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

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 several 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. Furthermore, for the fastest collective mode we showed that it spreads across both synchonous and incoherent oscillators.

Figure 1

Simulation data of the type I chimera, (a) and (c), and the type II chimera state, (b) and (d), with $N=256$ oscillators. Trajectories of coherent and incoherent oscillators are shown in dark red (dark gray) and orange (light gray), respectively. Top: Oscillator trajectories in the complex plane. Dots represent a snapshot of the oscillator states. Bottom: Real part of the oscillator states as a function of time.

Figure 2

Lyapunov spectra of a type I chimera state in Eq. (1) and a type II chimera state in Eq. (2) for a system size of $N=16$ oscillators.

Figure 3

Lyapunov spectra of type I and type II chimera states for increasing ensemble sizes from $N=32$ to $N=256$.

Figure 4

(a) Lyapunov dimensions type I and a type II chimera states as a function of $N$. (b) Effective Lyapunov dimension as a function of the reduced number of oscillators $N_{\text{inc}}+1$.

Figure 5

Distribution of the magnitude of the entries of the leading CVL at an arbitray instant in time for type I and type II chimeras in systems of 128 oscillators. The large peaks marked by arrows correspond to magnitude intervals which contain the magnitude entries of the synchronized oscillators.

Figure 6

Temporal distributions of the IPRs for type I [(a) and (c)] and type II (b) chimeras of the 128 oscillator ensembles. The blue (dark gray) IPR distributions were from all entries of the CLV, the orange (light gray) ones from the entries corresponding to the incoherent oscillators only (which were normalized to 1 before calculating the IPRs).

Figure 7

Localization indices for synchronized and incoherent groups as defined in Eq. (11) for type I and type II chimeras of the 128 oscillator ensembles.

Figure 8

Median localization of covariant Lyapunov vectors.