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

Dynamical Transitions in Large Systems of Mean Field-Coupled Landau-Stuart Oscillators: Extensive Chaos and Clumped States