  Distributed delays modeled by 'weak generic kernels' are introduced in the
well-known coupled Landau-Stuart system, as well as a chaotic van der
Pol-Rayleigh system with parametric forcing. The systems are close via the
'linear chain trick'. Linear stability analysis of the systems and conditions
for Hopf bifurcation which initiates oscillations are investigated, including
deriving the normal form at bifurcation, and deducing the stability of the
resulting limit cycle attractor. The value of the delay parameter $a=a_{Hopf}$
at Hopf bifurcation picks out the onset of Amplitude Death(AD) in all three
systems, with oscillations at larger values (corresponding to weaker delay). In
the Landau-Stuart system, the Hopf-generated limit cycles for $a>a_{Hopf}$ turn
out to be remarkably stable under very large variations of all other system
parameters beyond the Hopf bifurcation point, and do not undergo further
symmetry breaking, cyclic-fold, flip, transcritical or Neimark-Sacker
bifurcations. This is to be expected as the corresponding undelayed systems are
robust oscillators over wide ranges of their respective parameters. Numerical
simulations reveal strong distortion and rotation of the limit cycles in phase
space as the parameters are pushed far into the post-Hopf regime, and reveal
other features, such as how the oscillation amplitudes and time periods of the
physical variables on the limit cycle attractor change as the delay and other
parameters are varied. For the chaotic system, very strong delays may still
lead to the cessation of oscillations and the onset of AD (even for relatively
large values of the system forcing which tends to oppose this phenomenon).
Varying of the other important system parameter, the parametric excitation,
leads to a rich sequence of dynamical behaviors, with the bifurcations leading
from one regime (or type of attractor) into the next being carefully tracked.
