Vivek Kohar

Taming Explosive Growth through Dynamic Random Links

We study the dynamics of a collection of nonlinearly coupled limit cycle oscillators relevant to a wide class of systems, ranging from neuronal populations to electrical circuits, over network topologies varying from a regular ring to a random network. We find that for sufficiently strong coupling strengths the trajectories of the system escape to infinity in the regular ring network. However when a fraction of the regular connections are dynamically randomized, the unbounded growth is suppressed and the system remains bounded. Further, we find a scaling relation between the critical fraction of random links necessary for successful prevention of explosive behavior and the network rewiring time-scale. These results suggest a mechanism by which blow-ups may be controlled in extended oscillator systems.

Noise enhanced activity in a complex network

We consider the influence of local noise on a generalized network of populations having positive and negative feedbacks. The population dynamics at the nodes is nonlinear, typically chaotic, and allows cessation of activity if the population falls below a threshold value. We investigate the global stability of this large interactive system, as indicated by the average number of nodal populations that manage to remain active. Our central result is that the probability of obtaining active nodes in this network is significantly enhanced under fluctuations. Further, we find a sharp transition in the number of active nodes as noise strength is varied, along with clearly evident scaling behaviour near the critical noise strength. Lastly, we also observe noise induced temporal coherence in the active sub-network, namely, there is an enhancement in synchrony among the nodes at an intermediate noise strength.

Reduction of Additive Colored Noise Using Coupled Dynamics

We study the effect of additive colored noise on the evolution of maps and demonstrate that the deviations caused by such noise can be reduced using coupled dynamics. We consider both Ornstein–Uhlenbeck process as well as 1/fα noise in our numerical simulations. We observe that though the variance of deviations caused by noise depends on the correlations in the noise, under optimal coupling strength, it decreases by a factor equal to the number of coupled elements in the array as compared to the variance of deviations in a single isolated map. This reduction in noise levels occurs in chaotic as well as periodic regime of the maps. Lastly, we examine the effect of colored noise in chaos computing and find that coupling the chaos computing elements enhances the robustness of chaos computing.

Small-world networks exhibit pronounced intermittent synchronization

We report the phenomenon of temporally intermittently synchronized and desynchronized dynamics in Watts-Strogatz networks of chaotic Rössler oscillators. We consider topologies for which the master stability function (MSF) predicts stable synchronized behaviour, as the rewiring probability (p) is tuned from 0 to 1. MSF essentially utilizes the largest non-zero Lyapunov exponent transversal to the synchronization manifold in making stability considerations, thereby ignoring the other Lyapunov exponents. However, for an N-node networked dynamical system, we observe that the difference in its Lyapunov spectra (corresponding to the N - 1 directions transversal to the synchronization manifold) is crucial and serves as an indicator of the presence of intermittently synchronized behaviour. In addition to the linear stability-based (MSF) analysis, we further provide global stability estimate in terms of the fraction of state-space volume shared by the intermittently synchronized state, as p is varied from 0 to 1. This fraction becomes appreciably large in the small-world regime, which is surprising, since this limit has been otherwise considered optimal for synchronized dynamics. Finally, we characterize the nature of the observed intermittency and its dominance in state-space as network rewiring probability (p) is varied.