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 $f' (0)=1$, 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 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 $b\neq 0$. 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 $b \neq 0$ 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.