We study the synchronization of coupled dynamical systems on networks. The dynamics is governed by a local nonlinear oscillator for each node of the network and interactions connecting different nodes via the links of the network. We consider existence and stability conditions for both single- and multi-cluster synchronization. For networks with time-varying topology we compare the synchronization properties of these networks with the corresponding time-average network. We find that if the different coupling matrices corresponding to the time-varying networks commute with each other then the stability of the synchronized state for both the time-varying and the time-average topologies are approximately the same. On the other hand, for non-commuting coupling matrices the stability of the synchronized state for the time-varying topology is in general better than the time-average topology.

We find prominent similarities in the features of the time series for the (model earthquakes or) overlap of two Cantor sets when one set moves with uniform relative velocity over the other and time series of stock prices. An anticipation method for some of the crashes have been proposed here, based on these observations.

We analyse the origin of the multiple long time scales associated with the long time decay observed in non-polar solvation dynamics by linear stability analysis of solvent density modes where the effects of compressibility and solvent structure are systematically incorporated. The coupling of the solute–solvent interactions at both ground and excited states of the solute with the compressibility and solvent structure is found to have important effects on the time scales. The present theory suggests that the relatively longer time constant is controlled by the solvent compressibility, while the solvent structure at the nearest-neighbour length scale dominates the shorter time constant.

We discuss the equivalence between kinetic wealth-exchange models, in which agents exchange wealth during trades, and mechanical models of particles, exchanging energy during collisions. The universality of the underlying dynamics is shown both through a variational approach based on the minimization of the Boltzmann entropy and a microscopic analysis of the collision dynamics of molecules in a gas. In various relevant cases, the equilibrium distribution is well-approximated by a gamma-distribution with suitably defined temperature and number of dimensions. This in turn allows one to quantify the inequalities observed in the wealth distributions and suggests that their origin should be traced back to very general underlying mechanisms, for instance, the fact that smaller the fraction of the relevant quantity (e.g. wealth) that agent can exchange during an interaction, the closer the corresponding equilibrium distribution is to a fair distribution.

In the human lung, the gas transfer between air and blood is achieved in terminal units that are called `acini'. Whereas convection is still the predominant transport phenomenon at the acinus entrance, most of the acinar surface is in fact accessed by diffusion. The transition between convection and diffusion, and thus the size of the diffusion unit, depends on the air velocity at the acinus entrance. In this paper, we present a gas transport model which takes into account both the diffusion into the acinus and the diffusion across the alveolar membrane. It is shown that the physiological sizes of the diffusion unit in the lung, at rest or at exercise, can be explained by physical arguments. In that sense, diffusion is the `dimensioning criterion' of the lung at the acinar level. This approach shows that, due to diffusional screening at inspiration and at rest, there exists a permanent spatial inhomogeneity of oxygen and carbon dioxide partial pressure which reduces the effective surface efficiency of the human acinus to a value of only 30 to 40%. This model casts a new light on the properties of this physiological transport system. It permits in particular to understand how several diseases among which pulmonary edema may remain asymptomatic in their early stages.

We review a recent asymptotic weak noise approach to the Kardar–Parisi–Zhang equation for the kinetic growth of an interface in higher dimensions. The weak noise approach provides a many-body picture of a growing interface in terms of a network of localized growth modes. Scaling in 1d is associated with a gapless domain wall mode. The method also provides an independent argument for the existence of an upper critical dimension.

Users of the peer-to-peer system join and leave the network randomly, which makes the overlay network dynamic and unstable in nature. In this paper, we propose an analytical framework to assess the robustness of p2p networks in the face of user churn. We model the peer churn through degree-independent as well as degree-dependent node failure. Lately, superpeer networks are becoming the most widely used topology among the p2p networks. Therefore, we perform the stability analysis of superpeer networks as a case study. We validate the analytically derived results with the help of simulation.

We study the adaptation dynamics of an initially maladapted population evolving via the elementary processes of mutation and selection. The evolution occurs on rugged fitness landscapes which are defined on the multi-dimensional genotypic space and have many local peaks separated by low fitness valleys. We mainly focus on the Eigen’s model that describes the deterministic dynamics of an infinite number of self-replicating molecules. In the stationary state, for small mutation rates such a population forms a quasispecies which consists of the fittest genotype and its closely related mutants. The quasispecies dynamics on rugged fitness landscape follow a punctuated (or step-like) pattern in which a population jumps from a low fitness peak to a higher one, stays there for a considerable time before shifting the peak again and eventually reaches the global maximum of the fitness landscape. We calculate exactly several properties of this dynamical process within a simplified version of the quasispecies model.

We discuss exact enumeration technique and its application to polymers and biopolymers. Using this method one can obtain phase diagram in thermodynamic limit. The method works quite well in describing the outcomes of single molecule force spectroscopy results where finite size effects play a crucial role.

We obtain ratchet effect in inertial structureless systems in symmetric periodic potentials where the asymmetry comes from the non-uniform friction offered by the medium and driven by symmetric periodic forces. In the adiabatic limit the calculations are done by extending the matrix continued fraction method and also by numerically solving the appropriate Langevin equation. For finite frequency field drive the ratchet effect is obtained only numerically. In the transient time scales the system shows dispersionless behaviour as reported earlier when a constant force is applied. In the periodic drive case the dispersion behaviour is more complex. In this brief communication we report some of the results of our work.

During the last phase of cell division in bacteria, a polymeric ring forms at the division site. The ring, made of intracellular proteins, anchors to the cell wall and starts to contract. That initiates a dividing septum to close in, like the shutter of a camera, eventually guillotining the cell into two daughters. All through, the ring remains at the leading edge of the septum and seems to power its closure. It is not understood why does the ring contract. We propose a theoretical model to explain this. It is worth mentioning that a similar contraction phenomenon occurs for the actin ring in eukaryotes, but there it is due to motor proteins, which however, are absent in bacteria.

We consider navigation or search schemes on networks which have a degree distribution of the form $P(k) \propto$ exp($−k^{\gamma}$). In addition, the linking probability is taken to be dependent on social distances and is governed by a parameter 𝜆. The searches are realistic in the sense that not all search chains can be completed. An estimate of $\mu = \rho/s_{d}$, where 𝜌 is the success rate and $s_{d}$ the dynamic path length, shows that for a network of 𝑁 nodes, $\mu \propto N^{-\delta}$ in general. Dynamic small world effect, i.e., $\delta \simeq 0$ is shown to exist in a restricted region of the $\lambda - \gamma$ plane.

Bootstrap percolation transition may be first order or second order, or it may have a mixed character where a first-order drop in the order parameter is preceded by critical fluctuations. Recent studies have indicated that the mixed transition is characterized by power-law avalanches, while the continuous transition is characterized by truncated avalanches in a related sequential bootstrap process. We explain this behaviour on the basis of an analytical and numerical study of the avalanche distributions on a Bethe lattice.

Several networks occurring in real life have modular structures that are arranged in a hierarchical fashion. In this paper, we have proposed a model for such networks, using a stochastic generation method. Using this model we show that, the scaling relation between the clustering and degree of the nodes is not a necessary property of hierarchical modular networks, as had previously been suggested on the basis of a deterministically constructed model. We also look at dynamics on such networks, in particular, the stability of equilibria of network dynamics and of synchronized activity in the network. For both of these, we find that, increasing modularity or the number of hierarchical levels tends to increase the probability of instability. As both hierarchy and modularity are seen in natural systems, which necessarily have to be robust against environmental fluctuations, we conclude that additional constraints are necessary for the emergence of hierarchical structure, similar to the occurrence of modularity through multi-constraint optimization as shown by us previously.

Modelling the evolution of a financial index as a stochastic process is a problem awaiting a full, satisfactory solution since it was first formulated by Bachelier in 1900. Here it is shown that the scaling with time of the return probability density function sampled from the historical series suggests a successful model. The resulting stochastic process is a heteroskedastic, non-Markovian martingale, which can be used to simulate index evolution on the basis of an autoregressive strategy. Results are fully consistent with volatility clustering and with the multiscaling properties of the return distribution. The idea of basing the process construction on scaling, and the construction itself, are closely inspired by the probabilistic renormalization group approach of statistical mechanics and by a recent formulation of the central limit theorem for sums of strongly correlated random variables.

The dynamics of information traffic over scale-free networks has been investigated systematically. A series of routing strategies of data packets have been proposed, including the local routing strategy, the next-nearest-neighbour routing strategy, and the mixed routing strategy based on local static and dynamic information. The capacity of the network can be quantified by the phase transition from free flow state to congestion state. The optimal parameter values of each model leading to the highest efficiency of scale-free networked traffic systems have been found. Moreover, we have found hysteretic loop in networked traffic systems with finite packets delivering ability. Such hysteretic loop indicates the existence of the bi-stable state in the traffic dynamics over scale-free networks.

The cytoplasm of a living cell is crowded with several macromolecules of different shapes and sizes. Molecular diffusion in such a medium becomes anomalous due to the presence of macromolecules and diffusivity is expected to decrease with increase in macromolecular crowding. Moreover, many cellular processes are dependent on molecular diffusion in the cell cytosol. The enzymatic reaction rate has been shown to be affected by the presence of such macromolecules. A simple numerical model is proposed here based on percolation and diffusion in disordered systems to study the effect of macromolecular crowding on the enzymatic reaction rates. The model qualitatively explains some of the experimental observations.

Proteins are important biomolecules, which perform diverse structural and functional roles in living systems. Starting from a linear chain of amino acids, proteins fold to different secondary structures, which then fold through short- and long-range interactions to give rise to the final three-dimensional shapes useful to carry out the biophysical and biochemical functions. Proteins are defined as having a common `fold' if they have major secondary structural elements with same topological connections. It is known that folding mechanisms are largely determined by a protein's topology rather than its interatomic interactions. The native state protein structures can, thus, be modelled, using a graph-theoretical approach, as coarse-grained networks of amino acid residues as `nodes' and the inter-residue interactions/contacts as `links'. Using the network representation of protein structures and their 2D contact maps, we have identified the conserved contact patterns (groups of contacts) representing two typical folds – the EF-hand and the ubiquitin-like folds. Our results suggest that this direct and computationally simple methodology can be used to infer about the presence of specific folds from the protein's contact map alone.

This communication reports the photophysical characterization of self-assembled layer-by-layer (LbL) films of DNA (deoxyribonucleic acid) fabricated at different temperatures by electrostatic interaction with a polycation, poly(allylamine hydrochloride). It was observed that there was a successful incorporation of DNA molecules in DNA–PAH LbL films at room temperature as well as after melting temperature. An abrupt increase in intensity was observed in the absorption spectra of the films fabricated at high temperature which is an indication of the immobilization of unzipped DNA after melting of DNA. The films were observed to remain unaffected even after 250 h of film fabrication. The total electrostatic interaction time between DNA and PAH is about 15 min, that is, no PAH binding site is free.

Intrinsic curvature of biopolymers is emerging as an essential feature in various biological phenomena. Examples of polymers with intrinsic curvature are microtubule in eukaryotic cells or FtsZ filaments in prokaryotic cells. We consider the general model for polymers with intrinsic curvature. We aim to study both equilibrium and dynamic properties of such polymers. Here we report preliminary results on the equilibrium distribution function $P({\mathbf{R}})$ of the end-to-end distance ${\mathbf{R}}$. We employ transfer matrix method for this study.

Empirical study shows that many real networks in nature and society share two generic properties: they are scale-free and they display a high degree of clustering. Quite often they are modular in nature also, implying occurrences of several small tightly linked groups which are connected in a hierarchical manner among themselves. Recently, we have introduced a model of spatial scale-free network where nodes pop-up at randomly located positions in the Euclidean space and are connected to one end of the nearest link of the existing network. It has been already argued that the large scale behaviour of this network is like the Barabási–Albert model. In the present paper we briefly review these results as well as present additional results on the study of non-trivial correlations present in this model which are found to have similar behaviours as in the real-world networks. Moreover, this model naturally possesses the hierarchical characteristics lacked by most of the models of the scale-free networks.

In this paper we review the quenching dynamics of a quantum $XY$ spin-1/2 chain in the presence of a transverse field, when the transverse field or the anisotropic interaction is quenched at a slow but uniform rate. We also extend the results to the cases in which the system starts with any arbitrary initial condition as opposed to the initial fully magnetically aligned state which has been extensively studied earlier. The evolution is non-adiabatic in the time interval when the parameters are close to their critical values, and is adiabatic otherwise. The density of defects produced due to non-adiabatic transitions is calculated by mapping the many-particle system to an equivalent Landau–Zener problem. We show that in one dimension the density of defects in the final state scales as $1/\sqrt{\tau}$ irrespective of the initial condition, where 𝜏 is the quenching time-scale. However, the magnitude of density of defects is found to depend on the initial condition.

We show how the theory of large deviations in the coin toss experiment can give some insight into nonequilibrium fluctuation theorems and intermittency in turbulence.

Collective behaviour in multicell systems arises from exchange of chemicals/signals between cells and may be different from their intrinsic behaviour. These chemicals are products of regulated networks of biochemical pathways that underlie cellular functions, and can exhibit a variety of dynamics arising from the non-linearity of the reaction processes. We have addressed the emergent synchronization properties of a ring of cells, diffusively coupled by the end product of an intracellular model biochemical pathway exhibiting non-robust birhythmic behaviour. The aim is to examine the role of intercellular interaction in stabilizing the non-robust dynamics in the emergent collective behaviour in the ring of cells. We show that, irrespective of the inherent frequencies of individual cells, depending on the coupling strength, the collective behaviour does synchronize to only one type of oscillations above a threshold number of cells. Using two perturbation analyses, we also show that this emergent synchronized dynamical state is fairly robust under external perturbations. Thus, the inherent plasticity in the oscillatory phenotypes in these model cells may get suppressed to exhibit collective dynamics of a single type in a multicell system, but environmental influences can sometimes expose this underlying plasticity in its collective dynamics.