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This paper addresses the behavior of large systems of heterogeneous, globallycoupled oscillators each of which is described by the generic Landau-Stuartequation, which incorporates both phase and amplitude dynamics of individualoscillators. One goal of our paper is to investigate the effect of a spread inthe amplitude growth parameter of the oscillators and of the effect of ahomogeneous nonlinear frequency shift. Both of these effects are of potentialrelevance to recently reported experiments. Our second goal is to gain furtherunderstanding of the macroscopic system dynamics at large coupling strength,and its dependence on the nonlinear frequency shift parameter. It is proventhat at large coupling strength, if the nonlinear frequency shift parameter isbelow a certain value, then there is a unique attractor for which theoscillators all clump at a single amplitude and uniformly rotating phase (wecall this a single-cluster "locked state"). Using a combination of analyticaland numerical methods, we show that at higher values of the nonlinear frequencyshift parameter, the single-cluster locked state attractor continues to exist,but other types of coexisting attractors emerge. These include two-clusterlocked states, periodic orbits, chaotic orbits, and quasiperiodic orbits.

Phase and Amplitude Dynamics in Large Systems of Coupled Oscillators: Growth Heterogeneity, Nonlinear Frequency Shifts and Cluster States