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Two-sided approximation to periodic solutions of ordinary differential equations. (English) Zbl 0799.65077
A two-sided approximation to the periodic orbit of an autonomous ordinary differential equation system is presented. It is proved that the implicit and explicit Euler difference methods approximate the periodic solution from inner and outer sides separately, provided the periodic orbit is convex and stable.
A numerical example is given to illustrate the result and to point out that the convex condition is necessary for the difference methods in a one-sided approximation and the numerical orbit no longer possesses a one-sided approximation property, if the periodic orbit is not convex.

##### MSC:
 65L05 Numerical methods for initial value problems involving ordinary differential equations 65L12 Finite difference and finite volume methods for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 34C25 Periodic solutions to ordinary differential equations
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##### References:
 [1] Arnold, V.I. (1983): Geometrical methods in the theory of ordinary differential equation. Springer Berlin Heidelberg New York · Zbl 0507.34003 [2] Coddington, E.A., Levison, N. (1955): Theory of ordinary differential equations. McGraw-Hill, New York · Zbl 0064.33002 [3] Deuflhard, P. (1984): Computation of periodic solutions of nonlinear ODEs. BIT24, 456-466 · Zbl 0567.65049 [4] Doan, H.T. (1985): Invariant curves for numerical methods. Quart. Appl. Math.43, 385-393 · Zbl 0591.65055 [5] Jepson, A.D., Keller, H.P. (1984): Steady state and periodic solution paths: their bifurcations and computation. In: T. Küppar, H.D. Mittelmann, H. Weser eds., Numerical methods for bifurcation problems, ISNM 70, Birkhauser, Basel, pp. 219-246 · Zbl 0579.65057 [6] Langford, W.F. (1977): Numerical solution of bifurcation problems for ODEs. Numer. Math.28, 171-190 · Zbl 0344.65042 [7] Seydel, R. (1981): Numerical computation of periodic orbits that bifurcate from stationary solution of ODEs. Appl. Math. Comput.9, 257-271 · Zbl 0491.65049 [8] Zhang, L. (1991): Convergence of difference method for periodic solutions of ordinary differential equations. J. South China Univ. Tech.19(1), 73-82 [9] Zhang, L. (1992): Spline collocation approximation to periodic solutions of ODEs. J. Comput. Math.10, 147-154 · Zbl 0776.65051 [10] Zhang, L. (1991): Asymptotic expansion and geometric properties of the spline collocation periodic solution of an ODE system. Appl. Math. & Comput.42, 209-221 · Zbl 0728.65075 [11] Zhang, L. (1988): Spline approximation to periodic solutions of ODEs. Ph.D. Dissertation of Zhongshan University, China
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