Recently we have proved that Naimark-Sacker bifurcation (Hopf bifurcations for maps) occur in the Euler method applied to the delay differential equation x'(t) = -µ*x(t-1)[1-x(t)²] (BIT 39, pp. 110-115). By slightly modifying the proof, it is verified that the same result holds, e.g., for the delay differenial equation x'(t) = -µ*tanh[x(t-1)], which is obtained from the equation above 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 latter as a weak resonance phenomenon 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, 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.

MINISIMPOSIUM SESSION: 3. Numerical methods for delay equations (2 sessions)