Characterization of spatiotemporal dynamics of coupled oscillatory systems can be done by computing the Lyapunov exponents. We study the spatiotemporal dynamics of randomly coupled network of Kuramoto oscillators and find that the spectral statistics obtained from the Lyapunov exponent spectrum show interesting sensitivity to the coupling matrix. Our results indicate that in the weak coupling limit the gap distribution of the Lyapunov spectrum is Poissonian, while in the limit of strong coupling the gap distribution shows level repulsion. Moreover, the oscillators settle to an inhomogeneous oscillatory state, and it is also possible to infer the random network properties from the Lyapunov exponent spectrum.

The Hamiltonian mean-field model is a system of fully coupled rotators which exhibits a second-order phase transition at some critical energy in its canonical ensemble. We investigate the case where the interaction between the rotors is governed by a timedependent coupling matrix. Our numerical study reveals a shift in the critical point due to the temporal modulation.The shift in the critical point is shown to be independent of the modulation frequency above some threshold value, whereas the impact of the amplitude of modulation is dominant. In the microcanonical ensemble, the system with constant coupling reaches a quasi-stationary state at an energy near the critical point. Our result indicates that the quasi-stationary state subsists in presence of such temporal modulation of the coupling parameter.

Spectral statistics of the Lyapunov exponents computed for coupled map networks bear strong signatures of different phases emergent from the spatiotemporal dynamics. We find that the distributions of gaps in the Lyapunov spectrum for the chaotic and the synchronized phases show Poisson and GOE statistics, respectively, in agreement with the universal predictions of the random matrix theory. The presence of quenched disorder in coupled map networks generates a non-trivial chaotic Griffiths phase for intermediate coupling strengths. The Lyapunov spectral statistics obtained for the chaotic Griffiths phase show strong suppression of gaps and the Lyapunov vectors indicate a unique intermittent dynamical localization.