We introduce an approach based on algebraic topological methods that allow an accurate characterization of jamming in dynamical systems with queues. As a prototype system, we analyze the traffic of information packets with navigation and queuing at nodes on a network substrate in distinct dynamical regimes. A temporal sequence of traffic density fluctuations is mapped onto a mathematical graph in which each vertex denotes one dynamical state of the system. The coupling complexity between these states is revealed by classifying agglomerates of high-dimensional cliques that are intermingled at different topological levels and quantified by a set of geometrical and entropy measures. The free-flow, jamming, and congested traffic regimes result in graphs of different structure, while the largest geometrical complexity and minimum entropy mark the edge of the jamming region.

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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 the 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 that are most likely to cause congestion. The addition of assortative connections to hubs of high CBC relieves congestion very efficiently.

The connectivity properties of a weight-bearing network are exploited to enhance its capacity. We study a 2D 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. In the first strategy the reconnections start from the last layer upwards and stop when no new sites are added. In the second case, the reconnections start from the top layer and go all the way down to the last layer. 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.

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.

Two chaotic orbits can be synchronized by driving one of them by the other. Some of the variables of the driven orbit are set continuously to the corresponding variables of the drive orbit. It has been seen that synchronization can be achieved if the subsystem Lyapunov exponents corresponding to the remaining or response variables are all negative. We find that a procedure where the drive variable is set at discrete times can also achieve synchronization.However, the synchronization criterion is altered by the effect of the drive being set at finite time steps. An important consequence of this is found in the Lorenz system where synchronization can be achieved with z as the drive variable despite the existence of a marginal subsystem Lyapunov exponent. We also find that synchronization can be achieved for the Rossler attractor with z as the drive, even though the largest subsystem Lyapunov exponent is positive.In addition, we find that there is an optimal time step corresponding to the fastest rate of convergence for both cases above. Our synchronization criterion reduces to the usual subsystem-Lyapunov-exponent criterion in the limit of the time step tending to zero.PACS number(s): 05.45. +b

We study spatiotemporal intermittency (STI) 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 (DP) class, as well as regimes which show pure spatial intermittency (where the temporal behavior is regular) which do not belong to the DP class. Thus both DP and non-DP behavior can be seen in the same system. The signature of DP and non-DP behavior can be seen in the dynamic characterizers, 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 behavior. These results have implications for the manner in which correlations build up in extended systems.

We consider a lattice of coupled circle maps, a popular model for the study of mode-locked phenomena. We find that the onset of spatiotemporal intermittency (STI) in this system is analogous to directed percolation (DP), with the transition being to a 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 provide an example in support of the conjecture that the onset of STI in deterministic models belongs to the DP universality class. Nonuniversal spreading exponents are seen only for the cases where the initial state is homogeneous with symmetrically placed seeds leading to strictly symmetric spreading. However, very small departures from homogeneity are sufficient to restore the DP exponents.