Autonomous systems of ordinary differential equations (x′=f(x) with x(0) = x₀) are considered. It is assumed that one-step integration methods (x(t+h)=x(t)+hf_h(x(t)), t=kh, k=0,1, . . . ) are used to obtain approximations to the solution. Furthermore, it is assumed that the system of ordinary differential equations has a hyperbolic periodic orbit. The main purpose is to find out how the behavior of the exact solution is reflected in the numerical solution by the recursion defined by the integration method. It is proved that the one step method has a closed invariant curve, which may be parametrized. It is also proved that this curve converges to the periodic orbit, the speed of convergence being O(h^r), where r is the order of the one step method used. The theoretical results are illustrated graphically by plotting the numerical solutions of a simple model (for a food chain) that are calculated by both the Euler method and a Runge-Kutta method of order four, using several stepsizes in each case. The relation of the results obtained in this paper to other results obtained in this direction is discussed. The extension of the results to a variable stepsize mesh is briefly described.