Computability in planar dynamical systems
- 113 Downloads
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.
KeywordsComputability 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.
- Aubin J-P, Cellina A (1984) Differential inclusions: set-valued maps and viability theory. Number 364 in Grundlehren der Mathematischen Wissenschaften. Springer, BerlinGoogle Scholar
- 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
- Deimling K (1984) Multivalued differential equations. Number 1 in de Gruyter series in nonlinear analysis and applications. Walter de Gruyter and Co, BerlinGoogle Scholar
- 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
- Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Springer, New YorkGoogle Scholar
- Ko K-I (1991) Computational complexity of real functions. Birkhäuser, BostonGoogle Scholar
- Odifreddi P (1989) Classical recursion theory, vo 1. Elsevier, AmsterdamGoogle Scholar
- Perko L (2001) Differential equations and dynamical systems, 3rd edn. Springer, New YorkGoogle Scholar
- 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