On invariant closed curves for one-step methods (Q1822214)

Autonomous differential equations \(x'=f(x)\) are considered which possess a hyperbolic closed orbit \(\gamma\). It is shown that the application of a one-step method (e.g. an explicit Runge-Kutta method) yields an invariant closed curve \(\gamma_ h\) for sufficiently small (constant) step size h. This closed curve converges for \(h\to 0\) to the periodic orbit of the system with the order of the method and it inherits the stability of the periodic orbit. Several numerical experiments confirm the theoretical results.