# American Institute of Mathematical Sciences

## 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. [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. [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. [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. [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. [6] B. Fiedler, V. Flunkert, P. 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. [7] B. Fiedler and J. Mallet-Paret, Connections between Morse sets for delay differential equations, J. Reine Angew. Math., 397 (1989), 23-41. [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. [9] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. [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. [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. [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. [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. [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. [15] T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hung., 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x. [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. [17] A. López Nieto, Heteroclinic connections in delay equations, Master's Thesis, Freie Universität Berlin, 2017. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [28] R. G. Nussbaum, Differential-Delay Equations with Two Time Lags, Mem. Am. Math. Soc., 205, Providence, RI, 1978. doi: 10.1090/memo/0205. [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. [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. [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. [32] H. -O. Walther, Bifurcation from periodic solutions in functional differential equations, Math. Z., 182 (1983), 269-289. doi: 10.1007/BF01175630. [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. [34] H. -O. Walther, Topics in delay differential equations, Jahresber. DMV, 116 (2014), 87-114. doi: 10.1365/s13291-014-0086-6. [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. [36] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [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.

show all references

##### References:
 [1] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963. [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. [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. [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. [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. [6] B. Fiedler, V. Flunkert, P. 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. [7] B. Fiedler and J. Mallet-Paret, Connections between Morse sets for delay differential equations, J. Reine Angew. Math., 397 (1989), 23-41. [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. [9] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. [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. [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. [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. [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. [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. [15] T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hung., 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x. [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. [17] A. López Nieto, Heteroclinic connections in delay equations, Master's Thesis, Freie Universität Berlin, 2017. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [28] R. G. Nussbaum, Differential-Delay Equations with Two Time Lags, Mem. Am. Math. Soc., 205, Providence, RI, 1978. doi: 10.1090/memo/0205. [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. [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. [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. [32] H. -O. Walther, Bifurcation from periodic solutions in functional differential equations, Math. Z., 182 (1983), 269-289. doi: 10.1007/BF01175630. [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. [34] H. -O. Walther, Topics in delay differential equations, Jahresber. DMV, 116 (2014), 87-114. doi: 10.1365/s13291-014-0086-6. [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. [36] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [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.
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]
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
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
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 $\tilde{\Omega} = \varepsilon\omega_k = 1$
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 $\beta = 1/b <0$
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
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
 [1] D. Q. Cao, Y. R. Yang, Y. M. Ge. Characteristic equation approach to stability measures of linear neutral systems with multiple time delays. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 95-105. doi: 10.3934/dcds.2007.17.95 [2] Elena Braverman, Karel Hasik, Anatoli F. Ivanov, Sergei I. Trofimchuk. A cyclic system with delay and its characteristic equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-29. doi: 10.3934/dcdss.2020001 [3] Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058 [4] P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220 [5] Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369 [6] Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 823-837. doi: 10.3934/cpaa.2005.4.823 [7] Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385 [8] Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021 [9] Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793 [10] Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028 [11] Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii, Qingwen Hu. Selective Pyragas control of Hamiltonian systems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2019-2034. doi: 10.3934/dcdss.2019130 [12] Shangbing Ai. Multiple positive periodic solutions for a delay host macroparasite model. Communications on Pure & Applied Analysis, 2004, 3 (2) : 175-182. doi: 10.3934/cpaa.2004.3.175 [13] Benjamin B. Kennedy. Multiple periodic solutions of state-dependent threshold delay equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1801-1833. doi: 10.3934/dcds.2012.32.1801 [14] Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655 [15] Eduardo Cerpa. Null controllability and stabilization of the linear Kuramoto-Sivashinsky equation. Communications on Pure & Applied Analysis, 2010, 9 (1) : 91-102. doi: 10.3934/cpaa.2010.9.91 [16] Benjamin B. Kennedy. A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1633-1650. doi: 10.3934/dcdsb.2013.18.1633 [17] A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701 [18] Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic & Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205 [19] Andrei Fursikov, Lyubov Shatina. Nonlocal stabilization by starting control of the normal equation generated by Helmholtz system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1187-1242. doi: 10.3934/dcds.2018050 [20] Eugen Stumpf. On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3317-3340. doi: 10.3934/dcdsb.2017139

2017 Impact Factor: 0.561

## Tools

Article outline

Figures and Tables