Amplitude death, oscillation death, and periodic regimes in dynamically coupled Landau–Stuart oscillators with and without distributed delay

Abstract

The effects of a distributed ’weak generic kernel’ delay on dynamically coupled Landau–Stuart limit cycle oscillators are considered. The system is closed via the ’linear chain trick’ and the linear stability analysis of the system and conditions for Hopf bifurcations that initiate oscillations are investigated. After reformulation, the effect of the weak generic kernel delay turns out to be mathematically similar to the linear augmentation scheme considered earlier for coupled oscillator systems. For our coupled Landau–Stuart oscillators the distributed delay produces transitions from amplitude death (AD) or oscillation death (OD) to periodic behavior via Hopf Bifurcations, with the delayed limit cycle shrinking or growing as the delay is varied towards or away from the bifurcation point respectively. The transition from AD to OD occurs here via a supercritical pitchfork bifurcation, as also seen previously for some other couplings. In addition, the dynamically coupled Landau–Stuart system has a secondary limit cycle emerging within the OD parameter regime, in contrast to other couplings and systems studied earlier.The various transitions among AD, OD and periodic domains that we observe are more intricate than the simple AD states, and the rough boundaries of the parameter regimes where they occur, predicted by linear stability analysis and experimentally verified in earlier work on dynamically coupled systems. Also, although our delayed system mathematically resembles linear augmentation schemes used earlier, the numerical searches here reveal more complex transitions among various dynamical regimes than the direct AD and OD found in those studies.