# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021151
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## 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

Received  November 2020 Revised  April 2021 Early access May 2021

Fund Project: This work was partially supported by JSPS Kakenhi Grant Number JP17H02859

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.

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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021151
##### References:
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##### 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.  Google Scholar [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. Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, 2$^{nd}$ edition, Springer-Verlag, Berlin, 1993.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] E. Schöll and H. G. Schuster (eds.), Handbook of Chaos Control, 2$^{nd}$ edition, Wiley-VCH, Weinheim, 2008.  Google Scholar [16] J. Sieber, Generic stabilizability for time-delayed feedback control, Proc. R. Soc. A, 472 (2015), 20150593. doi: 10.1098/rspa.2015.0593.  Google Scholar [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.  Google Scholar
Phase portrait of (1.1) with $k_{jl} = 0$, $j,l = 1,2$
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)
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)
Stability region of the origin in (1.1) for $\alpha = 0.02$, $k_{11} = k_{22} = 0.02$ and $k_{12} = -0.02$
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
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 '$\bullet$'
. The locus of the stable periodic orbit is also represented by the symbol '$\bullet$'">Figure 5.  Unstable invariant torus on the Poincaré section $\{t = 0\mod 2\pi/\omega\}$ in (1.6) for $\gamma = -5$ and $\rho_1 = \rho_2 = 0.15$. The other parameter values are the same as those of Fig. 4. The locus of the stable periodic orbit is also represented by the symbol '$\bullet$'
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