Periodic orbits in the Euler method for a class of delay differential equations

Recently, we have proved that Naimark-Sacker bifurcations occur in the Euler method applied to a delay differential equation [1]. By slightly modifying the proof, it is verified that the same result holds, e.g., for another equation obtained from the equation by a change of the dependent variable. However, in computer experiments, the Euler method presents different behavior for the two equations: an invariant circle is observed in the former case; a stable periodic orbit is observed in the latter case. We show that it is reasonable to consider the behavior in the latter case as a weak resonance in the Naimark-Sacker bifurcation. More specifically, we study a class of delay differential equations which includes the second equation, paying attention to special periodic solutions found by Kaplan and Yorke [2], and prove constructively that the Euler method applied to each equation of the class has at least two periodic orbits. Numerical experiments are presented which indicate that one orbit is stable and the other orbit is unstable, whose unstable manifold forms an invariant circle.