  Oscillatory systems with long-range or global coupling offer promising
insight into the interplay between high-dimensional (or microscopic) chaotic
motion and collective interaction patterns. Within this paper, we use Lyapunov
analysis to investigate whether chimera states in globally coupled
Stuart-Landau (SL) oscillators exhibit collective degrees of freedom. We
compare two types of chimera states, which emerge in SL ensembles with linear
and nonlinear global coupling, respectively, the latter introducing a
constraint that conserves the oscillation of the mean. Lyapunov spectra reveal
that for both chimera states the Lyapunov exponents split into different groups
with different convergence properties in the limit of large system size.
Furthermore, in both cases the Lyapunov dimension is found to scale extensively
and the localization properties of covariant Lypunov vectors manifest the
presence of collective Lyapunov modes. Here, however, we find qualitative
differences between the two types of chimera states: Whereas the ones in the
system under nonlinear global coupling exhibit only slow collective modes
corresponding to Lyapunov exponents equal or close to zero, those which
experience the linear mean-field coupling exhibit also faster collective modes
associated with Lyapunov exponents with large positive or negative values.
