Simultaneous and Sequential Synchronisation in Arrays

We discuss the possibility of simultaneous and sequential synchronisation in vertical and horizontal arrays of unidirectionally coupled discrete systems. This is realized for the specific case of two dimensional Gumowski-Mira maps. The synchronised state can be periodic,thereby bringing in control of chaos, or chaotic for carefully chosen parameters of the participating units. The synchronised chaotic state is further characterised using variation of the time of synchronisation with coupling coefficient, size of the array etc. In the case of the horizontal array, the total time of synchronisation can be controlled by increasing the coupling coefficient step wise in small bunch of units.Comment: 10 pages 12 figures submitted to European Physical Journa

Mechanisms for tuning clustering and degree-correlations in directed networks

With complex networks emerging as an effective tool to tackle multidisciplinary problems, models of network generation have gained an importance of their own. These models allow us to extensively analyze the data obtained from real-world networks, study their relevance and corroborate theoretical results. In this work, we introduce methods, based on degree preserving rewiring, that can be used to tune the clustering and degree-correlations in directed networks with random and scale-free topologies. They provide null-models to investigate the role of the mentioned properties along with their strengths and limitations. We find that in the case of clustering, structural relationships, that are independent of topology and rewiring schemes are revealed, while in the case of degree-correlations, the network topology is found to play an important role in the working of the mechanisms. We also study the effects of link-density on the efficiency of these rewiring mechanisms and find that in the case of clustering, the topology of the network plays an important role in determining how link-density affects the rewiring process, while in the case of degree-correlations, the link-density and topology, play no role for sufficiently large number of rewiring steps. Besides the intended purpose of tuning network properties, the proposed mechanisms can also be used as a tool to reveal structural relationships and topological constraints.Comment: 8 pages, 11 figures, submitted to Physical Review

Suppression of dynamics and frequency synchronization in coupled slow and fast dynamical systems

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.Comment: 9 pages, 20 figures, submitted to European Physical Journal

Anticipatory synchronization with variable time delay and reset

A method to synchronize two chaotic systems with anticipation or lag, coupled in the drive response mode, is proposed. The coupling involves variable delay with three time scales. The method has the advantage that synchronization is realized with intermittant information about the driving system at intervals fixed by a reset time. The stability of the synchronization manifold is analyzed with the resulting discrete error dynamics. The numerical calculations in standard systems like the Rossler and Lorenz systems are used to demonstrate the method and the results of the analysis.Comment: 11 pages, 9 figures. submitted to Phys. Rev.

Bubbling and bistability in two parameter discrete systems

We present a graphical analysis of the mechanisms underlying the occurrences of bubbling sequences and bistability regions in the bifurcation scenario of a special class of one dimensional two parameter maps. The main result of the analysis is that whether it is bubbling or bistability is decided by the sign of the third derivative at the inflection point of the map function.Comment: LaTeX v2.09, 14 pages with 4 PNG figure