This paper considers the invariant sets of numerical one-step integration methods in a neighbourhood of a hyperbolic periodic solution of a nonlinear ODE. Using results from the dynamical systems theory it was possible to show that for the usual one-step methods the invariant sets areCk-circles (closed curves) for small enough stepsizeh. Here we give a direct proof for that and also show that they areO(hp)Ck-close to the true periodic trajectory, wherep is the order of the method.

Invariant curves of one-step methods

We generalize a result of Kirchgraber (1986) on multistep methods. We show that every strictly stable general linear method is essentially conjugate to a one step method of the same order. This result may be used to show that general properties of one step methods carry over to general linear methods. As examples we treat the existence of invariant curves and the construction of attracting sets.

General linear methods: connection to one step methods and invariant curves

It is shown that a discrete multi-time method can be constructed to obtain approximations to the periodic solutions of a special class of second-order nonlinear difference equations containing a small parameter Three examples illustrating the method are presented

Periodic solutions of second-order nonlinear difference equations containing a small parameter. IV: Multi-discrete time method

This paper deals with the long-time behaviour of numerical solutions of delay differential equations that have asymptotically stable periodic orbits. It is shown that Runge-Kutta discretizations of such equations have attractive invariant curves which approximate the periodic orbit with the order of the method.

Periodic orbits of delay differential equations under discretization

Two concepts for numerical periodic solutions of ODE's

The eigenvalues of the matrix A are 1 =b hi. Hence all the iterates 2/1,2/2, ■ • • , of (2) spiral away from the origin. On the other hand, every non-trivial solution of (1) is periodic with period 2ir. The essential difficulty with the differential equation (1) is that the equilibrium solution x = x = 0 is a center. Equivalently, the periodic solutions of (I) are not orbitally asymptotically stable, and thus they may be destroyed under an arbitrary small perturbation. The situation is very different, however, when x = <fi(t) (x = (a:, , ■ • • , x„)) is an orbitally asymptotically stable (unstable) solution of the differential equation

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Periodic solutions of finite difference equations

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Periodic solutions of finite difference equations