arXiv subjects

Edward Ott

Publications and source records attributed to Edward Ott.

At least 109 records · Page 6Linked to original sources

arXiv2008

Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators

It is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics. In particular, we derive an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered. For example, an exact, closed form solution for the nonlinear time evolution of the Kuramoto problem with a Lorentzian oscillator frequency distribution function is obtained. Low dimensional behavior is also demonstrated for several prototypical extensions of the Kuramoto model, and time-delayed coupling is also considered.

nlin.CD↗

arXiv2008

Adaptive synchronization of dynamics on evolving complex networks

We study the problem of synchronizing a general complex network by means of an adaptive strategy in the case where the network topology is slowly time varying and every node receives at each time only one aggregate signal from the set of its neighbors. We introduce an appropriately defined potential that each node seeks to minimize in order to reach/maintain synchronization. We show that our strategy is effective in tracking synchronization as well as in achieving synchronization when appropriate conditions are met.

cond-mat.dis-nn↗

arXiv2008

Synchronization in networks of networks: the onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators

The onset of synchronization in networks of networks is investigated. Specifically, we consider networks of interacting phase oscillators in which the set of oscillators is composed of several distinct populations. The oscillators in a given population are heterogeneous in that their natural frequencies are drawn from a given distribution, and each population has its own such distribution. The coupling among the oscillators is global, however, we permit the coupling strengths between the members of different populations to be separately specified. We determine the critical condition for the onset of coherent collective behavior, and develop the illustrative case in which the oscillator frequencies are drawn from a set of (possibly different) Cauchy-Lorentz distributions. One motivation is drawn from neurobiology, in which the collective dynamics of several interacting populations of oscillators (such as excitatory and inhibitory neurons and glia) are of interest.

math.DS↗

arXiv2007

Network synchronization of groups

In this paper we study synchronized motions in complex networks in which there are distinct groups of nodes where the dynamical systems on each node within a group are the same but are different for nodes in different groups. Both continuous time and discrete time systems are considered. We initially focus on the case where two groups are present and the network has bipartite topology (i.e., links exist between nodes in different groups but not between nodes in the same group). We also show that group synchronous motions are compatible with more general network topologies, where there are also connections within the groups.

cond-mat.dis-nn↗

arXiv2007

Approximating the largest eigenvalue of network adjacency matrices

The largest eigenvalue of the adjacency matrix of a network plays an important role in several network processes (e.g., synchronization of oscillators, percolation on directed networks, linear stability of equilibria of network coupled systems, etc.). In this paper we develop approximations to the largest eigenvalue of adjacency matrices and discuss the relationships between these approximations. Numerical experiments on simulated networks are used to test our results.

cond-mat.dis-nn↗

arXiv2007

Estimating the State of Large Spatiotemporally Chaotic Systems

Data assimilation refers to the process of obtaining an estimate of a system's state using a model for the system's time evolution and a time series of measurements that are possibly noisy and incomplete. However, for practical reasons, the high dimensionality of large spatiotemporally chaotic systems prevents the use of classical data assimilation techniques. Here, via numerical computations on the paradigmatic example of large aspect ratio Rayleigh-Benard convection, we demonstrate the applicability of a recently developed data assimilation method designed to circumvent this difficulty. In addition, we describe extensions of the algorithm for estimating unknown system parameters. Our results suggest the potential usefulness of our data assimilation technique to a broad class of situations in which there is spatiotemporally chaotic behavior.

nlin.CD↗

arXiv2007

Weighted percolation on directed networks

We present an analysis of the percolation transition for general node removal strategies valid for locally tree-like directed networks. On the basis of heuristic arguments we predict that, if the probability of removing node $i$ is $p_i$, the network disintegrates if $p_i$ is such that the largest eigenvalue of the matrix with entries $A_{ij}(1-p_i)$ is less than 1, where $A$ is the adjacency matrix of the network. The knowledge or applicability of a Markov network model is not required by our theory, thus making it applicable to situations not covered by previous works. We test our predicted percolation criterion against numerical results for different networks and node removal strategies.

cond-mat.soft↗

arXiv2006

Dynamics on Complex Networks and Applications

At the eight-year anniversary of Watts & Strogatz's work on the collective dynamics of small-world networks and seven years after Barabasi & Albert's discovery of scale-free networks, the area of dynamical processes on complex networks is at the forefront of the current research on nonlinear dynamics and complex systems. This volume brings together a selection of original contributions in complementary topics of statistical physics, nonlinear dynamics and biological sciences, and is expected to provide the reader with a comprehensive up-to-date representation of this rapidly developing area.

cond-mat.dis-nn↗

arXiv2006

Experimental Test of Universal Conductance Fluctuations by means of Wave-Chaotic Microwave Cavities

The mathematical equivalence of the time-independent Schrodinger equation and the Helmholtz equation is exploited to provide a novel means of studying universal conductance fluctuations in ballistic chaotic mesoscopic systems using a two-dimensional microwave-cavity. The classically chaotic ray trajectories within a suitably-shaped microwave cavity play a role analogous to that of the chaotic dynamics of non-interacting electron transport through a ballistic quantum dot in the absence of thermal fluctuations. The microwave cavity is coupled through two single-mode ports and the effect of non-ideal coupling between the ports and cavity is removed by a previously developed method based on the measured radiation impedance matrix. The Landauer-Buttiker formalism is applied to obtain the conductance of a corresponding mesoscopic quantum-dot device. We find good agreement for the probability density functions (PDFs) of the experimentally derived surrogate conductance, as well as its mean and variance, with the theoretical predictions of Brouwer and Beenakker. We also observe a linear relation between the quantum dephasing parameter and the cavity ohmic loss parameter.

cond-mat.mes-hall↗

arXiv2006

Scale dependence of branching in arterial and bronchial trees

Although models of branching in arterial and bronchial trees often predict a dependence of bifurcation parameters on the scale of the bifurcating vessels, direct verifications of this dependence with data are uncommon. We compare measurements of bifurcation parameters in airways and arterial trees of different mammals as a function of scale to general features predicted by theoretical models. We find that the size dependence is more complex than existing theories based solely on energy minimization explain, and suggest additional factors that may govern the branching at different scales.

physics.bio-ph↗

arXiv2006

The emergence of coherence in complex networks of heterogeneous dynamical systems

We present a general theory for the onset of coherence in collections of heterogeneous maps interacting via a complex connection network. Our method allows the dynamics of the individual uncoupled systems to be either chaotic or periodic, and applies generally to networks for which the number of connections per node is large. We find that the critical coupling strength at which a transition to synchrony takes place depends separately on the dynamics of the individual uncoupled systems and on the largest eigenvalue of the adjacency matrix of the coupling network. Our theory directly generalizes the Kuramoto model of equal strength, all-to-all coupled phase oscillators to the case of oscillators with more realistic dynamics coupled via a large heterogeneous network.

cond-mat.dis-nn↗

arXiv2006

Universal Properties of 2-Port Scattering, Impedance and Admittance Matrices of Wave Chaotic Systems

Statistical fluctuations in the eigenvalues of the scattering, impedance and admittance matrices of 2-Port wave-chaotic systems are studied experimentally using a chaotic microwave cavity. These fluctuations are universal in that their properties are dependent only upon the degree of loss in the cavity. We remove the direct processes introduced by the non-ideally coupled driving ports through a matrix-normalization process that involves the radiation-impedance matrix of the two driving ports. We find good agreement between the experimentally obtained marginal probability density functions (PDFs) of the eigenvalues of the normalized impedance, admittance and scattering matrix and those from Random Matrix Theory (RMT). We also experimentally study the evolution of the joint PDF of the eigenphases of the normalized scattering matrix as a function of loss. Experimental agreement with the theory by Brouwer and Beenakker for the joint PDF of the magnitude of the eigenvalues of the normalized scattering matrix is also shown.

cond-mat.dis-nn↗

arXiv2006

Delayed Feedback and Chaos on the Driven Diode-Terminated Transmission Line

A simple model of a distributed, non-linear circuit that produces chaos at GHz frequencies is introduced and tested experimentally. The model circuit is a driven diode-terminated transmission line with the transmission line impedance mismatched to that of the source. Experiments were performed with sinusoidal driving frequencies of 10 MHz to 1.2 GHz, driving powers of -30 to +50 dBm and transmission line delay times from 3 to 20 ns. Diode reverse recovery times ranged from 4 to 100 ns. As a result of many experiments, it was found that chaotic behavior was strongly dependent on the unbiased, small-signal reactance of the system as seen by the driving source, and influenced by an applied DC voltage-bias across the diode. In the experiments that showed period-doubling and / or chaos, the reverse recovery times of the diodes were on the order of both the driving period and the delay time of the circuits. Comparisons between theory and experiment are in general agreement. Chaos produced with a driving frequency of 1.105 GHz has been observed experimentally.

nlin.CD↗

arXiv2006

Characterization of Fluctuations of Impedance and Scattering Matrices in Wave Chaotic Scattering

In wave chaotic scattering, statistical fluctuations of the scattering matrix $S$ and the impedance matrix $Z$ depend both on universal properties and on nonuniversal details of how the scatterer is coupled to external channels. This paper considers the impedance and scattering variance ratios, $VR_z$ and $VR_s$, where $VR_z=Var[Z_{ij}]/\{Var[Z_{ii}]Var[Z_{jj}] \}^{1/2}$, $VR_s=Var[S_{ij}]/\{Var[S_{ii}]Var[S_{jj}] \}^{1/2}$, and $Var[.]$ denotes variance. $VR_z$ is shown to be a universal function of distributed losses within the scatterer. That is, $VR_z$ is independent of nonuniversal coupling details. This contrasts with $VR_s$ for which universality applies only in the large loss limit. Explicit results are given for $VR_z$ for time reversal symmetric and broken time reversal symmetric systems. Experimental tests of the theory are presented using data taken from scattering measurements on a chaotic microwave cavity.

cond-mat.mes-hall↗

arXiv2005

Synchronization in large directed networks of coupled phase oscillators

We extend recent theoretical approximations describing the transition to synchronization in large undirected networks of coupled phase oscillators to the case of directed networks. We also consider extensions to networks with mixed positive/negative coupling strengths. We compare our theory with numerical simulations and find good agreement.

cond-mat.stat-mech↗

arXiv2005

Formation of Multifractal Population Patterns from Reproductive Growth and Local Resettlement

We consider the general character of the spatial distribution of a population that grows through reproduction and subsequent local resettlement of new population members. We present several simple one and two-dimensional point placement models to illustrate possible generic behavior of these distributions. We show, numerically and analytically, that these models all lead to multifractal spatial distributions of population. Additionally, we make qualitative links between our models and the example of the Earth at Night image, showing the Earth's nighttime man-made light as seen from space. The Earth at Night data suffer from saturation of the sensing photodetectors at high brightness (`clipping'), and we account for how this influences the determined dimension spectrum of the light intensity distribution.

nlin.PS↗

arXiv2005

Aspects of the Scattering and Impedance Properties of Chaotic Microwave Cavities

We consider the statistics of the impedance of a chaotic microwave cavity coupled to a single port. We remove the non-universal effects of the coupling from the experimental data using the radiation impedance obtained directly from the experiments. We thus obtain the normalized impedance whose Probability Density Function (PDF) is predicted to be universal in that it depends only on the loss (quality factor) of the cavity. We find that impedance fluctuations decrease with increasing loss. The results apply to scattering measurements on any wave chaotic system.

nlin.CD↗

arXiv2005

Intermittency in Two-Dimensional Turbulence with Drag

We consider the enstrophy cascade in forced two-dimensional turbulence with a linear drag force. In the presence of linear drag, the energy wavenumber spectrum drops with a power law faster than in the case without drag, and the vorticity field becomes intermittent, as shown by the anomalous scaling of the vorticity structure functions. Using a previous theory, we compare numerical simulation results with predictions for the power law exponent of the energy wavenumber spectrum and the scaling exponents of the vorticity structure functions $ζ_{2q}$ obtained in terms of the distribution of finite time Lyapunov exponents. We also study, both by numerical experiment and theoretical analysis, the multifractal structure of the viscous enstrophy dissipation in terms of its Rényi dimension spectrum $D_q$ and singularity spectrum $f(α)$. We derive a relation between $D_q$ and $ζ_{2q}$, and discuss its relevance to a version of the refined similarity hypothesis. In addition, we obtain and compare theoretically and numerically derived results for the dependence on separation $r$ of the probability distribution of $δ_{\V{r}}ω$, the difference between the vorticity at two points separated by a distance $r$. Our numerical simulations are done on a $4096 \times 4096$ grid.

cond-mat.soft↗