Refuting the Odd-Number Limitation of Time-Delayed Feedback Control

We refute an often invoked theorem which claims that a periodic orbit with an odd number of real Floquet multipliers greater than unity can never be stabilized by time-delayed feedback control in the form proposed by Pyragas. Using a generic normal form, we demonstrate that the unstable periodic orbit generated by a subcritical Hopf bifurcation, which has a single real unstable Floquet multiplier, can in fact be stabilized. We derive explicit analytical conditions for the control matrix in terms of the amplitude and the phase of the feedback control gain, and present a numerical example. Our results are of relevance for a wide range of systems in physics, chemistry, technology, and life sciences, where subcritical Hopf bifurcations occur.

Figure 1

Pyragas (red dashed) and Hopf (black solid) curves in the ($\lambda$, $\tau$) plane: (a) Hopf bifurcation curves $n=0,\dots ,10$, (b) Hopf bifurcation curves $n=0$, 1 in an enlarged scale. Yellow (light gray) shading marks the domains of unstable $z=0$ and numbers in parentheses denote the dimension of the unstable manifold of $z=0$ ($\gamma =-10$, $b_0=0.3$, and $\beta =\pi /4$).

Figure 2

(a) Top: Real part of Floquet exponents $\Lambda$ of the periodic orbit vs feedback amplitude $b_0$. Bottom: Real part of eigenvalue $\eta$ of the steady state vs feedback amplitude $b_0$. (b) Floquet multipliers $\mu =\exp (\Lambda \tau )$ in the complex plane with the feedback amplitude $b_0\in [0,0.3]$ as a parameter. (c) Radii of periodic orbits. Solid (dashed) lines correspond to stable (unstable) orbits. ($\lambda =-0.005$, $\gamma =-10$, $\tau =\frac{2\pi }{1-\gamma \lambda }$, $\beta =\pi /4$).

Figure 3

Domain of control in the plane of the complex feedback gain $b=b_0{e}^{i\beta }$ for three different values of the bifurcation parameter $\lambda$. The black solid curves indicate the boundary of stability in the limit $\lambda \nearrow 0$; see (18, 19). The color-shading shows the magnitude of the largest (negative) real part of the Floquet exponents of the periodic orbit ($\gamma =-10$, $\tau =\frac{2\pi }{1-\gamma \lambda }$).