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

TL;DR

This paper investigates stabilizing rapid oscillations in a delay differential equation using a small resonant delay feedback, providing rigorous conditions for stabilization even in highly unstable regimes.

Contribution

It introduces a novel stabilization method for high-dimensional unstable oscillatory solutions in delay equations via narrow control parameter regions.

Findings

Stabilization of high unstable dimension solutions is achieved.

Control regions are precisely characterized for large oscillation frequencies.

The method extends previous feedback control approaches with a simplified delay structure.

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.