A solution of a system of m autonomous differential equations defines a trajectory in m-dimensional space and, in particular, may give a closed orbital path. Typical trajectories are described by a model nonlinear problem introduced in this article. For this problem, a trajectory lies on a surface characterized by a real symmetric matrix. It is shown that some Runge-Kutta methods possess a property which ensures that, for this model problem, the numerical solution lies on the same surface as the trajectory. When m = 2, the numerical solution lies on the trajectory. This property is related to algebraic stability. A weaker property suffices for normalized differential systems.