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Periodic orbits in the Euler method for a class of delay differential equations. (English) Zbl 1161.65367

Summary: Recently, we have proved that Naimark-Sacker bifurcations occur in the Euler method applied to a delay differential equation [BIT 39, No. 1, 110–115 (1999; Zbl 0918.65054)]. 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 J. L. Kaplan and J. A. Yorke [J. Math. Anal. Appl. 48, 317–324 (1974; Zbl 0293.34102)], 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.

MSC:
65P30Bifurcation problems (numerical analysis)
34K13Periodic solutions of functional differential equations
34K18Bifurcation theory of functional differential equations
37G15Bifurcations of limit cycles and periodic orbits