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The effects of a distributed 'weak generic kernel' delay on cyclicallycoupled limit cycle and chaotic oscillators are considered. For coupled Van derPol oscillators (and in fact, other oscillators as well) the delay can producetransitions from amplitude death(AD) or oscillation death (OD) to Hopfbifurcation-induced periodic behavior, with the delayed limit cycle shrinkingor growing as the delay is varied towards or away from the bifurcation pointrespectively. The transition from AD to OD is mediated here via a pitchforkbifurcation, as seen earlier for other couplings as well. Also, the cyclicallycoupled undelayed van der Pol system here is already in a state of AD/OD, andintroducing the delay allows both oscillations and AD/OD as the delay parameteris varied. This is in contrast to other limit cycle systems, where diffusivecoupling alone does not result in the onset of AD/OD. For systems where theindividual oscillators are chaotic, such as a Sprott oscillator system or acoupled van der Pol-Rayleigh system with parametric forcing, the delay mayproduce AD/OD (as in the Sprott case), with the AD to OD transition nowoccurring via a transcritical bifurcation instead. However, this may not bepossible, and the delay might just vary the attractor shape. In either of thesesituations however, increased delay strength tends to cause the system to havesimpler behavior, streamlining the shape of the attractor, or shrinking it incases with oscillations.

Delay Effects on Amplitude Death, Oscillation Death, and Renewed Limit Cycle Behavior in Cyclically Coupled Oscillators