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

Figure 1.1.  Supercritical Hopf bifurcations of (1.3) at $\lambda = \lambda_k $. Note the strict unstable dimensions $ E(\lambda_k) = k $ of the trivial equilibrium, in parentheses $ (k) $, and the inherited unstable dimensions $ [k] $, in brackets, of the local branches of bifurcating periodic orbits with constant minimal period $ p_k = 4/(2k+1) $. All branches consist of unstable rapidly oscillating periodic solutions, except for the stable slowly oscillating branch $ k = 0$. See [8]

Figure 1.2.  Additional Hopf curves (colored solid), zero eigenvalue (red dashed), and Takens-Bogdanov bifurcations (TB, black) at fixed $ \lambda = \lambda_k $, for odd $ k = 9 $, (a) top, and even $ k = 10 $, (b) bottom. The Hopf curves are generated by the control parameters $ \vartheta $ and $ b $ of the delayed feedback terms in (1.2). The more stable side is found towards smaller $ |b| $, at red Hopf branches, and towards larger $ |b| $, at blue branches. The same statement holds true at the zero eigenvalue; see the red dashed line. See [8] for further details

Figure 1.3.  Control induced Hopf curves in parameters $ (\vartheta, b) $, as in fig. 1.2, near $ \vartheta = 0 $. (a) $ k = 10 $, (b) zoom into $ k = 10 $, (c) $ k = 50 $, (d) zoom into $ k = 50 $. Vertical coordinates are $ B $, in (a), (c), and $ -\log(-B) $, for the zooms (b), (d), with scaled $ B = \frac{1}{2} b \omega_k $. Pyragas regions $ \mathcal{P} $ are indicated in green. Hopf curves $ \mu = i\tilde{\omega} $ with Hopf frequencies $ 0 < \tilde{\omega} < \omega_k $ are dashed (red), and Hopf curves with $ \tilde{\omega} > \omega_k $ are solid (red, blue). For color coding see fig. 1.2. Unstable dimensions $ E(b, \vartheta) $ of $ x\equiv 0 $, and of bifurcating periodic orbits, are indicated in parentheses

Figure 2.1.  Purely imaginary eigenvalues $\mu = i \tilde{\omega} = i\tilde{\omega}_{0, j}^\pm$ and Hopf control parameters $B = B_{0, j}^\pm$ at $\varepsilon = \omega_k^{-1} = ((k+\tfrac{1}{2})\pi)^{-1}$. The horizontal axis is $-1 \leq \Omega = \tilde{\Omega}-\Omega_0 \leq 0$, with $\Omega_0 = 1$. Left: odd $k$. Right: even $k$. Top row: hashing $\tilde{\Omega} = \varepsilon \tilde{\omega}$ alias $\Omega = \varepsilon(\omega + \ldots)$ according to lemma 2.3, (2.22)-(2.25) and (3.45). Note how $\tilde{\Omega} = \tilde{\Omega}_{0, j}^\pm = \varepsilon\tilde{\omega}_{0, 1}^\pm$ enumerate the Hopf frequencies defined by the intersections of the slanted hashing lines, of slope $1/\varepsilon$, with the relations $\tilde{\omega} = \tilde{\omega}^\pm (\tilde{\Omega})$, induced by the 2-scale characteristic equation; see lemma 2.4 and (3.49). Bottom row: the resulting control parameters $B = B_{0, j}^\pm = B^\pm (\tilde{\Omega}_{0, j}^\pm)$, also induced by the 2-scale characteristic equation according to lemma 2.4. Solid dots $\bullet$ indicate transverse Hopf bifurcations, where the Hopf pair $\mu = \pm i\omega$ crosses towards the stable side for decreasing $|B|$, see lemma 2.4(iv). Note the zero real eigenvalue $\square$ at "Hopf" frequency $\tilde{\omega} = 0$, for $B = (-1)^k$. Also note the non-crossing trivial Hopf pair $\blacksquare$ at $\mu = \pm i\omega_k$, which terminates the curves $B^-(\tilde{\Omega})$ at $\tilde{\Omega} = \varepsilon\omega_k = 1$

Figure 2.2.  Purely imaginary eigenvalues $ \mu = i\, \tilde{\omega} = i\, \tilde{\omega}_{m, j}^\pm $, two top rows, and Hopf control parameters $ B = B_{m, j}^\pm <0 $, bottom rows, at $ \varepsilon = \omega_k^{-1} = ((k+\tfrac{1}{2})\pi)^{-1} $. The horizontal axis is $ -1 < \underline{\Omega}_m \leq \Omega = \tilde{\Omega}-\Omega_m \leq 0 $ with $ \Omega_m = 2m+1 $. Left: even $ m $. Right: odd $ m $. Layout and legends as in figure 2.1. Again, solid dots $ \bullet $ indicate transverse Hopf stabilization towards smaller control parameters $ |B| $, i.e. towards larger $ B <0 $, at $ B_{m, j}^+ $. Circles $ \circ $, in contrast, indicate transverse Hopf destabilization towards the same side, at $ B_{m, j}^- $. Note how destabilization by each $ B_{m, j}^- <0 $ is annihilated when $ B<0 $ increases through the subsequent stabilization at $ B_{m, j}^+ <0 $. See theorem 3.4(iv). Only for odd $ m $ and $ j = 1 $, the subsequent stabilization at $ B_{m, 1}^+ = 0, \ \diamond $, fails to occur at any finite control amplitude $ \beta = 1/b <0 $

Figure 3.1.  Hopf curves $ \tilde{\omega} \mapsto (\varepsilon(i\tilde{\omega}), B(i\tilde{\omega})) $, oriented along increasing $ \tilde{\omega} $. Note the resulting unstable dimensions $ E $, in parantheses, to the left, and $ E+2 $ to the right, of the Hopf curves

Figure 4.1.  Stability windows (hashed) between intervals $ I_{m, j} \; = \; (B_{m, j}^-, \; B_{m, j}^+) $ of Hopf-induced unstable eigenvalues with imaginary parts in the disjoint intervals designed by $ m, j $. Note how the first, leftmost, stability window between $ I_{m, j_{m}+1} $ and $ I_{m, j_m} $ contains the only Pyragas region $ \mathcal{P} = (B_{0, 1}^+, B_{1, 1}^-) $ of stable supercritical Hopf bifurcation, for any $ m $ such that $ I_{m, j_m+1} $ still exists