Beginning with the effective scheme of stabilization of periodic orbits of minimal period \(T\) by a feedback control with time delay \(\tau=n T\) suggested by K. Pyragas [Continuous control of chaos by self-containing feedback, Phys. Lett. A170, 421 (1992)], the authors review the subsequent achievements, give some general symmetry and stability considerations and apply their developments to the normal form of a subcritical Hopf bifurcation with a time-delayed feedback term \(\dot z(t)=[\lambda+i+(1+i\gamma)| z(t)|^2] z(t)+ b[z(t-\tau)-z(t)], z\in \mathbb{C}\) and real parameters \(\lambda\) and \(\gamma\). Here the Hopf frequency is normalized to unity, the feedback matrix \(B\) is represented by multiplication with a complex number \(b=b_R+i b_I= b_0 e^{i\beta}, b_0>0\) and the nonlinearity \(f(\lambda, z(t))=[\lambda+i+(1+i\gamma)| z(t)|^2] z(t)\) and control \(B=b\) commute with complex rotations. Therefore \(z(t)\exp(i\theta)\) solves this equation for any fixed \(\theta\) whenever \(z(t)\) does. In particular, nonresonant Hopf bifurcations from the trivial solution \(z\equiv 0\) at simple imaginary eigenvalues \(\eta=i \omega \not =0\) produce rotating wave solutions \(z(t)=z(0) \exp(i\frac{2\pi}{T} t)\) with minimal period \(T=\frac{2\pi}{\omega}\) even in the nonlinear case and with delay terms.
The authors derive explicit analytical conditions for stabilization of the periodic orbit generated by a subcritical Hopf bifurcations in terms of the amplitude and the phase of the feedback control gain. The obtained results emphasize the crucial role of a non-vanishing phase of the control signal for stabilization of periodic orbits which violate the odd-number limitation.
For the entire collection see [Zbl 1141.37001].