Summary
We show that a one-step method as applied to a dynamical system with a hyperbolic periodic orbit, exhibits an invariant closed curve for sufficiently small step size. This invariant curve converges to the periodic orbit with the order of the method and it inherits the stability of the periodic orbit. The dynamics of the one-step method on the invariant curve can be described by the rotation number for which we derive an asymptotic expression. Our results complement those of [2, 3] where one-step methods were shown to create invariant curves if the dynamical system has a periodic orbit which is stable in either time direction or if the system undergoes a Hopf bifurcation.
Similar content being viewed by others
References
Beyn, W.-J.: On the numerical approximation of phase portraits near stationary points. SIAM J. Numer. Anal. (to appear)
Brezzi, F., Ushiki, S., Fujii, H.: Real and ghost bifurcation dynamics in difference schemes for ODEs, in Numerical Methods for Bifurcation Problems (T. Küpper, H.D. Mittelmann, H. Weber, eds.). pp. 79–104. Boston: Birkhäuser 1984
Braun, M., Hershenov, J.: Periodic solutions of finite difference equations. Quart. Appl. Math.35, 139–147 (1977)
Doan, H.T.: Invariant curves for numerical methods. Quart. Appl. Math.43, 385–393 (1985)
Eirola, T.: Two concepts for numerical periodic solutions of ODE's. Appl. Math. Comput. (to appear)
Eirola, T.: Invariant circles of one-step methods. BIT (Submitted)
Grigorieff, R.D.: Numerik gewöhnlicher Differentialgleichungen 1. Stuttgart: Teubner 1972
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Appl. Math. Sci.42. Berlin, Heidelberg, New York: Springer 1983
Hale, J.: Ordinary Differential Equations. New York: John Wiley 1969
Iooss, G.: Bifurcation of Maps and Applications. Math. Stud.36. Amsterdam: North Holland 1979
Irwin, M.C.: Smooth Dynamical Systems. New York: Academic Press 1980
Kloeden, P.E., Lorenz, J.: Stable attracting sets in dynamical systems and their one step discretizations. SIAM J. Numer. Anal.23, 986–995 (1986)
Sell, G.R.: What is a dynamical system? In: Studies in Ordinary Differential Equations (J. Hale, ed.), pp. 32–51. Math. Assoc. Am. 1977
Stetter, H.J.: Analysis of Discretization Methods for Ordinary Differential Equations. Berlin, Heidelberg, New York: Springer 1973
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Beyn, WJ. On invariant closed curves for one-step methods. Numer. Math. 51, 103–122 (1987). https://doi.org/10.1007/BF01399697
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01399697