Natural Computing

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

Computability in planar dynamical systems

  • Daniel GraçaEmail author
  • Ning Zhong


In this paper we explore the problem of computing attractors and their respective basins of attraction for continuous-time planar dynamical systems. We consider C 1 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.


Computability Planar dynamical systems Equilibrium points Limit cycles Basins of attraction 



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.


  1. Aubin J-P, Cellina A (1984) Differential inclusions: set-valued maps and viability theory. Number 364 in Grundlehren der Mathematischen Wissenschaften. Springer, BerlinGoogle Scholar
  2. Blondel VD, Bournez O, Koiran P, Tsitsiklis JN (2001) The stability of saturated linear dynamical systems is undecidable. J Comput Syst Sci 62:442–462MathSciNetzbMATHCrossRefGoogle Scholar
  3. Braverman M, Yampolsky M (2006) Non-computable Julia sets. J Am Math Soc 19(3):551–0578MathSciNetzbMATHCrossRefGoogle Scholar
  4. Buescu J, Graça DS, Zhong N (2006) Computability and dynamical systems. In Pinto A, Peixoto M, Rand D (eds) Dynamics and games in science I and II. Springer, New YorkGoogle Scholar
  5. Collins P (2005) Continuity and computability of reachable sets. Theor Comput Sci 341:162–195zbMATHCrossRefGoogle Scholar
  6. Collins P, Graça DS (2009) Effective computability of solutions of differential inclusions—the ten thousand monkeys approach. J Univ Comput Sci 15(6):1162–1185MathSciNetzbMATHGoogle Scholar
  7. Deimling K (1984) Multivalued differential equations. Number 1 in de Gruyter series in nonlinear analysis and applications. Walter de Gruyter and Co, BerlinGoogle Scholar
  8. Graça DS, Zhong N (2009) Computing domains of attraction for planar dynamics. In: Calude CS, Costa JF, Dershowitz N, Freire E, Rozenberg G (eds) 8th international conference on unconventional computation (UC 2009) (LNCS), vol 5715. Springer, New York, pp 179–190. Google Scholar
  9. Graça DS, Zhong N, Buescu J (2009) Computability, noncomputability and undecidability of maximal intervals of IVPs. Trans Am Math Soc 361(6):2913–2927MathSciNetzbMATHCrossRefGoogle Scholar
  10. Graça DS, Campagnolo ML, Buescu J (2009) Computational bounds on polynomial differential equations. Appl Math Comput 215(4):1375–1385MathSciNetzbMATHCrossRefGoogle Scholar
  11. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Springer, New YorkGoogle Scholar
  12. Hirsch MW, Smale S (1974) Differential equations, dynamical systems, and linear algebra. Academic Press, New YorkzbMATHGoogle Scholar
  13. Hoyrup M (2007) Dynamical systems: stability and simulability. Math Struct Comput Sci 17:247–259MathSciNetzbMATHCrossRefGoogle Scholar
  14. Hubbard JH, West BH (1995) Differential equations: a dynamical systems approach—higher-dimensional systems. Springer, New YorkCrossRefGoogle Scholar
  15. Ko K-I (1991) Computational complexity of real functions. Birkhäuser, BostonGoogle Scholar
  16. Lorenz EN (1963) Deterministic non-periodic flow. J Atmos Sci 20:130–141CrossRefGoogle Scholar
  17. Moore C (1990) Unpredictability and undecidability in dynamical systems. Phys Rev Lett 64(20):2354–2357MathSciNetzbMATHCrossRefGoogle Scholar
  18. Odifreddi P (1989) Classical recursion theory, vo 1. Elsevier, AmsterdamGoogle Scholar
  19. Perko L (2001) Differential equations and dynamical systems, 3rd edn. Springer, New YorkGoogle Scholar
  20. Pour-El MB, Richards JI (1989) Computability in analysis and physics. Springer, New YorkzbMATHGoogle Scholar
  21. Puri A, Borkar V, Varaiya P (1995) Epsilon-approximation of differential inclusions. In: Proceedings of the 34th IEEE conference on decision and control, New Orleans, pp 2892–2897Google Scholar
  22. Rettinger R, Weihrauch K, Zhong N (2009) Topological complexity of blowup problems. J Univ Comput Sci 15(6):1301–1316MathSciNetzbMATHGoogle Scholar
  23. Viana M (2000) What’s new on Lorenz strange attractors?. Math Intell 22(3):6–19MathSciNetzbMATHCrossRefGoogle Scholar
  24. Weihrauch K (2000) Computable analysis: an introduction. Springer, New YorkzbMATHGoogle Scholar
  25. Zhong N (2009) Computational unsolvability of domain of attractions of nonlinear systems. Proc Am Math Soc 137:2773–2783MathSciNetzbMATHCrossRefGoogle Scholar
  26. Zhong N, Weihrauch K (2003) Computability theory of generalized functions. J ACM (JACM) 50(4):469–505MathSciNetCrossRefGoogle Scholar

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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