  The effects of a distributed 'weak generic kernel' delay on cyclically
coupled limit cycle and chaotic oscillators are considered. For coupled Van der
Pol oscillators (and in fact, other oscillators as well) the delay can produce
transitions from amplitude death(AD) or oscillation death (OD) to Hopf
bifurcation-induced periodic behavior, 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 is mediated here via a pitchfork
bifurcation, as seen earlier for other couplings as well. Also, the cyclically
coupled undelayed van der Pol system here is already in a state of AD/OD, and
introducing the delay allows both oscillations and AD/OD as the delay parameter
is varied. This is in contrast to other limit cycle systems, where diffusive
coupling alone does not result in the onset of AD/OD. For systems where the
individual oscillators are chaotic, such as a Sprott oscillator system or a
coupled van der Pol-Rayleigh system with parametric forcing, the delay may
produce AD/OD (as in the Sprott case), with the AD to OD transition now
occurring via a transcritical bifurcation instead. However, this may not be
possible, and the delay might just vary the attractor shape. In either of these
situations however, increased delay strength tends to cause the system to have
simpler behavior, streamlining the shape of the attractor, or shrinking it in
cases with oscillations.
