Numerische Mathematik

Numer Math 66,399^09 (1993) V'i '• ü

Numerische Mathematik

© Springer Verlag 1993

Two - sided approximation to periodic solutions of ordinary differential equations*

Liqing Zhang

Automation Department, South Chma University of Technology, Guangzhou, Post Code 510641, P R China

Received September 28, 1990/Revised version received March 24, 1993

Summary . A two-sided approximation to the periodic orbit of an autonomous ordinary differential equation system is considered First some results about variational equation systems for periodic solutions are obtained in Sect 2 Then it IS proved that if the periodic orbit is convex and stable, the explicit difference solution approximates the periodic orbit from the outer part and the implicit one from the inner part respectively Finally a numerical example is given to illustrate our result and to point out that the numerical solution no longer has a one-sided approximation property, if the periodic orbit is not convex

Mathematical Subject Classification (1991) 65L10

1 . Introduction

The numerical approximation of periodic solutions of autonomous differential equation systems has been a topic of considerable interest in recent years Many numerical methods, like the hnear multi-step method, Galerkm method, the multi- shooting method and the collocation method, have been employed to approximate the periodic solutions [3-9] But there seems to be very httle hterature on the numerical analysis of the geometric properties of ODEs

The questions we are interested in here are

Ql What geometric properties of a periodic solution can a numerical method preserve*^

Q2 Can we find two numerical methods which approximate a limit cycle from one side separately*^ III [10]» we have proved that the cubic spline collocation solution preserves the convexness of the periodic solution invariant, and approximates the periodic solution from one side under certain conditions But this condition is rather complicated because the 5-th derivative of the periodic solution is involved

* The Work is supported by the National Natural Science Foundation of China