Basins of attraction estimation through symbolic graphical computing techniques
In the present paper an elementary symbolic method for the estimation of basins of attraction in second order nonlinear dynamical systems is formulated and its implementation using Mathematica® is shown. The estimation algorithm is based upon the construction of positively invariant compact boxes that trap the trajectories with unbounded initial conditions. We obtain such boxes through a Lyapunov function whose orbital derivative is bounded by a bounding function that can be represented as the addition of two scalar functions. The detection of persistent oscillating behaviors deserves a prominent place between all possible applications of our tool, as it will be shown by considering Fitzhugh Equations as a case study.
KeywordsDissipative Dynamical Systems Basins of Attraction Lyapunov Functions Nonlinear Oscillations Symbolic Computing
Unable to display preview. Download preview PDF.
- [FiR]Fitzhugh, R., Mathematical Models of Excitation and Propagation in Nerve, in Biological Engineering, Schwan, G. (Editor), 1–85, McGraw-Hill Publishing Company, New York, 1969.Google Scholar
- [H-C]Hale, J. and KoÇak, H., Dynamics and Bifurcations, Springer-Verlag, New York, 1991.Google Scholar
- [PlV]Pliss, V., Nonlocal Problems of the Theory of Oscillations, Academic Press, New York, 1966.Google Scholar
- [RM1]Rodríguez-Millán, J., Multiparameter Hopf Bifurcations in Fitzhugh Equation, Proceedings of the Eleventh International Conference on Nonlinear Oscillations, 479–481, Budapest, August 17–23, 1987.Google Scholar
- [RM2]Rodríguez-Millán, J., A Topological Approach to the Global Dynamics of Fitzhugh Equation Without Diffusion, MSc Dissertation, Facultad de Ciencias, Universidad Central de Venezuela, Caracas, 1992. (In Spanish)Google Scholar
- [RM3]Rodríguez-Millán et al, PAH, A Symbolic-Graphical Tool for the Study of Nonlinear Oscillations in Second Order Dynamical Systems, Program and Abstracts of the First European Nonlinear Oscillations Conference, Hamburg, August 16–20, 1993.Google Scholar
- [RuW]Rudin, W., Real and Complex Analysis, Second Edition, McGraw-Hill Publishing Company, New York, 1979.Google Scholar
- [Troy]Troy, Bifurcation Phenomena in Fitzhugh's Nerve Conduction Equations, J. Math. Anal. Appl. 54, 678–690, 1976Google Scholar