Neelima M Gupte

We investigate the statistics and dynamics of failure in a two-dimensional load-bearing network with branching hierarchical structure, and its variants. The variants strengthen the original lattice by using connectivity strategies which add new sites to the maximal cluster in top-to-bottom or bottom-to-top versions. We study the load-bearing capacity and the failure tolerance of all versions, as well as that of the strongest realization of the original lattice, the V lattice. The average number of failures as a function of the test load shows power-law behavior with power 5/2 for the V lattice, but sigmoidal behavior for all other versions. Thus the V lattice turns out to be the critical case of the load-bearing lattices. The distribution of failures is Gaussian for the original lattice, the V lattice, and the bottom-to-top strategy, but is non-Gaussian for the top-to-bottom one. The bottom-to-top strategy leads to stable and strong lattices, and can resist failure even when tested with weights which greatly exceed the capacity of its backbone. We also examine the behavior of asymmetric lattices and discover that the mean failure rates are minimized if the probability of connection p is symmetric with respect to both neighbors. Our results can be of relevance in the context of realistic networks. © 2011 American Physical Society.

Brain hubs are best connected central nodes in the human connectome that play a critical role in integrated brain dynamics. How the hubs perform their function vs different dynamical processes and the role of their higher-order connections involving different brain regions remains elusive. Here we simulate the phase synchronisation processes on the human connectome core network consisting of the eight brain hubs and all attached simplexes of different sizes. The leading pairwise interactions among neighbouring nodes are assumed, taking into account the natural weights of edges. Our results reveal that increasing the positive pairwise couplings promotes a continuous synchronisation while a weak partial synchronisation occurs for a wide range of negative couplings. The weights of edges stabilise the synchronisation process supporting the absence of hysteresis. Furthermore, the time evolution of the order parameter shows cyclic fluctuations induced by the concurrent evolution of phases associated with different groups of nodes. We show that these oscillations exhibit long-range temporal correlations and multifractality. The asymmetrical singularity spectra are determined, which vary with the time scale and depend on the weights of edges. These findings suggest a possible way that the brain functional geometry maintains a desirable low-level synchrony through complex patterns of phase fluctuations.

We propose a new method for the creation of coherent structures in two-dimensional area-preserving maps. The coherent structures are created by applying a parameter perturbation in the neighbourhood of the periodic points of the map. We demonstrate the success of our method in the case of the standard map and in two cases of maps which represent realistic fluid flows, the blinking vortex map of Aref and the journal bearing map of Chaiken. We also find the signatures of the coherent structures in quantifiers like the local exit time distribution and of the local Lyapunov exponents of the system. Our method may provide useful insights into the ways in which coherent structures are created in natural systems. © 2007 Elsevier B.V. All rights reserved.

We study network traffic dynamics in a two dimensional communication network with regular nodes and hubs. If the network experiences heavy message traffic, congestion occurs due to finite capacity of the nodes. We discuss strategies to manipulate hub capacity and hub connections to relieve congestion and define a coefficient of betweenness centrality (CBC), a direct measure of network traffic, which is useful for identifying hubs which are most likely to cause congestion. The addition of assortative connections to hubs of high CBC relieves congestion very efficiently.

We study the dynamics of inertial particles in two-dimensional incompressible flows. The particle dynamics is modeled by four-dimensional dissipative bailout embedding maps of the base flow which is represented by 2âd area preserving maps. The phase diagram of the embedded map is rich and interesting both in the aerosol regime, where the density of the particle is larger than that of the base flow, as well as the bubble regime, where the particle density is less than that of the base flow. The embedding map shows three types of dynamic behavior, periodic orbits, chaotic structures, and mixed regions. Thus, the embedding map can target periodic orbits as well as chaotic structures in both the aerosol and bubble regimes at certain values of the dissipation parameter. The bifurcation diagram of the 4âd map is useful for the identification of regimes where such structures can be found. An attractor merging and widening crisis is seen for a special region for the aerosols. At the crisis, two period-10 attractors merge and widen simultaneously into a single chaotic attractor. Crisis induced intermittency is seen at some points in the phase diagram. The characteristic times before bursts at the crisis show power-law behavior as functions of the dissipation parameter. Although the bifurcation diagram for the bubbles looks similar to that of aerosols, no such crisis regime is seen for the bubbles. Our results can have implications for the dynamics of impurities in diverse application contexts. © 2007 The American Physical Society.

We describe adaptive control algorithms whereby a spatially extended nonlinear system can be steered to a target state with desired spatiotemporal characteristics. Specifically we implement our control on a two-dimensional coupled map lattice, and successfully direct the system to desired targets ranging from spatiotemporal fixed points and regular spatial patterns to spatiotemporal chaos. The proposed methodology entails monitoring the local neighborhood of only one (arbitrary) site in order to regulate the entire lattice. Further, knowledge of the system’s governing equations is not required. We also demonstrate the success of this method in controlling an unstable elastic array, a system of interest in engineering applications. © 1998 The American Physical Society.

We study the existence of different types of chimera states in a globally coupled sine circle map lattice with different strengths of intergroup and intragroup coupling. Some of the typical chimera phase configurations that can be observed in this system are aperiodic chimera states, splay chimera states and chimera states with spatiotemporally intermittent behaviour in the desynchronised group. These states are seen in different regions of the parameter space for three distinct kinds of initial conditions. We obtain the phase diagram containing the third type of chimera state, viz. the one with spatiotemporally intermittent regions, using complex order parameters. We construct an equivalent cellular automaton (CA) and reproduce the phase diagram in the region of interest by solving the mean field equation obtained for the CA.

We study the efficiency of the gradient mechanism of message transfer in a two-dimensional communication network of regular nodes and randomly distributed hubs. Each hub on the network is assigned some randomly chosen capacity and hubs with lower capacities are connected to the hubs with maximum capacity. The average travel times of single messages traveling on the lattice decrease rapidly as the number of hubs increase. The functional dependence of the average travel times on the hub density shows q -exponential behavior with a power-law tail. We also study the relaxation behavior of the network when a large number of messages are created simultaneously at random locations and travel on the network toward their designated destinations. For this situation, in the absence of the gradient mechanism, the network can show congestion effects due to the formation of transport traps. We show that if hubs of high betweenness centrality are connected by the gradient mechanism, efficient decongestion can be achieved. The gradient mechanism is less prone to the formation of traps than other decongestion schemes. We also study the spatial configurations of transport traps and propose minimal strategies for their elimination. © 2008 The American Physical Society.