doi: 10.3934/dcdss.2020068

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

Institut für Mathematik, Freie Universität Berlin, Arnimallee 3-7, 14195 Berlin, Germany

* Corresponding author: Bernold Fiedler

Dedicated to Professor Jürgen Scheurle in gratitude and friendship

Received  January 2018 Revised  April 2018 Published  April 2019

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
$ \mathcal{P} $
of scalar 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 [30], 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 in [8] which required control terms
$ b^{-1} (x(t-\vartheta) + x(t-\vartheta -p/2)) $
with a third delay
$ \vartheta $
near 1.
Citation: Bernold Fiedler, Isabelle Schneider. Stabilized rapid oscillations in a delay equation: Feedback control by a small resonant delay. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020068
References:
[1] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963. Google Scholar
[2]

O. Diekmann, S. A. van Gils, S. M. Verduyn-Lunel and H.-O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, App. Math. Sci. 110, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. Google Scholar

[3]

P. Dormayer, Smooth bifurcation of symmetric periodic solutions of functional differential equations, J. Differ. Equations., 82 (1989), 109-155. doi: 10.1016/0022-0396(89)90170-8. Google Scholar

[4]

B. Fiedler, V. Flunkert, M. Georgi, P. Hövel and E. Schöll, Refuting the odd number limitation of time-delayed feedback control, Phys. Rev. Lett. 98 (2007), 114101. doi: 10.1103/PhysRevLett.98.114101. Google Scholar

[5]

B. Fiedler, V. Flunkert, M. Georgi, P. Hövel and E. Schöll, Beyond the odd-number limitation of time-delayed feedback control, In Handbook of Chaos Control, (E. Schöll et al., eds.), Wiley-VCH, Weinheim, (2008), 73–84.Google Scholar

[6]

B. FiedlerV. FlunkertP. Hövel and E. Schöll, Delay stabilization of periodic orbits in coupled oscillator systems, Phil. Trans. Roy. Soc. A., 368 (2010), 319-341. doi: 10.1098/rsta.2009.0232. Google Scholar

[7]

B. Fiedler and J. Mallet-Paret, Connections between Morse sets for delay differential equations, J. Reine Angew. Math., 397 (1989), 23-41. Google Scholar

[8]

B. Fiedler and S. Oliva, Delayed feedback control of a delay equation at Hopf bifurcation, J. Dyn. Differ. Equations, 28 (2016), 1357-1391. doi: 10.1007/s10884-015-9456-8. Google Scholar

[9]

J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. Google Scholar

[10]

J. K. Hale and S. M. Verduyn-Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar

[11]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, In Handbook of Differential Equations: Ordinary Differential Equations, Vol. III. (A. Cañada, P. Drbek and A. Fonda eds.), Elsevier/North-Holland, Amsterdam, (2006), 435–545. doi: 10.1016/S1874-5725(06)80009-X. Google Scholar

[12]

W. Just, B. Fiedler, V. Flunkert, M. Georgi, P. Hövel and E. Schöll, Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control, Phys. Rev. E., 76 (2007), 026210, 11pp. doi: 10.1103/PhysRevE.76.026210. Google Scholar

[13]

J. L. Kaplan and J. A. Yorke, Ordinary differential equations which yield periodic solutions of differential delay equations, J. Math. Analysis Appl., 48 (1974), 317-324. doi: 10.1016/0022-247X(74)90162-0. Google Scholar

[14]

V. Kolmanovski and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0. Google Scholar

[15]

T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hung., 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x. Google Scholar

[16]

J. Kurzweil, Small delays don't matter, In Proc. Symp. Differential Equations and Dynamical Systems, Warwick 1969 (D. Chillingworth ed.), Springer-Verlag Berlin, 1971, 47–49.Google Scholar

[17]

A. López Nieto, Heteroclinic connections in delay equations, Master's Thesis, Freie Universität Berlin, 2017.Google Scholar

[18]

J. Mallet-Paret, Morse decompositions for differential delay equations, J. Differ. Equations, 72 (1988), 270-315. doi: 10.1016/0022-0396(88)90157-X. Google Scholar

[19]

J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time-lags: Ⅰ, Arch. Ration. Mech. Analysis, 120 (1992), 99-146. doi: 10.1007/BF00418497. Google Scholar

[20]

J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time-lags: Ⅱ, J. Reine Angew. Math., 477 (1996), 129-197. doi: 10.1515/crll.1996.477.129. Google Scholar

[21]

J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time-lags: Ⅲ, J. Differ. Equations, 189 (2003), 640-692. doi: 10.1016/S0022-0396(02)00088-8. Google Scholar

[22]

J. Mallet-Paret and R. D. Nussbaum, Stability of periodic solutions of state-dependent delay-differential equations, J. Differ. Equations, 250 (2011), 4085-4103. doi: 10.1016/j.jde.2010.10.023. Google Scholar

[23]

J. Mallet-Paret and G. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Differ. Equations, 125 (1996), 385-440. doi: 10.1006/jdeq.1996.0036. Google Scholar

[24]

J. Mallet-Paret and G. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differ. Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037. Google Scholar

[25]

A. D. Myshkis, General theory of differential equations with retarded argument, AMS Translations, Ser. I, vol. 4. AMS, Providence (1962), Translated from Uspekhi Mat. Nauk (N.S.), 4 (1949), 99-141. Google Scholar

[26]

H. Nakajima, On analytical properties of delayed feedback control of chaos, Phys. Lett. A., 232 (1997), 207-210. doi: 10.1016/S0375-9601(97)00362-9. Google Scholar

[27]

H. Nakajima and Y. Ueda, Half-period delayed feedback control for dynamical systems with symmetries, Phys. Rev. E., 58 (1998), 1757-1763. doi: 10.1103/PhysRevE.58.1757. Google Scholar

[28]

R. G. Nussbaum, Differential-Delay Equations with Two Time Lags, Mem. Am. Math. Soc., 205, Providence, RI, 1978. doi: 10.1090/memo/0205. Google Scholar

[29]

R. G. Nussbaum, Functional differential equations, In Handbook of Dynamical Systems, Vol. Ⅱ. (B. Fiedler ed.), Elsevier/North-Holland, Amsterdam, (2002), 461–499. doi: 10.1016/S1874-575X(02)80031-5. Google Scholar

[30]

K. Pyragas, Continuous control of chaos by self-controlling feedback, Theoretical and Practical Methods in Non-linear Dynamics, (1996), 118-123. doi: 10.1016/B978-012396840-1/50038-2. Google Scholar

[31]

K. Pyragas, A twenty-year review of time-delay feedback control and recent developments, Int. Symp. Nonl. Th. Appl., Palma de Mallorca, 1 (2014), 683–686. doi: 10.15248/proc.1.683. Google Scholar

[32]

H. -O. Walther, Bifurcation from periodic solutions in functional differential equations, Math. Z., 182 (1983), 269-289. doi: 10.1007/BF01175630. Google Scholar

[33]

H.-O. Walther, The 2-dimensional attractor of $\dot{x}(t) = -\mu x(t) + f(x(t-1))$, Mem. Amer. Math. Soc., 113 (1995), vi+76 pp. doi: 10.1090/memo/0544. Google Scholar

[34]

H. -O. Walther, Topics in delay differential equations, Jahresber. DMV, 116 (2014), 87-114. doi: 10.1365/s13291-014-0086-6. Google Scholar

[35]

E. M. Wright, On a non-linear differential-difference equation, J. Reine Angew. Math., 194 (1955), 66-87. doi: 10.1515/crll.1955.194.66. Google Scholar

[36]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. Google Scholar

[37]

J. Yu and Z. Guo, A survey on the periodic solutions to Kaplan-Yorke type delay differential equation-Ⅰ, Ann. Differ. Equations, 30 (2014), 97-114. Google Scholar

show all references

References:
[1] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963. Google Scholar
[2]

O. Diekmann, S. A. van Gils, S. M. Verduyn-Lunel and H.-O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, App. Math. Sci. 110, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. Google Scholar

[3]

P. Dormayer, Smooth bifurcation of symmetric periodic solutions of functional differential equations, J. Differ. Equations., 82 (1989), 109-155. doi: 10.1016/0022-0396(89)90170-8. Google Scholar

[4]

B. Fiedler, V. Flunkert, M. Georgi, P. Hövel and E. Schöll, Refuting the odd number limitation of time-delayed feedback control, Phys. Rev. Lett. 98 (2007), 114101. doi: 10.1103/PhysRevLett.98.114101. Google Scholar

[5]

B. Fiedler, V. Flunkert, M. Georgi, P. Hövel and E. Schöll, Beyond the odd-number limitation of time-delayed feedback control, In Handbook of Chaos Control, (E. Schöll et al., eds.), Wiley-VCH, Weinheim, (2008), 73–84.Google Scholar

[6]

B. FiedlerV. FlunkertP. Hövel and E. Schöll, Delay stabilization of periodic orbits in coupled oscillator systems, Phil. Trans. Roy. Soc. A., 368 (2010), 319-341. doi: 10.1098/rsta.2009.0232. Google Scholar

[7]

B. Fiedler and J. Mallet-Paret, Connections between Morse sets for delay differential equations, J. Reine Angew. Math., 397 (1989), 23-41. Google Scholar

[8]

B. Fiedler and S. Oliva, Delayed feedback control of a delay equation at Hopf bifurcation, J. Dyn. Differ. Equations, 28 (2016), 1357-1391. doi: 10.1007/s10884-015-9456-8. Google Scholar

[9]

J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. Google Scholar

[10]

J. K. Hale and S. M. Verduyn-Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar

[11]

F. Hartung, T. Krisztin, H.-O. Walther and J. Wu, Functional differential equations with state-dependent delays: Theory and applications, In Handbook of Differential Equations: Ordinary Differential Equations, Vol. III. (A. Cañada, P. Drbek and A. Fonda eds.), Elsevier/North-Holland, Amsterdam, (2006), 435–545. doi: 10.1016/S1874-5725(06)80009-X. Google Scholar

[12]

W. Just, B. Fiedler, V. Flunkert, M. Georgi, P. Hövel and E. Schöll, Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control, Phys. Rev. E., 76 (2007), 026210, 11pp. doi: 10.1103/PhysRevE.76.026210. Google Scholar

[13]

J. L. Kaplan and J. A. Yorke, Ordinary differential equations which yield periodic solutions of differential delay equations, J. Math. Analysis Appl., 48 (1974), 317-324. doi: 10.1016/0022-247X(74)90162-0. Google Scholar

[14]

V. Kolmanovski and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0. Google Scholar

[15]

T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hung., 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x. Google Scholar

[16]

J. Kurzweil, Small delays don't matter, In Proc. Symp. Differential Equations and Dynamical Systems, Warwick 1969 (D. Chillingworth ed.), Springer-Verlag Berlin, 1971, 47–49.Google Scholar

[17]

A. López Nieto, Heteroclinic connections in delay equations, Master's Thesis, Freie Universität Berlin, 2017.Google Scholar

[18]

J. Mallet-Paret, Morse decompositions for differential delay equations, J. Differ. Equations, 72 (1988), 270-315. doi: 10.1016/0022-0396(88)90157-X. Google Scholar

[19]

J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time-lags: Ⅰ, Arch. Ration. Mech. Analysis, 120 (1992), 99-146. doi: 10.1007/BF00418497. Google Scholar

[20]

J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time-lags: Ⅱ, J. Reine Angew. Math., 477 (1996), 129-197. doi: 10.1515/crll.1996.477.129. Google Scholar

[21]

J. Mallet-Paret and R. D. Nussbaum, Boundary layer phenomena for differential-delay equations with state-dependent time-lags: Ⅲ, J. Differ. Equations, 189 (2003), 640-692. doi: 10.1016/S0022-0396(02)00088-8. Google Scholar

[22]

J. Mallet-Paret and R. D. Nussbaum, Stability of periodic solutions of state-dependent delay-differential equations, J. Differ. Equations, 250 (2011), 4085-4103. doi: 10.1016/j.jde.2010.10.023. Google Scholar

[23]

J. Mallet-Paret and G. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Differ. Equations, 125 (1996), 385-440. doi: 10.1006/jdeq.1996.0036. Google Scholar

[24]

J. Mallet-Paret and G. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differ. Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037. Google Scholar

[25]

A. D. Myshkis, General theory of differential equations with retarded argument, AMS Translations, Ser. I, vol. 4. AMS, Providence (1962), Translated from Uspekhi Mat. Nauk (N.S.), 4 (1949), 99-141. Google Scholar

[26]

H. Nakajima, On analytical properties of delayed feedback control of chaos, Phys. Lett. A., 232 (1997), 207-210. doi: 10.1016/S0375-9601(97)00362-9. Google Scholar

[27]

H. Nakajima and Y. Ueda, Half-period delayed feedback control for dynamical systems with symmetries, Phys. Rev. E., 58 (1998), 1757-1763. doi: 10.1103/PhysRevE.58.1757. Google Scholar

[28]

R. G. Nussbaum, Differential-Delay Equations with Two Time Lags, Mem. Am. Math. Soc., 205, Providence, RI, 1978. doi: 10.1090/memo/0205. Google Scholar

[29]

R. G. Nussbaum, Functional differential equations, In Handbook of Dynamical Systems, Vol. Ⅱ. (B. Fiedler ed.), Elsevier/North-Holland, Amsterdam, (2002), 461–499. doi: 10.1016/S1874-575X(02)80031-5. Google Scholar

[30]

K. Pyragas, Continuous control of chaos by self-controlling feedback, Theoretical and Practical Methods in Non-linear Dynamics, (1996), 118-123. doi: 10.1016/B978-012396840-1/50038-2. Google Scholar

[31]

K. Pyragas, A twenty-year review of time-delay feedback control and recent developments, Int. Symp. Nonl. Th. Appl., Palma de Mallorca, 1 (2014), 683–686. doi: 10.15248/proc.1.683. Google Scholar

[32]

H. -O. Walther, Bifurcation from periodic solutions in functional differential equations, Math. Z., 182 (1983), 269-289. doi: 10.1007/BF01175630. Google Scholar

[33]

H.-O. Walther, The 2-dimensional attractor of $\dot{x}(t) = -\mu x(t) + f(x(t-1))$, Mem. Amer. Math. Soc., 113 (1995), vi+76 pp. doi: 10.1090/memo/0544. Google Scholar

[34]

H. -O. Walther, Topics in delay differential equations, Jahresber. DMV, 116 (2014), 87-114. doi: 10.1365/s13291-014-0086-6. Google Scholar

[35]

E. M. Wright, On a non-linear differential-difference equation, J. Reine Angew. Math., 194 (1955), 66-87. doi: 10.1515/crll.1955.194.66. Google Scholar

[36]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. Google Scholar

[37]

J. Yu and Z. Guo, A survey on the periodic solutions to Kaplan-Yorke type delay differential equation-Ⅰ, Ann. Differ. Equations, 30 (2014), 97-114. Google Scholar

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
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