Abstract. We present our study on the emergent states of two interacting nonlinear systems with differing dynamical time scales. We find that the inability of the interacting systems to fall in step leads to difference in phase as well as change in amplitude. If the mismatch is small, the systems settle to a frequency synchronized state with constant phase difference. But as mismatch in time scale increases, the systems have to compromise to a state of no oscillations. We illustrate this for standard nonlinear systems and identify the regions of quenched dynamics in the parameter plane. The transition curves to this state are studied analytically and confirmed by direct numerical simulations. As an important special case, we revisit the well-known model of coupled ocean atmosphere system used in climate studies for the interactive dynamics of a fast oscillating atmosphere and slowly changing ocean. Our study in this context indicates occurrence of multi stable periodic states and steady states of convection coexisting in the system, with a complex basin structure.

The interplay between time scales and structural properties of complex networks of nonlinear oscillators can generate many interesting phenomena, like amplitude death, cluster synchronization, frequency synchronization etc. We study the emergence of such phenomena and their transitions by considering a complex network of dynamical systems in which a fraction of systems evolves on a slower time scale on the network. We report the transition to amplitude death for the whole network and the scaling near the transitions as the connectivity pattern changes. We also discuss the suppression and recovery of oscillations and the cross over behavior as the number of slow systems increases. By considering a scale free network of systems with multiple time scales, we study the role of heterogeneity in link structure on dynamical properties and the consequent critical behaviors. In this case with hubs made slow, our main results are the escape time statistics for loss of complete synchrony as the slowness spreads on the network and the self-organization of the whole network to a new frequency synchronized state. Our results have potential applications in biological, physical, and engineering networks consisting of heterogeneous oscillators.Complexity of real world systems are studied mainly in terms of the nonlinearity in the intrinsic dynamics of their sub systems and the complex interaction patterns among them. In such systems, the variability and heterogeneity of the interacting sub systems can add a further level of complexity. In this context heterogeneity arising from differing dynamical time scales offers several challenges and has applications in diverse fields, ranging from biology, economy, sociology to physics and engineering. In our work, we study the interesting cooperative dynamics in interacting nonlinear systems of differing time scales using the frame work of complex networks.

Abstract. We study the occurrence of frequency synchronised states with tunable emergent frequencies in a network of connected systems. This is achieved by the interplay between time scales of nonlinear dynamical systems connected to form a network, where out of N systems, m evolve on a slower time scale. In such systems, in addition to frequency synchronised states, we also observe amplitude death, synchronised clusters and multifrequency states. We report an interesting cross over behaviour from fast to slow collective dynamics as the number of slow systems m increases. The transition to amplitude death is analysed in detail for minimal network configurations of 3 and 4 systems which actually form possible motifs of the full network.

An information-theoretic approach for detecting causality and information transfer was applied to phases and amplitudes of oscillatory components related to different time scales and obtained using the wavelet transform from a time series generated by the Epileptor model. Three main time scales and their causal interactions were identified in the simulated epileptic seizures, in agreement with the interactions of the model variables. An approach consisting of wavelet transform, conditional mutual information estimation, and surrogate data testing applied to a single time series generated by the model was demonstrated to be successful in the identification of all directional (causal) interactions between the three different time scales described in the model. Thus, the methodology was prepared for the identification of causal cross-frequency phase–phase and phase–amplitude interactions in experimental and clinical neural data.