-
Previous Article
Wave phenomena in a compartmental epidemic model with nonlocal dispersal and relapse
- DCDS-B Home
- This Issue
-
Next Article
The model of nutrients influence on the tumor growth
May
2022, 27(5): 2621-2634.
doi: 10.3934/dcdsb.2021151
Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions
Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan |
We study a normal form of the subcritical Hopf bifurcation subjected to time-delayed feedback. An unstable periodic orbit is born at the bifurcation in the normal form without the delay and it can be stabilized by the time-delayed feedback. We show that there exist finite time blow-up solutions for small initial functions, near the bifurcation point, when the feedback gains are small. This can happen even if the origin is stable or the unstable periodic orbit of the normal form is stabilized by the delay feedback. We give numerical examples to illustrate the theoretical result.
Mathematics Subject Classification:
Primary: 34K12, 37C35; Secondary: 34K13, 34K20, 34C29.
Citation:
Kazuyuki Yagasaki. Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions. Discrete and Continuous Dynamical Systems - B,
2022, 27
(5)
: 2621-2634.
doi: 10.3934/dcdsb.2021151
![]()
References:
| [1] |
G. Brown, C. M. Postlethwaite and M. Silber,
Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation, Physica D, 240 (2011), 859-871.
doi: 10.1016/j.physd.2010.12.011. |
| [2] |
E. Doedel and B. E. Oldeman, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, 2012. Available online from http://cmvl.cs.concordia.ca/auto. |
| [3] |
A. Eremin, E. Ishiwata, T. Ishiwata and Y. Nakata, Delay-induced blow-up in a limit-cycle oscillation model, submitted for publication, arXiv: 1803.07815. |
| [4] |
B. Fiedler, V. Flunkert, M. Grebogi, 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, P. Hövel and E. Schöll,
Delay stabilization of periodic orbits in coupled oscillator systems, Phil. Trans. R. Soc. A, 368 (2010), 319-341.
doi: 10.1098/rsta.2009.0232. |
| [6] |
B. Fiedler, V. Flunkert, P. Hövel and E. Schöll,
Beyond the odd number limitation of time-delayed feedback control of periodic orbits, Eur. Phys. J. Special Topics, 191 (2010), 53-70.
doi: 10.1140/epjst/e2010-01341-9. |
| [7] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
| [8] |
E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, 2$^{nd}$ edition, Springer-Verlag, Berlin, 1993. |
| [9] |
W. Just, B. Fiedler, M. Grebogi, V. Flunkert, P. Hövel and E. Schöll, Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control, Phys. Rev. E, 76 (2007), 026210.
doi: 10.1103/PhysRevE.76.026210. |
| [10] |
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. |
| [11] |
H. Nakajima and Y. Ueda,
Limitation of generalized delayed feedback control, Physica D, 111 (1998), 143-150.
doi: 10.1016/S0167-2789(97)80009-7. |
| [12] |
C. M. Postlethwaite, G. Brown and M. Silber, Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems, Phil. Trans. R. Soc. A, 371 (2013), 20120467.
doi: 10.1098/rsta.2012.0467. |
| [13] |
A. S. Purewal, C. M. Postlethwaite and B. Krauskopf,
A global bifurcation analysis of the subcritical Hopf normal form subject to Pyragas time-delay feedback control, SIAM J. Appl. Dyn. Syst., 13 (2014), 1879-1915.
doi: 10.1137/130949804. |
| [14] |
K. Pyragas,
Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 170 (1992), 421-428.
doi: 10.1016/B978-012396840-1/50038-2. |
| [15] |
E. Schöll and H. G. Schuster (eds.), Handbook of Chaos Control, 2$^{nd}$ edition, Wiley-VCH, Weinheim, 2008. |
| [16] |
J. Sieber, Generic stabilizability for time-delayed feedback control, Proc. R. Soc. A, 472 (2015), 20150593.
doi: 10.1098/rspa.2015.0593. |
| [17] |
J. E. S. Socolar, D. W. Sukow and D. J. Gauthier,
Stabilizing unstable periodic orbits in fast dynamical systems, Phys. Rev. E., 50 (1994), 3245-3248.
doi: 10.1103/PhysRevE.50.3245. |
show all references
References:
| [1] |
G. Brown, C. M. Postlethwaite and M. Silber,
Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation, Physica D, 240 (2011), 859-871.
doi: 10.1016/j.physd.2010.12.011. |
| [2] |
E. Doedel and B. E. Oldeman, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, 2012. Available online from http://cmvl.cs.concordia.ca/auto. |
| [3] |
A. Eremin, E. Ishiwata, T. Ishiwata and Y. Nakata, Delay-induced blow-up in a limit-cycle oscillation model, submitted for publication, arXiv: 1803.07815. |
| [4] |
B. Fiedler, V. Flunkert, M. Grebogi, 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, P. Hövel and E. Schöll,
Delay stabilization of periodic orbits in coupled oscillator systems, Phil. Trans. R. Soc. A, 368 (2010), 319-341.
doi: 10.1098/rsta.2009.0232. |
| [6] |
B. Fiedler, V. Flunkert, P. Hövel and E. Schöll,
Beyond the odd number limitation of time-delayed feedback control of periodic orbits, Eur. Phys. J. Special Topics, 191 (2010), 53-70.
doi: 10.1140/epjst/e2010-01341-9. |
| [7] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
| [8] |
E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, 2$^{nd}$ edition, Springer-Verlag, Berlin, 1993. |
| [9] |
W. Just, B. Fiedler, M. Grebogi, V. Flunkert, P. Hövel and E. Schöll, Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control, Phys. Rev. E, 76 (2007), 026210.
doi: 10.1103/PhysRevE.76.026210. |
| [10] |
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. |
| [11] |
H. Nakajima and Y. Ueda,
Limitation of generalized delayed feedback control, Physica D, 111 (1998), 143-150.
doi: 10.1016/S0167-2789(97)80009-7. |
| [12] |
C. M. Postlethwaite, G. Brown and M. Silber, Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems, Phil. Trans. R. Soc. A, 371 (2013), 20120467.
doi: 10.1098/rsta.2012.0467. |
| [13] |
A. S. Purewal, C. M. Postlethwaite and B. Krauskopf,
A global bifurcation analysis of the subcritical Hopf normal form subject to Pyragas time-delay feedback control, SIAM J. Appl. Dyn. Syst., 13 (2014), 1879-1915.
doi: 10.1137/130949804. |
| [14] |
K. Pyragas,
Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 170 (1992), 421-428.
doi: 10.1016/B978-012396840-1/50038-2. |
| [15] |
E. Schöll and H. G. Schuster (eds.), Handbook of Chaos Control, 2$^{nd}$ edition, Wiley-VCH, Weinheim, 2008. |
| [16] |
J. Sieber, Generic stabilizability for time-delayed feedback control, Proc. R. Soc. A, 472 (2015), 20150593.
doi: 10.1098/rspa.2015.0593. |
| [17] |
J. E. S. Socolar, D. W. Sukow and D. J. Gauthier,
Stabilizing unstable periodic orbits in fast dynamical systems, Phys. Rev. E., 50 (1994), 3245-3248.
doi: 10.1103/PhysRevE.50.3245. |
Figure 2.
Numerical simulations for (1.1) with $ \alpha = 0.02 $ , $ k_{11} = k_{22} = 0.02 $ , $ k_{12} = -0.02 $ , $ k_{21} = 0.06 $ , $ \phi_1 = \phi_2 = \frac{1}{2}\pi $ and $ \omega = 0.96 $ : (a) $ \gamma = 0 $ and $ \rho_1 = \rho_2 = 0.1 $ ; (b) $ \gamma = -0.5 $ and $ \rho_1 = \rho_2 = 0.1 $ ; (c) $ \gamma = -5 $ and $ \rho_1 = \rho_2 = 0.14 $ . The delay time is $ \tau = 2n\pi $ (resp. $ \tau = 2n\pi/0.99 $ ) with $ n = 10 $ and $ n = 23 $ for the blue and red lines, respectively, in Fig. (a) (resp. in Fig. (b)); $ \tau = 2n\pi/0.9 $ with $ n = 1 $ , $ 2 $ , $ 4 $ , $ 11 $ and $ 23 $ for orange, green, purple, blue and red lines, respectively, in Fig. (c). In each figure, the vertical line with the same color represents the delay time while the horizontal black line represents the boundary of the region (3.2)
Figure 3.
Condition (1.9) for $ \alpha = 0.02 $ , $ k_{11} = k_{22} = 0.02 $ , $ k_{12} = -0.02 $ and $ \phi = 0 $ when $ \rho_1 = \rho_2\ ( = \rho) $ . It holds above the curves. Figure (b) is an enlargement of Fig. (a)
Figure A.1.
Stability region of the origin in (1.1) for $ \alpha = 0.02 $ , $ k_{11} = k_{22} = 0.02 $ and $ k_{12} = -0.02 $
Figure B.1.
Stability regions of the periodic orbit (1.2) in (1.1) for $ \alpha = 0.02 $ and $ k_{11} = k_{22} = 0.02 $ : (a) $ \gamma = -0.5 $ ; (b) $ -1 $ ; (c) $ -5 $ . In each figure, the characteristic equation (B.5) has an eigenvalue $ \lambda = 1 $ of multiplicity two on the red line and a pair of complex eigenvalues with moduli one on the blue line. Here $ \tau = 2\pi n/\Omega $ with $ n = 1 $ and $ 2 $ are taken for the solid and dashed lines, respectively
Figure 4.
Periodic orbits in (1.6) for $ \alpha = 0.02 $ , $ \omega = 0.96 $ , $ k_{11} = k_{22} = 0.02 $ , $ k_{12} = -0.02 $ , $ k_{21} = 0.06 $ and $ \bar{\phi}_1 = \bar{\phi}_2 = \frac{1}{2}\pi $ . Stable and unstable periodic orbits are, respectively, plotted as solid and broken lines and torus bifurcations points are denoted by the symbol '$ \bullet $ '
| [1] |
Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134 |
| [2] |
Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006 |
| [3] |
José M. Arrieta, Raúl Ferreira, Arturo de Pablo, Julio D. Rossi. Stability of the blow-up time and the blow-up set under perturbations. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 43-61. doi: 10.3934/dcds.2010.26.43 |
| [4] |
István Győri, Yukihiko Nakata, Gergely Röst. Unbounded and blow-up solutions for a delay logistic equation with positive feedback. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2845-2854. doi: 10.3934/cpaa.2018134 |
| [5] |
Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391 |
| [6] |
Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052 |
| [7] |
Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 |
| [8] |
Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete and Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569 |
| [9] |
Yuya Tanaka, Tomomi Yokota. Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022075 |
| [10] |
Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1 |
| [11] |
Vincent Calvez, Thomas O. Gallouët. Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1175-1208. doi: 10.3934/dcds.2016.36.1175 |
| [12] |
Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113 |
| [13] |
Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831 |
| [14] |
Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903 |
| [15] |
Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, 2021, 29 (6) : 3581-3607. doi: 10.3934/era.2021052 |
| [16] |
Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations and Control Theory, 2022, 11 (2) : 515-536. doi: 10.3934/eect.2021011 |
| [17] |
Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure and Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243 |
| [18] |
Xingwu Chen, Jaume Llibre, Weinian Zhang. Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3953-3965. doi: 10.3934/dcdsb.2017203 |
| [19] |
Jaume Llibre, Clàudia Valls. Hopf bifurcation for some analytic differential systems in $\R^3$ via averaging theory. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 779-790. doi: 10.3934/dcds.2011.30.779 |
| [20] |
Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]