Forced synchronization of a delayed-feedback oscillator

Dynamics of a delayed-feedback oscillator with cubic nonlinearity driven by an external harmonic signal is considered. Regimes of forced synchronization and their stability conditions are studied analytically and numerically for the cases of single-frequency and multiple-frequency (self-modulation) operation of the free-running oscillator. Special attention is paid to synchronization of the oscillator with bistability of steady states which arises near the boundary of generation zone. Results of numerical simulation of bifurcation transitions to synchronization regime for different driving frequencies are presented. Main differences from the classical picture of synchronization for a system with one degree of freedom are discussed.

► Dynamics of a delayed-feedback oscillator with cubic nonlinearity driven by an external harmonic signal is considered. ► We plot resonance curves and synchronization tongues and investigate mechanisms of synchronization by numerical simulations. ► Synchronization tongues have a very complicated structure caused by resonances with the different eigenmodes. ► Transition to synchronous regime is accompanied by a complicated sequence of bifurcations including period doubling cascade. ► When the driving force amplitude is increased well above the synchronization threshold, synchronous regime becomes unstable.