Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions (Q2125195)

The paper considers a system of two nonlinear differential equations \( \dot x_1(t) = ..., \dot x_2(t) = ...\) with coupling terms of the form \(k_{ij} \cdot (x_j(t) - x_j(t-\tau))\) in the \(i\)-th equation, \( j = 1,2\), corresponding to what is known as Pyragas control. The system is chosen such that when the coupling coefficients \(k_{ij}\) are all zero, it becomes an ODE which is a normal form for subcritical Hopf bifurcation, producing an unstable limit cycle. In polar coordinates, the radial component of this ODE looks like \(\dot r = r\cdot (r^2- \alpha)\) with a positive \(\alpha\). As opposed to previous related work, in this paper the time delay appears in linear terms. It is shown that for suitable parameter values and initial functions, solutions can `blow-up' at a finite time \(T>0\), which can be smaller or larger than the delay \(\tau\). This can happen in the presence of both stability of zero, and stabilization of the periodic orbit by the delayed feedback, as is shown in the appendix with a combination of analytical and numerical methods. Important techniques are averaging, and an estimate of the type \(\dot r > r^3 - \beta r\) showing how the cubic nonlinearity causes the singularity in finite time, for appropriate initial functions.