Dynamics of a limit cycle oscillator with extended delay feedback (Q378990)

The authors study a Stuart-Landau oscillator with delayed feedback, defined as follows: NEWLINE\[NEWLINE\dot{z}(t) = (m + ni)z(t) - A|z(t)|^2z(t) + k(w(t-\tau) - z(t)),NEWLINE\]NEWLINE where the ``extended delayed feedback'' \(w(t)\) is recursively defined as NEWLINE\[NEWLINE w(t) = (1-p)z(t) + pw(t-\tau).NEWLINE\]NEWLINE The system is equivalent to a delay differential equation of neutral type for which the authors study Hopf-bifurcations of the equilibrium \(z\equiv 0\) and determine the direction of the bifurcations as well as the stability and coexistence of emerging periodic solutions.NEWLINENEWLINEFurther, they study the appearance of double-Hopf bifurcations in the two-parameter plane \((k,\tau)\) and present the corresponding unfoldings, revealing the existence of an unstable quasiperiodic solution for some parameter values.NEWLINENEWLINEFinally, they give numerical illustrations for the theoretical results, including examples for the excitation of periodic behavior, as well as its suppression by the feedback. Two-dimensional bifurcation diagrams are presented.