Triple equivalence of the oscillatory behavior for scalar delay differential equations

P. N. Nesterov ; J. I. Stavroulakis

Teoretičeskaâ i matematičeskaâ fizika, Tome 220 (2024) no. 1, pp. 113-136

Résumé

We study the oscillation of a first-order delay equation with negative feedback at the critical threshold $1/e$. We apply a novel center manifold method, proving that the oscillation of the delay equation is equivalent to the oscillation of a $2$-dimensional system of ordinary differential equations (ODEs) on the center manifold. It is well known that the delay equation oscillation is equivalent to the oscillation of a certain second-order ODE, and we furthermore show that the center manifold system is asymptotically equivalent to this same second-order ODE. In addition, the center manifold method has the advantage of being applicable to the case where the parameters oscillate around the critical value $1/e$, thereby extending and refining previous results in this case.

Détail

Keywords: delay differential equation, critical state, center manifold, asymptotics.
Mots-clés : oscillation problem

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