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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 |
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.
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. |





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