Fluctuations and simple chaotic dynamics

We describe the effects of fluctuations on the period-doubling bifurcation to chaos. We study the dynamics of maps of the interval in the absence of noise and numerically verify the scaling behavior of the Lyapunov characteristic exponent near the transition to chaos. As previously shown, fluctuations produce a gap in the period-doubling bifurcation sequence. We show that this implies a scaling behavior for the chaotic threshold and determine the associated critical exponent. By considering fluctuations as a disordering field on the deterministic dynamics, we obtain scaling relations between various critical exponents relating the effect of noise on the Lyapunov characteristic exponent. A rule is developed to explain the effects of additive noise at fixed parameter value from the deterministic dynamics at nearby parameter values.     

Fluctuations in airport arrival and departure traffic: A network analysis

Air traffic is a typical complex system,in which movements of traffic components(pilots,controllers,equipment,and environment),especially airport arrival and departure traffic,form complicated spatial and temporal dynamics.The fluctuations of airport arrival and departure traffic are studied from the point of view of networks as the special correlation between different airports.Our collected flow volume data on the time-dependent activity of US airport arrival and departure traffic indicate that the coupling between the average flux and the fluctuation of an individual airport obeys a certain scaling law with a wide variety of scaling exponents between 1/2 and 1.These scaling phenomena can explain the interaction between the airport internal dynamics(e.g.queuing at airports,a ground delay program and following flying traffic) and a change in the external(network-wide) traffic demand(e.g.an increase in traffic during peak hours every day),allowing us to further understand the mechanisms governing the collective behaviour of the transportation system.We separate internal dynamics from external fluctuations using a scaling law which is helpful for us to systematically determine the origin of fluctuations in airport arrival and departure traffic,uncovering the collective dynamics.Hot spot features are observed in airport traffic data as the dynamical inhomogeneity in the fluxes of individual airports.The intrinsic characteristics of airport arrival and departure traffic under severe weather is discussed as well.     

Fluctuations of time averages for Langevin dynamics in a binding force field

We derive a simple formula for the fluctuations of the time average x(t) around the thermal mean (eq) for overdamped brownian motion in a binding potential U(x). Using a backward Fokker-Planck equation, introduced by Szabo, Schulten, and Schulten in the context of reaction kinetics, we show that for ergodic processes these finite measurement time fluctuations are determined by the Boltzmann measure. For the widely applicable logarithmic potential, ergodicity is broken. We quantify the large nonergodic fluctuations and show how they are related to a superaging correlation function.     

Unlearning dynamics in Hopfield neural network

The dynamics of a convergent iterative unlearning algorithm proposed earlier [7, 8] is examined. A self-consistent system of equations of the spectral dynamics of a synaptic matrix is obtained at the thermodynamic limit. The unlearning intensity (which varies during the iteration process) that optimizes the algorithm's rate of convergence on the projector matrix is found. The synaptic-matrix spectrum dynamics for optimal unlearning is determined.Institute of Physicotechnical Problems, Moscow. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 9, pp. 1104–1115, September, 1994.     

A study of network dynamics

The dynamics of a discrete-time neural network model are investigated. First, a numerical survey of network power spectra is reported for networks of varying size with random weight matrices and initial states. The steepness of the logistic function and a symmetry measure of the weight matrix are taken as control parameters. Summary statistics are presented to give gross measures of the model's temporal activity in parameter space. Second, a detailed study of the dynamics of a particular network is described. Complex dynamical behavior is observed, including Hopf bifurcations, the Ruelle-Takens-Newhouse route to chaos (showing mode-locking at rational winding numbers and the destruction of an invariant torus), and the period-doubling route to chaos.     

Disease dynamics in a dynamic social network

We develop a framework for simulating a realistic, evolving social network (a city) into which a disease is introduced. We compare our results to prevaccine era measles data for England and Wales, and find that they capture the quantitative and qualitative features of epidemics in populations spanning two orders of magnitude. Our results provide unique insight into how and why the social topology of the contact network influences the propagation of the disease through the population. We argue that network simulation is suitable for concurrently probing contact network dynamics and disease dynamics in ways that prior modeling approaches cannot and it can be extended to the study of less well-documented diseases.     

Metabolic network dynamics in open chaotic flow

We have analyzed the dynamics of metabolically coupled replicators in open chaotic flows. Replicators contribute to a common metabolism producing energy-rich monomers necessary for replication. The flow and the biological processes take place on a rectangular grid. There can be at most one molecule on each grid cell, and replication can occur only at localities where all the necessary replicators (metabolic enzymes) are present within a certain neighborhood distance. Due to this finite metabolic neighborhood size and imperfect mixing along the fractal filaments produced by the flow, replicators can coexist in this fluid system, even though coexistence is impossible in the mean-field approximation of the model. We have shown numerically that coexistence mainly depends on the metabolic neighborhood size, the kinetic parameters, and the number of replicators coupled through metabolism. Selfish parasite replicators cannot destroy the system of coexisting metabolic replicators, but they frequently remain persistent in the system. (c) 2002 American Institute of Physics.     

Two-population dynamics in a growing network model

We introduce a growing network evolution model with nodal attributes. The model describes the interactions between potentially violent V and non-violent N agents who have different affinities in establishing connections within their own population versus between the populations. The model is able to generate all stable triads observed in real social systems. In the framework of rate equations theory, we employ the mean-field approximation to derive analytical expressions of the degree distribution and the local clustering coefficient for each type of nodes. Analytical derivations agree well with numerical simulation results. The assortativity of the potentially violent network qualitatively resembles the connectivity pattern in terrorist networks that was recently reported. The assortativity of the network driven by aggression shows clearly different behavior than the assortativity of the networks with connections of non-aggressive nature in agreement with recent empirical results of an online social system.     

Fluctuations, response and aging dynamics in a simple glass-forming liquid out of equilibrium

By means of molecular dynamics computer simulations we investigate the out of equilibrium relaxation dynamics of a simple glass former, a binary Lennard-Jones system, after a quench to low temperatures. We find that one-time quantities, such as the energy or the structure factor, show only a weak time dependence. By comparing the out of equilibrium structure factor with equilibrium data we find evidence that during the aging process the system remains in that part of phase space that mode-coupling theory classifies as liquid like. Two-times correlation functions show a strong time and waiting time dependence. For large and times corresponding to the early -relaxation regime the correlators approach the Edwards-Anderson value by means of a power-law in time. For large but fixed values of the relaxation dynamics in the -relaxation regime seems to be independent of the observable and temperature. The -relaxation shows a power-law dependence on time with an exponent which is independent of but depends on the observable. We find that at long times the correlation functions can be expressed as and compute the function h(t). This function is found to show a t-dependence which is a bit stronger than a logarithm and to depend on the observable considered. If the system is quenched to very low temperatures the relaxation dynamics at long times shows fast drops as a function of time. We relate these drops to relatively local rearrangements in which part of the sample relaxes its stress by a collective motion of 50-100 particles. Finally we discuss our measurements of the time dependent response function. We find that at long times the correlation functions and the response are not related by the usual fluctuation dissipation theorem but that this relation is similar to the one found for spin glasses with one step replica symmetry breaking. Received 17 May 1999     

Reconstruction of shared nonlinear dynamics in a network

We study a general physical network consisting of a collection of response systems with complex nonlinear dynamics, influenced by a common driver. The goal is to reconstruct dynamics, regular or chaotic, that are common to all of the response systems, working from simultaneous time series measured at the responses systems only. A fundamental theorem is stated concerning the reconstruction of the common driver. An algorithm is developed, based on the theorem, to carry out the reconstruction, and is demonstrated with several examples.     

Revealing network connectivity from response dynamics

We present a method to infer the complete connectivity of a network from its stable response dynamics. As a paradigmatic example, we consider networks of coupled phase oscillators and explicitly study their long-term stationary response to temporally constant driving. For a given driving condition, measuring the phase differences and the collective frequency reveals information about how the units are interconnected. Sufficiently many repetitions for different driving conditions yield the entire network connectivity (the absence or presence of each connection) from measuring the response dynamics only. For sparsely connected networks, we obtain good predictions of the actual connectivity even for formally underdetermined problems.     

Epidemic dynamics on an adaptive network

Many real-world networks are characterized by adaptive changes in their topology depending on the state of their nodes. Here we study epidemic dynamics on an adaptive network, where the susceptibles are able to avoid contact with the infected by rewiring their network connections. This gives rise to assortative degree correlation, oscillations, hysteresis, and first order transitions. We propose a low-dimensional model to describe the system and present a full local bifurcation analysis. Our results indicate that the interplay between dynamics and topology can have important consequences for the spreading of infectious diseases and related applications.     

Fluctuations, dynamics, and the stretch-coil transition of single actin filaments in extensional flows

Semiflexible polymers subject to hydrodynamic forcing play an important role in cytoskeletal motions in the cell, particularly when filaments guide molecular motors whose motions create flows. Near hyperbolic stagnation points, filaments experience a competition between bending elasticity and tension and are predicted to display suppressed thermal fluctuations in the extensional regime and a buckling instability under compression. Using a microfluidic cross-flow geometry, we verify these predictions in detail, including a fluctuation-rounded stretch-coil transition of actin filaments.     

Fluctuations and pseudo long range dependence in network flows: A non-stationary Poisson process model

In the study of complex networks (systems), the scaling phenomenon of flow fluctuations refers to a certain power-law between the mean flux (activity) < F_i> of the i-th node and its variance σ_i as F_{i ∝ <Fi^α. Such scaling laws are found to be prevalent both in natural and man-made network systems, but the understanding of their origins still remains limited. This paper proposes a non-stationary Poisson process model to give an analytical explanation of the non-universal scaling phenomenon: the exponent α varies between 1/2 and 1 depending on the size of sampling time window and the relative strength of the external/internal driven forces of the systems. The crossover behaviour and the relation of fluctuation scaling with pseudo long range dependence are also accounted for by the model. Numerical experiments show that the proposed model can recover the multi-scaling phenomenon.}     

Feedback between node and network dynamics can produce real-world network properties

Real-world networks are characterized by common features, including among others a scale-free degree distribution, a high clustering coefficient and a short typical distance between nodes. These properties are usually explained by the dynamics of edge and node addition and deletion.     

Neural network learning dynamics in a path integral framework

Using the local-density approximation, calculating the Hellmann-Feynman forces and applying the direct method, the phonon dispersion relations for the rutile-like structure of crystalline SnO₂ have been derived for the first time. The phonon frequencies at the point agree very well with Raman and infrared data and other phenomenological model calculations. The LO/TO splitting is estimated by calculating phonons from an elongated supercell. The computations under pressure reveal a soft mode of B_1g symmetry which leads to a ferroelastic phase transition. The pressure-dependence of the lattice constants and the Grüneisen parameters of the modes are calculated. Received 18 July 1999     

On the aging dynamics in an immune network model

Recently we have used a cellular automata model which describes the dynamics of a multi-connected network to reproduce the refractory behavior and aging effects obtained in immunization experiments performed with mice when subjected to multiple perturbations. In this paper we investigate the similarities between the aging dynamics observed in this multi-connected network and the one observed in glassy systems, by using the usual tools applied to analyze the latter. An interesting feature we show here, is that the model reproduces the biological aspects observed in the experiments during the long transient time it takes to reach the stationary state. Depending on the initial conditions, and without any perturbation, the system may reach one of a family of long-period attractors. The perturbations may drive the system from its natural attractor to other attractors of the same family. We discuss the different roles played by the small random perturbations ("noise") and by the large periodic perturbations ("immunizations").Received: 7 May 2003, Published online: 23 July 2003PACS: 87.18.Hf Spatiotemporal pattern formation in cellular populations - 87.10.+e General theory and mathematical aspects - 61.43.Fs GlassesD.A. Stariolo: Research Associate of the Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, Trieste, Italy     

Multimode dynamics in a network with resource mediated coupling

The purpose of this paper is to study the special forms of multimode dynamics that one can observe in systems with resource-mediated coupling, i.e., systems of self-sustained oscillators in which the coupling takes place via the distribution of primary resources that controls the oscillatory state of the individual unit. With this coupling, a spatially inhomogenous state with mixed high and low-amplitude oscillations in the individual units can arise. To examine generic phenomena associated with this type of interaction we consider a chain of resistively coupled electronic oscillators connected to a common power supply. The two-oscillator system displays antiphase synchronization, and it is interesting to note that two-mode oscillations continue to exist outside of the parameter range in which oscillations occur for the individual unit. At low coupling strengths, the multi-oscillator system shows high dimensional quasiperiodicity with little tendency for synchronization. At higher coupling strengths, one typically observes spatial clustering involving a few oscillating units. We describe three different scenarios according to which the cluster can slide along the chain as the bias voltage changes.     

Small-world network effect in competing Glauber- and Kawasaki-type dynamics

In this article, we investigate the competing Glauber-type and Kawasaki-type dynamics with small-world network (SWN) effect, in the framework of the Gaussian model. The Glauber-type single-spin transition mechanism with probability p simulates the contact of the system with a heat bath and the Kawasaki-type dynamics with probability 1-p simulates an external energy flux. Two different types of SWN effect are studied, one with the total number of links increased and the other with it conserved. The competition of the dynamics leads to an interesting self-organization process that can be characterized by a phase diagram with two identifiable temperatures. By studying the modification of the phase diagrams, the SWN effect on the two dynamics is analyzed. For the Glauber-type dynamics, more important is the altered average coordination number while the Kawasaki-type dynamics is enhanced by the long range spin interaction and redistribution.Received: 11 October 2003, Published online: 30 January 2004PACS: 89.75.-k Complex systems - 64.60.Ht Dynamic critical phenomena - 64.60.Cn Order-disorder transformations; statistical mechanics of model systems - 64.60.Fr Equilibrium properties near critical points, critical exponents     

Neural network with formed dynamics of activity

The problem of developing a neural network with a given pattern of the state sequence is considered. A neural network structure and an algorithm of forming its bond matrix which lead to an approximate but robust solution of the problem are proposed and discussed. Limiting characteristics of the serviceability of the proposed structure are studied. Various methods of visualizing dynamic processes in a neural network are compared. Possible applications of the results obtained for interpretation of neurophysiological data and in neuroinformatics systems are discussed.Institute of Neurocybernetics, State University, Rostov-on-Don. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 9, pp. 1065–1076, September, 1994.