Delayed Feedback Control of a Delay Equation at Hopf Bifurcation

Abstract

We embark on a case study for the scalar delay equation $$\begin{aligned} \dot{x} (t) = \lambda f(x(t-1)) + b^{-1} (x(t-\vartheta ) + x(t-\vartheta -p/2)) \end{aligned}$$ with odd nonlinearity f, real nonzero parameters \(\lambda , \, b\), and three positive time delays \(1,\, \vartheta ,\, p/2\). We assume supercritical Hopf bifurcation from \(x \equiv 0\) in the well-understood single-delay case \(b = \infty \). Normalizing \(f' (0)=1\), branches of constant minimal period \(p_k = 2\pi /\omega _k\) are known to bifurcate from eigenvalues \(i\omega _k = i(k+\tfrac{1}{2})\pi \) at \(\lambda _k = (-1)^{k+1}\omega _k\), for any nonnegative integer k. The unstable dimension is k, at the local branch k. We obtain stabilization of such branches, for arbitrarily large unstable dimension k. For \(p:= p_k\) the branch k of constant period \(p_k\) persists as a solution, for any \(b\ne 0\) and \(\vartheta \ge 0\). Indeed the delayed feedback term controlled by b vanishes on branch k: the feedback control is noninvasive there. Following an idea of Pyragas, we seek parameter regions \(\mathcal {P}\) of controls \(b \ne 0\) and delays \(\vartheta \ge 0\) such that the branch k becomes stable, locally at Hopf bifurcation. We determine rigorous expansions for \(\mathcal {P}\) in the limit of large k. The only two regions \(\mathcal {P} = \mathcal {P}^\pm \) which we were able to detect, in this setting, required delays \(\vartheta \) near 1, controls b near \((-1)^k \cdot 2/\omega _k\), and were of very small area of order \(k^{-4}\). Our analysis is based on a 2-scale covering lift for the frequencies involved.