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The hidden geometry of simplicial complexes can influence the collective dynamics of nodes in different ways depending on the simplex-based interactions of various orders and competition between local and global structural features. We study a system of phase oscillators attached to nodes of 4-dimensional simplicial complexes and interacting via positive/negative edges-based pairwise $K_1$ and triangle-based triple $K_2\geq 0$ couplings. Three prototypal simplicial complexes are grown by aggregation of 5-cliques, controlled by the chemical affinity parameter $\nu$, resulting in sparse, mixed, and compact architecture, all of which have 1-hyperbolic graphs but different spectral dimensions. By changing the interaction strength $K_1\in[-4,2]$ along the forward and backward sweeps, we numerically determine individual phases of each oscillator and a global order parameter to measure the level of synchronisation. Our results reveal how different architectures of simplicial complexes, in conjunction with the interactions and internal-frequency distributions, impact the shape of the hysteresis loop and lead to patterns of locally synchronised groups that hinder global network synchronisation., Comment: 9 pages, 7 figures; regular article, submitted

Recent studies of dynamic properties in complex systems point out the profound impact of hidden geometry features known as simplicial complexes, which enable geometrically conditioned many-body interactions. Studies of collective behaviours on the controlled-structure complexes can reveal the subtle interplay of geometry and dynamics. Here, we investigate the phase synchronisation dynamics under the competing interactions embedded on 1-simplex (edges) and 2-simplex (triangles) faces of a homogeneous 4-dimensional simplicial complex. Its underlying network is a 1-hyperbolic graph with the assortative correlations among the node's degrees and the spectral dimension that exceeds $d_s=4$. We determine the time-averaged system's order parameter to characterise the synchronisation level. In the absence of higher interactions, the complete synchrony is continuously reached with the increasing positive pairwise interactions ($K_1>0$), and a partial synchronisation for the negative couplings ($K_1<0$) with no apparent hysteresis. Similar behaviour occurs in the degree-preserved randomised network. In contrast, the synchronisation is absent for the negative pairwise coupling in the entirely random graph and simple scale-free networks. Increasing the strength $K_2\neq 0$ of the triangle-based interactions gradually hinders the synchronisation promoted by pairwise couplings, and the non-symmetric hysteresis loop opens with an abrupt desynchronisation transition towards the $K_1<0$ branch. However, for substantial triangle-based interactions, the frustration effects prevail, preventing the complete synchronisation, and the abrupt transition disappears. These findings shed new light on the mechanisms by which the high-dimensional simplicial complexes in natural systems, such as human connectomes, can modulate their native synchronisation processes., Comment: 9 pages, 6 figures included

Recent studies of networks representing complex systems from the brain to social graphs have revealed their higher-order architecture, which can be described by aggregates of simplexes (triangles, tetrahedrons, and higher cliques). Current research aims at quantifying these hidden geometries by the algebraic topology methods and deep graph theory and understanding the dynamic processes on simplicial complexes. Here, we use the recently introduced model for geometrical self-assembly of cliques to grow nano-networks of triangles and study the filed-driven spin reversal processes on them. With the antiferromagnetic interactions between the Ising spins attached to the nodes, this assembly ideally supports the geometric frustration, which is recognized as the origin of some new phenomena in condensed matter physics. In the dynamical model, a gradual switching from the pairwise to triangle-based interactions is controlled by a parameter. Thus, the spin frustration effects on each triangle give way to the mesoscopic ordering conditioned by a complex arrangement of triangles. We show how the balance between these interactions changes the shape of the hysteresis loop. Meanwhile, the fluctuations in the accompanying Barkhausen noise exhibit robust indicators of self-organized criticality, which is induced by the network geometry alone without any magnetic disorder., Comment: 6 pages, 6 figures

The identification of precursors of climatic phenomena has enormous practical importance. Recent work constructs a climate network based on surface air temperature data to analyze the El Ni\~no phenomena. We utilize microtransitions which occur before the discontinuous percolation transition in the network as well as other network quantities to identify a set of reliable precursors of El Ni\~no episodes. These precursors identify nine out of twelve El Ni\~no episodes occurring in the period of 1979 to 2018 with a lead time varying from six to ten months. We also find indicators of tipping events in the data.

We study the existence of chimera states, i.e. mixed states, in a globally coupled sine circle map lattice, with different strengths of inter-group and intra-group coupling. We find that at specific values of the parameters of the CML, a completely random initial condition evolves to chimera states, having a phase synchronised and a phase desynchronised group, where the space time variation of the phases of the maps in the desynchronised group shows structures similar to spatiotemporally intermittent regions. Using the complex order parameter we obtain a phase diagram that identifies the region in the parameter space which supports chimera states of this type, as well as other types of phase configurations such as globally phase synchronised states, two phase clustered states and fully phase desynchronised states. We estimate the volume of the basin of attraction of each kind of solution. The STI chimera region is studied in further detail via numerical and analytic stability analysis, and the Lyapunov spectrum is calculated. This state is identified to be hyperchaotic as the two largest Lyapunov exponents are found to be positive. The distributions of laminar and burst lengths in the incoherent region of the chimera show exponential behaviour. The average fraction of laminar/burst sites is identified to be the important quantity which governs the dynamics of the chimera. After an initial transient, these settle to steady values which can be used to reproduce the phase diagram in the chimera regime., Comment: The following article has been submitted to Chaos: An Interdisciplinary Journal of Nonlinear Science. After it is published, it will be found at https://publishing.aip.org/resources/librarians/products/journals/. (19 pages, 52 figures). arXiv admin note: substantial text overlap with arXiv:1810.03911

We study the existence of chimera states, i.e. mixed states, in a globally coupled sine circle map lattice, with different strengths of inter-group and intra-group coupling. We find that at specific values of the parameters of the CML, a completely random initial condition evolves to chimera states, having a phase synchronised and a phase desynchronised group, where the space time variation of the phases of the maps in the desynchronised group shows structures similar to spatiotemporally intermittent regions. Using the complex order parameter we obtain a phase diagram that identifies the region in the parameter space which supports chimera states of this type, as well as other types of phase configurations. An equivalent cellular automaton is obtained which shows space time behaviour similar to the CML. We also derive a mean field equation for this cellular automaton and compare its solutions with the corresponding phase configurations obtained in the parameter space, and show that it matches with the behavior seen for the CML., Comment: 39 pages, 57 figures

We analyze the time series obtained from different dynamical regimes of the logistic map by constructing their equivalent time series (TS) networks, using the visibility algorithm. The regimes analyzed include both periodic and chaotic regimes, as well as intermittent regimes and the Feigenbaum attractor at the edge of chaos. We use the methods of algebraic topology to define the simplicial characterizers, which can analyse the simplicial structure of the networks at both the global and local levels. The simplicial characterisers bring out the hierarchical levels of complexity at various topological levels. These hierarchical levels of complexity find the skeleton of the local dynamics embedded in the network which influence the global dynamical properties of the system, and also permit the identification of dominant motifs. We also analyze the same networks using conventional network characterizers such as average path lengths and clustering coefficients. We see that the simplicial characterizers are capable of distinguishing between different dynamical regimes and can pick up subtle differences in dynamical behavior, whereas the usual characterizers provide a coarser characterization. However, the two taken in conjunction, can provide information about the dynamical behavior of the time series, as well as the correlations in the evolving system. Our methods can therefore provide powerful tools for the analysis of dynamical systems., Comment: 14 pages, 8 figure, 4 tables

We use the visibility algorithm to construct the time series networks obtained from the time series of different dynamical regimes of the logistic map. We define the simplicial characterisers of networks which can analyse the simplicial structure at both the global and local levels. These characterisers are used to analyse the TS networks obtained in different dynamical regimes of the logisitic map. It is seen that the simplicial characterisers are able to distinguish between distinct dynamical regimes. We also apply the simplicial characterisers to time series networks constructed from fMRI data, where the preliminary results indicate that the characterisers are able to differentiate between distinct TS networks., Comment: 11 pages, 2 figures, 4 tables. Accepted for publication in Proceedings of the 4th International Conference on Applications in Nonlinear Dynamics (ICAND 2016)

The problem of synchronization of coupled Hamiltonian systems exhibits interesting features due to the non-uniform or mixed nature (regular and chaotic) of the phase space. We study these features by investigating the synchronization of unidirectionally coupled area-preserving maps coupled by the Pecora-Carroll method. We find that coupled standard maps show complete synchronization for values of the nonlinearity parameter at which regular structures are still present in phase space. The distribution of synchronization times has a power law tail indicating long synchronization times for at least some of the synchronizing trajectories. With the introduction of coherent structures using parameter perturbation in the system, this distribution crosses over to exponential behavior, indicating shorter synchronization times, and the number of initial conditions which synchronize increases significantly, indicating an enhancement in the basin of synchronization. On the other hand, coupled blinking vortex maps display both phase synchronization and phase slips, depending on the location of the initial conditions. We discuss the implication of our results., Comment: 15 pages, 10 figures, Perspectives in Nonlinear Dynamics 2016 (24th - 29th July 2016), Berlin, Germany. The paper is a part of the proceedings to appear in Pramana - Journal of Physics (Indian Academy of Sciences); distributed in print outside India and online worldwide by Springer

Mapping time series onto graphs and the use of graph theory methods opens up the possibility to study the structure of the phase space manifolds underlying the fluctuations of a dynamical variable. Here, we propose to go beyond the standard graph measures and analyze the higher-order structures such as triangles, tetrahedra and higher-order cliques and their complexes occurring in the time-series networks, which are detectable by the algebraic topology methods. We apply the methodology to the signal of Barkhausen noise accompanying the domain-wall dynamics on the hysteresis loop of disordered ferromagnets driven by the external field. Our analysis demonstrates how the appearance of the complexes with cliques of a high order correlates to the enhanced collective fluctuations in the central part of the hysteresis loop, in contrast to the fractional Gaussian noise fluctuations at the beginning of the loop, which correspond to the graph of a simpler topology. The multifractal analysis of the corresponding segments of these time series confirms that we deal with different types of the stochastic process., Comment: 7 pages, 6 figures

We discuss transport on load bearing branching hierarchical networks which can model diverse systems which can serve as models of river networks, computer networks, respiratory networks and granular media. We study avalanche transmissions and directed percolation on these networks, and on the V lattice, i.e., the strongest realization of the lattice. We find that typical realizations of the lattice show multimodal distributions for the avalanche transmissions, and a second order transition for directed percolation. On the other hand, the V lattice shows power - law behavior for avalanche transmissions, and a first order (explosive) transition to percolation. The V lattice is thus the critical case of hierarchical networks. We note that small perturbations to the V lattice destroy the power-law behavior of the distributions, and the first order nature of the percolation. We discuss the implications of our results., Comment: 10 Pages, 11 Figures, Published in (Chapter 17) International Conference on Theory and Application in Nonlinear Dynamics (ICAND 2012), Understanding Complex Systems

We study the dynamics of inertial particles in three dimensional incompressible maps, as representations of volume preserving flows. The impurity dynamics has been modeled, in the Lagrangian framework, by a six-dimensional dissipative bailout embedding map. The fluid-parcel dynamics of the base map is embedded in the particle dynamics governed by the map. The base map considered for the present study is the Arnold-Beltrami-Childress (ABC) map. We consider the behavior of the system 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 phase spaces in both the regimes show rich and complex dynamics with three type of dynamical behaviors - chaotic structures, regular orbits and hyperchaotic regions. In the one-action case, the aerosol regime is found to have periodic attractors for certain values of the dissipation and inertia parameters. For the aerosol regime of the two-action ABC map, an attractor merging and widening crises is identified using the bifurcation diagram and the spectrum of Lyapunov exponents. After the crisis an attractor with two parts is seen, and trajectories hop between these parts with period 2. The bubble regime of the embedded map shows strong hyperchaotic regions as well as crisis induced intermittency with characteristic times between bursts that scale as a power law behavior as a function of the dissipation parameter. Furthermore, we observe riddled basin of attraction and unstable dimension variability in the phase space in the bubble regime. The bubble regime in one-action shows similar behavior. This study of a simple model of impurity dynamics may shed light upon the transport properties of passive scalars in three dimensional flows. We also compare our results with those seen earlier in two dimensional flows., Comment: 28 pages, 37 figures; Submitted to Phys. Rev. E

We analyze an idealized model for the transmission or flow of particles, or discrete packets of information, in a weight bearing branching hierarchical 2-D networks, and its variants. The capacities add hierarchically down the clusters. Each node can accommodate a limited number of packets, depending on its capacity and the packets hop from node to node, following the links between the nodes. The statistical properties of this system are given by the Maxwell - Boltzmann distribution. We obtain analytical expressions for the mean occupation numbers as functions of capacity, for different network topologies. The analytical results are shown to be in agreement with the numerical simulations. The traffic flow in these models can be represented by the site percolation problem. It is seen that the percolation transitions in the 2-D model and in its variant lattices are continuous transitions, whereas the transition is found to be explosive (discontinuous) for the V- lattice, the critical case of the 2-D lattice. We discuss the implications of our analysis., Comment: 24 pages, 41 figures

The strength and stability properties of hierarchical load bearing networks and their strengthened variants have been discussed in recent work. Here, we study the avalanche time distributions on these load bearing networks. The avalanche time distributions of the V- lattice, a unique realization of the networks, show power-law behavior when tested with certain fractions of its trunk weights. All other avalanche distributions show Gaussian peaked behavior. Thus the V- lattice is the critical case of the network. We discuss the implications of this result., Comment: 7 pages, 10 figures, To appear in the Proceedings of "Perspectives in Nonlinear Dynamics 2010", Accepted in a special issue of Pramana - journal of physics

Systems of coupled oscillators have been seen to exhibit chimera states, i.e. states where the system splits into two groups where one group is phase locked and the other is phase randomized. In this work, we report the existence of chimera states in a system of two interacting populations of sine circle maps. This system also exhibits the clustered chimera behavior seen earlier in delay coupled systems. Rich spatio-temporal behavior is seen in different regimes of the phase diagram.We carry out a detailed analysis of the stability regimes and map out the phase diagram using numerical and analytic techniques., Comment: 10 pages, 5 pictures

We study message transport on a $1-d$ ring of nodes and randomly distributed hubs. Messages are deposited on the network at a constant rate. When the rate at which messages are deposited on the lattice is very high, messages start accumulating after a critical time and the average load per node starts increasing. The power-spectrum of the load time-series shows $1/f$ like noise similar to the scenario of the Internet traffic. The inter-arrival time distribution of messages for the $1-d$ ring network shows stretched exponential behavior, which crosses over to power-law behavior if assortative connections are added to the hubs. The distribution of travel times in a related double ring geometry is shown to be bimodal with one peak corresponding to initial congestion and another peak to later decongestion., Comment: 3 pages, 6 figures, Submitted to IEEE

We study the transport properties of passive inertial particles in a $2-d$ incompressible flows. Here the particle dynamics is represented by the $4-d$ dissipative embedding map of $2-d$ area-preserving standard map which models the incompressible flow. The system is a model for impurity dynamics in a fluid and is characterized by two parameters, the inertia parameter $\alpha$, and the dissipation parameter $\gamma$. We obtain the statistical characterisers of transport for this system in these dynamical regimes. These are, the recurrence time statistics, the diffusion constant, and the distribution of jump lengths. The recurrence time distribution shows a power law tail in the dynamical regimes where there is preferential concentration of particles in sticky regions of the phase space, and an exponential decay in mixing regimes. The diffusion constant shows behaviour of three types - normal, subdiffusive and superdiffusive, depending on the parameter regimes. Phase diagrams of the system are constructed to differentiate different types of diffusion behaviour, as well as the behaviour of the absolute drift. We correlate the dynamical regimes seen for the system at different parameter values with the transport properties observed at these regimes, and in the behaviour of the transients. This system also shows the existence of a crisis and unstable dimension variability at certain parameter values. The signature of the unstable dimension variability is seen in the statistical characterisers of transport. We discuss the implications of our results for realistic systems., Comment: 28 pages, 14 figures, To Appear in Phys. Rev. E; Vol. 79 (2009)

The dynamics of inertial particles in $2-d$ incompressible flows can be modeled by $4-d$ bailout embedding maps. The density of the inertial particles, relative to the density of the fluid, is a crucial parameter which controls the dynamical behaviour of the particles. We study here the dynamical behaviour of aerosols, i.e. particles heavier than the flow. An attractor widening and merging crisis is seen the phase space in the aerosol case. Crisis induced intermittency is seen in the time series and the laminar length distribution of times before bursts gives rise to a power law with the exponent $\beta=-1/3$. The maximum Lyapunov exponent near the crisis fluctuates around zero indicating unstable dimension variability (UDV) in the system. The presence of unstable dimension variability is confirmed by the behaviour of the probability distributions of the finite time Lyapunov exponents., Comment: 7 pages, 4 figures, appeared in the Proceeding of the conference Perspectives in Nonlinear Dynamics 2007, a special issue of the Journal Pramana

We study the dynamics of inertial particles in two dimensional incompressible flows. The Maxey-Riley equation describing the motion of inertial particles is used to construct a four dimensional dissipative bailout embedding map. This map models the dynamics of the inertial particles while the base flow is represented by a 2-d area preserving map. The dynamics of particles heavier than the fluid, the aerosols, as well as that of bubbles, particles lighter than the fluid, can be classified into 3 main dynamical regimes - periodic orbits, chaotic structures and mixed regions. A phase diagram in the parameter space is constructed with the Lyapunov characteristic exponents of the 4-d map in which these dynamical regimes are distinctly identified. The embedding map can target periodic orbits, as well as chaotic structures, in both the aerosol and bubble regimes, at suitable values of the dissipation parameter., Comment: 11 pages, 4 figures. Published in the proceedings of the conference "Recent Developments in Nonlinear Dynamics", held at the Bharathidasan University, Thiruchirapalli, India, in February 7-9, 2008

We identify the statistical characterizers of congestion and decongestion for message transport in model communication lattices. These turn out to be the travel time distributions, which are Gaussian in the congested phase, and log-normal in the decongested phase. Our results are demonstrated for two dimensional lattices, such the Waxman graph, and for lattices with local clustering and geographic separations, gradient connections, as well as for a 1-d ring lattice with random assortative connections. The behavior of the distribution identifies the congested and decongested phase correctly for these distinct network topologies and decongestion strategies. The waiting time distributions of the systems also show identical signatures of the congested and decongested phases., Comment: Submitted to Phys. Rev. E (Rapid)

Studies of the phase diagram of the coupled sine circle map lattice have identified the presence of two distinct universality classes of spatiotemporal intermittency viz. spatiotemporal intermittency of the directed percolation class with a complete set of directed percolation exponents, and spatial intermittency which does not belong to this class. We show that these two types of behavior are special cases of a spreading regime where each site can infect its neighbors permitting an initial disturbance to spread, and a non-spreading regime where no infection is possible, with the two regimes being separated by a line, the infection line. The coupled map lattice can be mapped on to an equivalent cellular automaton which shows a transition from a probabilistic cellular automaton to a deterministic cellular automaton at the infection line. The origins of the spreading-non-spreading transition in the coupled map lattice, as well as the probabilistic to deterministic transition in the cellular automaton lie in a dynamical phenomenon, an attractor-widening crisis at the infection line. Indications of unstable dimension variability are seen in the neighborhood of the infection line. This may provide useful pointers to the spreading behavior seen in other extended systems., Comment: 20 pages, 9 figures

We study synchronization in the context of network traffic on a $2-d$ communication network with local clustering and geographic separations. The network consists of nodes and randomly distributed hubs where the top five hubs ranked according to their coefficient of betweenness centrality (CBC) are connected by random assortative and gradient mechanisms. For multiple message traffic, messages can trap at the high CBC hubs, and congestion can build up on the network with long queues at the congested hubs. The queue lengths are seen to synchronize in the congested phase. Both complete and phase synchronization is seen, between pairs of hubs. In the decongested phase, the pairs start clearing, and synchronization is lost. A cascading master-slave relation is seen between the hubs, with the slower hubs (which are slow to decongest) driving the faster ones. These are usually the hubs of high CBC. Similar results are seen for traffic of constant density. Total synchronization between the hubs of high CBC is also seen in the congested regime. Similar behavior is seen for traffic on a network constructed using the Waxman random topology generator. We also demonstrate the existence of phase synchronization in real Internet traffic data., Comment: 13 Pages, 15 figures

The phase diagram of the coupled sine circle map lattice exhibits a variety of interesting phenomena including spreading regions with spatiotemporal intermittency, non-spreading regions with spatial intermittency, and coherent structures termed solitons. A cellular automaton mapping of the coupled map lattice maps the spreading to non-spreading transition to a transition from a probabilistic to a deterministic cellular automaton. The solitonic sector of the map shows spatiotemporal intermittency with soliton creation, propagation and annihilation. A probabilistic cellular automaton mapping is set up for this sector which can identify each one of these phenomena., Comment: 6 pages, 2 figures. Submitted to Pramana-Journal of Physics. To appear in the Proceedings of "Perspectives in Nonlinear Dynamics 2007"

We study message transfer in a $2-d$ communication network of regular nodes and randomly distributed hubs. We study both single message transfer and multiple message transfer on the lattice. The average travel time for single messages travelling between source and target pairs of fixed separations shows $q-$exponential behaviour as a function of hub density with a characteristic power-law tail, indicating a rapid drop in the average travel time as a function of hub density. This power-law tail arises as a consequence of the log-normal distribution of travel times seen at high hub densities. When many messages travel on the lattice, a congestion-decongestion transition can be seen. The waiting times of messages in the congested phase show a Gaussian distribution, whereas the decongested phase shows a log-normal distribution. Thus, the congested or decongested behaviour is encrypted in the behaviour of the waiting time distributions., Comment: 7 Pages, 6 figure, to appear in the Proceeding of the conference Perspectives in Nonlinear Dynamics 2007, a special issue of the Journal Pramana

We study the efficiency of the gradient mechanism of message transfer in a $2-d$ 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 time of single messages traveling on this lattice, plotted as a function of hub density, shows q-exponential behavior. At high hub densities, this distribution can be fitted well by a power law. 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 towards 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., Comment: 24 pages, 25 figures

We study the dynamics of inertial particles in two dimensional incompressible flows. The particle dynamics is modelled by four dimensional dissipative bailout embedding maps of the base flow which is represented by 2-d 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 behaviour, 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 4-d 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 behaviour 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., Comment: 16 pages, 9 figures, submitted for publication

The phase diagram of the coupled sine circle map lattice shows spatio-temporal intermittency of two distinct types: spatio-temporal intermittency of the directed percolation (DP) class, and spatial intermittency which does not belong to this class. These two types of behaviour are seen to be special cases of the spreading and non-spreading regimes seen in the system. In the spreading regime, each site can infect its neighbours permitting an initial disturbance to spread, whereas in the non-spreading regime no infection is possible. The two regimes are separated by a line which we call the infection line. The coupled map lattice can be mapped on to an equivalent cellular automaton which shows a transition from a probabilistic cellular automaton (PCA) to a deterministic cellular automaton (DCA) at the infection line. Thus the existence of the DP and non-DP universality classes in the same system is signalled by the PCA to DCA transition. We also discuss the dynamic origin of this transition., Comment: 11 pages, 6 figures

We study spatio-temporal intermittency (STI) in a system of coupled sine circle maps. The phase diagram of the system shows parameter regimes with STI of both the directed percolation (DP) and non-DP class. STI with synchronized laminar behaviour belongs to the DP class. The regimes of non-DP behaviour show spatial intermittency (SI), where the temporal behaviour of both the laminar and burst regions is regular, and the distribution of laminar lengths scales as a power law. The regular temporal behaviour for the bursts seen in these regimes of spatial intermittency can be periodic or quasi-periodic, but the laminar length distributions scale with the same power-law, which is distinct from the DP case. STI with traveling wave (TW) laminar states also appears in the phase diagram. Soliton-like structures appear in this regime. These are responsible for cross-overs with accompanying non-universal exponents. The soliton lifetime distributions show power law scaling in regimes of long average soliton life-times, but peak at characteristic scales with a power-law tail in regimes of short average soliton life-times. The signatures of each type of intermittent behaviour can be found in the dynamical characterisers of the system viz. the eigenvalues of the stability matrix. We discuss the implications of our results for behaviour seen in other systems which exhibit spatio-temporal intermittency., Comment: 25 pages, 11 figures. Submitted to Phys. Rev. E

Systems of coupled sine circle maps show regimes of spatiotemporally intermittent behaviour with associated scaling exponents which belong to the DP class, as well as regimes of spatially intermittent behaviour (with associated regular dynamical behaviour) which do not belong to the DP class. Both types of behaviour are seen along the bifurcation boundaries of the synchronized solutions, and contribute distinct signatures to the dynamical characterizers of the system, viz. the distribution of eigenvalues of the one step stability matrix. Within the spatially intermittent (SI) class, the temporal behaviour of the burst solutions can be quasi-periodic or travelling wave. The usual characterizers of bifurcations, i.e. the eigenvalues of the stability matrix crossing the unit circle, pick up the bifurcation from the synchronized solution to SI with quasi-periodic bursts but are unable to pick up the bifurcation of the synchronized solution to SI with TW bursts. Other characterizers, such as the Shannon entropy of the eigenvalue distribution, and the rate of change of the largest eigenvalue with parameter are required to pick up this bifurcation. This feature has also been seen for other bifurcations in this system, e.g. that from the synchronized solution to kink solutions. We therefore conjecture that in the case of high dimensional systems, entropic characterizers provide better signatures of bifurcations from one ordered solution to another., Comment: 9 pages, 6 figures, Presented at the International Conference on "Statistical Mechanics of Plasticity and Related Instabilities", (SMPRI 2005), at IISc, Bangalore

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., Comment: 9 pages, 5 figures, the Proceedings of the 3rd International Conference NEXT-SigmaPhi

We study spatiotemporal intermittency in a system of coupled sine circle maps. The phase diagram of the system shows parameter regimes where the STI lies in the directed percolation class, as well as regimes which show pure spatial intermittency (where the temporal behaviour is regular) which do not belong to the DP class. Thus, both DP and non-DP behaviour can be seen in the same system. The signature of DP and non-DP behaviour can be seen in the dynamic characterisers, viz. the spectrum of eigenvalues of the linear stability matrix of the evolution equation, as well as in the multifractal spectrum of the eigenvalue distribution. The eigenvalue spectrum of the system in the DP regimes is continuous, whereas it shows evidence of level repulsion in the form of gaps in the spectrum in the non-DP regime. The multifractal spectrum of the eigenvalue distribution also shows the signature of DP and non-DP behaviour. These results have implications for the manner in which correlations build up in extended systems., Comment: 24 pages, 9 figures. To be published in Phys. Rev. E

We discuss the spatiotemporal intermittency (STI) seen in coupled map lattices (CML-s). We identify the types of intermittency seen in such systems in the context of several specific CML-s. The Chat\'e-Manneville CML is introduced and the on-going debate on the connection of the spatiotemporal intermittency seen in this model with the problem of directed percolation is summarised. We also discuss the STI seen in the sine circle map model and its connection with the directed percolation problem, as well as the inhomogenous logistic map lattice which shows the novel phenomenon of spatial intermittency and other types of behaviour not seen in the other models. The connection of the bifurcation behaviour in this model with STI is touched upon. We conclude with a discussion of open problems., Comment: 23 pages, 6 figures, To appear in the Proceedings of Indian National Science Academy, Part A

We study the phase diagram of the sine circle map lattice with random initial conditions and identify the various types of dynamical behaviour which appear here. We focus on the regions which show spatio-temporal intermittency and characterise the accompanying scaling behaviour. Directed percolation exponents are seen at some points in the parameter space in the neighbourhood of bifurcation boundaries. We discuss this behaviour as well as other types of behaviour seen in the parameter space in the context of the phase diagram., Comment: 4 pages, 4 eps figures. Minor formatting changes

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 hub congestion. We find that the betweenness centrality (BC) criterion is useful for identifying hubs which are most likely to cause congestion, and that the addition of assortative connections to hubs of high BC relieves congestion most efficiently., Comment: submitted to Phys Rev Lett, 3 figures

We address the problem of message transfer in a communication network. The network consists of nodes and links, with the nodes lying on a two dimensional lattice. Each node has connections with its nearest neighbours, whereas some special nodes, which are designated as hubs, have connections to all the sites within a certain area of influence. The degree distribution for this network is bimodal in nature and has finite variance. The distribution of travel times between two sites situated at a fixed distance on this lattice shows fat fractal behaviour as a function of hub-density. If extra assortative connections are now introduced between the hubs so that each hub is connected to two or three other hubs, the distribution crosses over to power-law behaviour. Cross-over behaviour is also seen if end-to-end short cuts are introduced between hubs whose areas of influence overlap, but this is much milder in nature. In yet another information transmission process, namely, the spread of infection on the network with assortative connections, we again observed cross-over behaviour of another type, viz. from one power-law to another for the threshold values of disease transmission probability. Our results are relevant for the understanding of the role of network topology in information spread processes., Comment: 12 figures

The connectivity properties of a weight-bearing network are exploited to enhance it's capacity. We study a 2-d network of sites where the weight-bearing capacity of a given site depends on the capacities of the sites connected to it in the layers above. The network consists of clusters viz. a set of sites connected with each other with the largest such collection of sites being denoted as the maximal cluster. New connections are made between sites in successive layers using two distinct strategies. The key element of our strategies consists of adding as many disjoint clusters as possible to the sites on the trunk $T$ of the maximal cluster. The new networks can bear much higher weights than the original networks and have much lower failure rates. The first strategy leads to a greater enhancement of stability whereas the second leads to a greater enhancement of capacity compared to the original networks. The original network used here is a typical example of the branching hierarchical class. However the application of strategies similar to ours can yield useful results in other types of networks as well., Comment: 17 pages, 3 EPS files, 5 PS files, Phys. Rev. E (to appear)

We consider a lattice of coupled circle maps, a model arising naturally in descriptions of solid state phenomena such as Josephson junction arrays. We find that the onset of spatiotemporal intermittency (STI) in this system is analogous to directed percolation (DP), with the transition being to an unique absorbing state for low nonlinearities, and to weakly chaotic absorbing states for high nonlinearities. We find that the complete set of static exponents and spreading exponents at all critical points match those of DP very convincingly. Further, hyperscaling relations are fulfilled, leading to independent controls and consistency checks of the values of all the critical exponents. These results lend strong support to the conjecture that the onset of STI in deterministic models belongs to the DP universality class., Comment: Submitted to Physical Review E

We investigate the spatiotemporal dynamics of coupled circle map lattices, evolving under synchronous (parallel) updating on one hand and asynchronous (random) updating rules on the other. Synchronous evolution of extended spatiotemporal systems, such as coupled circle map lattices, commonly yields multiple co-existing attractors, giving rise to phenomena strongly dependent on the initial lattice. By marked contrast numerical evidence here strongly indicates that asynchronous evolution eliminates most of the attractor states arising from special sets of initial conditions in synchronous systems, and tends to yield more global attractors. Thus the phenomenology arising from asynchronous evolution is more generic and robust in that it is obtained from many different classes of initial states. Further we show that in parameter regions where both asynchronous and synchronous evolution yield spatio-temporal intermittency, asynchronicity leads to better scaling behaviour., Comment: 15 pages (including 5 figures), submitted to Phys. Rev. Letts

We study the phenomenon of intermittency in inhomogeneous lattices of coupled map where inhomogeneity appears in the form of different values of map parameters at adjacent sites.The system exhibits spatiotemporal intermittency in various regions of parameter space.We identify the types of co-dimension two bifurcations which give rise to spatio-temporal or purely spatial intermittency and study the mechanism in one case, that of synchronised fixed point Power law distributions of laminar lengths is observed in vicinity of such co-dim 2 points and the exponents fall in three ranges for three different types of bifurcations. Three of the exponents seen in this model show good agreement with those observed in fluid experiments with quasi-one dimensional geometries., Comment: 8 pages,4 postscript figures

We describe adaptive control algorithms whereby a chaotic dynamical system can be steered to a target state with desired characteristics. A specific implementation considered has the objective of directing the system to a state which is more chaotic or mixed than the uncontrolled one. This methodology is easy to implement in discrete or continuous dynamical systems. It is robust and efficient, and has the additional advantage that knowledge of the detailed behaviour of the system is not required., Comment: 3 postscript figures; to appear in Phys Rev E

We study spatially periodic orbits for a coupled map lattice of sine circle maps with nearest neighbour coupling and periodic boundary conditions. The stability analysis for an arbitrary spatial period k is carried out in terms of the independent variables of the problem and the stability matrix is reduced to a neat block diagonal form. For a lattice of size kN, we show that the largest eigenvalue for the stability matrix of size $kN \times kN$ is the same as that for the basic spatial period k matrix of size $k \times k$. Thus the analysis for a kN lattice case can be reduced to that for a k lattice case. We illustrate this explicitly for a spatial period two case. Our formalism is general and can be extended to any coupled map lattice. We also obtain the stability regions of solutions which have the same spatial and temporal period numerically. Our analysis shows that such regions form a set of Arnold tongues in the $\Omega-\epsilon-K$ space. The tongues corresponding to higher spatial periods are contained within the tongues seen in the temporally periodic spatial period one or synchronised case. We find an interesting new bifurcation wherein the the spatially synchronised and temporal period one solution undergoes a bifurcation to a spatio-temporal period two travelling wave solution. The edges of the stability interval of this solution are analytically obtained., Comment: 17 pages, REVTEX, and 9 ps figures. Submitted to Physical Review E

We study the spatially synchronized and temporally periodic solutions of a 1-d lattice of coupled sine circle maps. We carry out an analytic stability analysis of this spatially synchronized and temporally periodic case and obtain the stability matrix in a neat block diagonal form. We find spatially synchronized behaviour over a substantial range of parameter space. We have also extended the analysis to higher spatial periods with similar results. Numerical simulations for various temporal periods of the synchronized solution, reveal that the entire structure of the Arnold tongues and the devil's staircase seen in the case of the single circle map can also be observed for the synchronized coupled sine circle map lattice. Our formalism should be useful in the study of spatially periodic behaviour in other coupled map lattices., Comment: uuencoded, 1 rextex file 14 pages, 3 postscript figures