  In this paper, we study dynamical systems in which a large number $N$ of
identical Landau-Stuart oscillators are globally coupled via a mean-field.
Previously, it has been observed that this type of system can exhibit a variety
of different dynamical behaviors including clumped states in which each
oscillator is in one of a small number of groups for which all oscillators in
each group have the same state which is different from group to group, as well
as situations in which all oscillators have different states and the
macroscopic dynamics of the mean field is chaotic. We argue that this second
type of behavior is $^{\backprime}$extensive$^{\prime}$ in the sense that the
chaotic attractor in the full phase space of the system has a fractal dimension
that scales linearly with $N$ and that the number of positive Lyapunov
exponents of the attractor also scales with linearly $N$. An important focus of
this paper is the transition between clumped states and extensive chaos as the
system is subjected to slow adiabatic parameter change. We observe explosive
(i.e., discontinuous) transitions between the clumped states (which correspond
to low dimensional dynamics) and the extensively chaotic states. Furthermore,
examining the clumped state, as the system approaches the explosive transition
to extensive chaos, we find that the oscillator population distribution between
the clumps continually evolves so that the clumped state is always marginally
stable. This behavior is used to reveal the mechanism of the explosive
transition. We also apply the Kaplan-Yorke formula to study the fractal
structure of the extensively chaotic attractors.
