and show that stability is in general necessary (but may not be sufficient) to attain computability. In particular, we show that (a) the problem of determining the number of attractors in a given compact set is in general undecidable, even for analytic systems and (b) the attractors are semi-computable for stable systems. We also show that the basins of attraction are semi-computable if and only if the system is stable.

Keywords

Computability Planar dynamical systems Equilibrium points Limit cycles Basins of attraction 

Notes

Acknowledgments

D. Graça was partially supported by Fundação para a Ciência e a Tecnologia and EU FEDER POCTI/POCI via CLC, SQIG - Instituto de Telecomunicações. DG was also attributed a Taft Research Collaboration grant which made possible a research visit to U. Cincinnati. N. Zhong was partially supported by the 2009 Taft Summer Research Fellowship.

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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.DM/FCT, Universidade do AlgarveFaroPortugal
  2. 2.SQIG, Instituto de TelecomunicaçõesLisbonPortugal
  3. 3.DMSUniversity of CincinnatiCincinnatiUSA

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