Stabilized rapid oscillations in a delay equation: Feedback control by a small resonant delay

Abstract

We study scalar delay equations $$\dot{x}(t)=\lambda f(x(t-1))+b^{-1}(x(t)+x(t-p/2))$$ with odd nonlinearity $f$, real nonzero parameters $\lambda ,b$, and two positive time delays $1,p/2$. We assume supercritical Hopf~bifurcation from $x\equiv 0$ in the well-understood single-delay case $b=\infty$. Normalizing $f^{\prime }(0)=1$, branches of constant minimal period $p_k=2\pi /{\omega }_k$ are known to bifurcate from eigenvalues $i{\omega }_k=i(k+\frac{1}{2})\pi$ at ${\lambda }_k=(-1{)}^{k+1}{\omega }_k$, for any nonnegative integer $k$. The unstable dimension of these rapidly oscillating periodic solutions is $k$, at the local branch $k$. We obtain stabilization of such branches, for arbitrarily large unstable dimension $k$, and for, necessarily, delicately narrow regions of control amplitudes $b<0$. For $p$:= $p_k$ the branch $k$ of constant period $p_k$ persists as a solution, for any $b\ne 0$. Indeed the delayed feedback term controlled by $b$ vanishes on branch $k$: the feedback control is noninvasive there. Following an idea of Pyragas (1992), we seek parameter regions $\mathcal{P}=({\underline{b}}_k,{\overline{b}}_k)$ of controls $b\ne 0$ such that the branch $k$ becomes stable, locally at Hopf~bifurcation. We determine rigorous expansions for $\mathcal{P}$ in the limit of large $k$. Our analysis is based on a 2-scale covering lift for the slow and rapid frequencies involved. These results complement earlier results by Fiedler and Oliva (2016) which required control terms $$b^{-1}(x(t-\vartheta )+x(t-\vartheta -p/2))$$ with a third delay $\vartheta$ near 1.

arXiv comment49 pages, 7 figures