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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

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.

Mathematics Subject Classification: Primary: 34K12, 37C35; Secondary: 34K13, 34K20, 34C29.

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• Figure 1.  Phase portrait of (1.1) with $k_{jl} = 0$, $j,l = 1,2$

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 '$\bullet$'

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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