Open AccessExistence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions
Author(s) -
Kazuyuki Yagasaki
Publication year - 2022
Publication title -
discrete and continuous dynamical systems. series b
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.864
H-Index - 53
eISSN - 1553-524X
pISSN - 1531-3492
DOI - 10.3934/dcdsb.2021151
Subject(s) - hopf bifurcation , mathematics , bifurcation , orbit (dynamics) , control theory (sociology) , mathematical analysis , stability (learning theory) , physics , nonlinear system , computer science , control (management) , quantum mechanics , artificial intelligence , engineering , aerospace engineering , machine learning
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.