  This paper addresses the behavior of large systems of heterogeneous, globally
coupled oscillators each of which is described by the generic Landau-Stuart
equation, which incorporates both phase and amplitude dynamics of individual
oscillators. One goal of our paper is to investigate the effect of a spread in
the amplitude growth parameter of the oscillators and of the effect of a
homogeneous nonlinear frequency shift. Both of these effects are of potential
relevance to recently reported experiments. Our second goal is to gain further
understanding of the macroscopic system dynamics at large coupling strength,
and its dependence on the nonlinear frequency shift parameter. It is proven
that at large coupling strength, if the nonlinear frequency shift parameter is
below a certain value, then there is a unique attractor for which the
oscillators all clump at a single amplitude and uniformly rotating phase (we
call this a single-cluster "locked state"). Using a combination of analytical
and numerical methods, we show that at higher values of the nonlinear frequency
shift parameter, the single-cluster locked state attractor continues to exist,
but other types of coexisting attractors emerge. These include two-cluster
locked states, periodic orbits, chaotic orbits, and quasiperiodic orbits.
