Lyapunov spectra and collective modes of chimera states in globally coupled Stuart-Landau oscillators

Abstract

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.

Attracting Poisson Chimeras in Two-population Networks

“…8 (b,d)). Therefore, unlike the globally coupled Poisson chimeras, which are neutrally stable, the incoherent population of nonlocal Poisson chimeras is stable, as suggested by the fact that all pairs of the incoherent Lyapunov exponents are definitely negative, except for the two zero exponents which are connected to the continuous symmetries: the phase shift (v ps = (δ φ 0 , ..., δ φ 0 ) where |δ φ 0 | 1) and the time shift (v ts ∝ φch = f(φ ch )), respectively, which in fact do not affect the stability of the trajectory 33 . For large-size Poisson chimeras in Fig.…”

“…Therefore, a CLV is a collective mode if IPR (i) (N) ∼ 1 N as N increases, whereas a CLV is localized when IPR (i) (N) ∼ const. as N increases 32,33 .…”

“…As a last step of our analysis of the dynamics of Poisson chimeras, we investigate whether some of the CLVs correspond to collective perturbations, or modes. Therefore, we calculate the time-averaged inverse participation ratio (IPR) for various system sizes according to 32,33…”

“…Our main method to access the properties of the various chimera trajectories is Lyapunov spectral analysis, which yields the spectra of the Lyapunov exponents (LEs) and the covariant Lyapunov vectors (CLVs) [29][30][31][32][33] . The analysis reveals Here, only the connections from the first oscillator are fully depicted.…”

Dynamics of a nonlinear quantum oscillator under non-Markovian pumping

“…It should be mentioned that this model can be considered a particular case of the Stuart-Landau oscillator [43][44][45][46]. Autocorrelation function of fluctuations η(t) is determined by the Keldysh component of self-energy.…”