Erik M. Bollt

Sufficient Conditions for Fast Switching Synchronization in Time-Varying Network Topologies

In previous work [J. D. Skufca and E. Bollt, Mathematical Biosciences and Engineering, 1 (2004), pp. 347-359], empirical evidence indicated that a time-varying network could propagate sufficient in...

Local method for detecting communities.

We propose a method of community detection that is computationally inexpensive and possesses physical significance to a member of a social network. This method is unlike many divisive and agglomerative techniques and is local in the sense that a community can be detected within a network without requiring knowledge of the entire network. A global application of this method is also introduced. Several artificial and real-world networks, including the famous Zachary karate club, are analyzed.

Synchronization in Random Weighted Directed Networks

We assess synchronization of oscillators that are coupled via a time-varying stochastic network, modeled as a weighted directed random graph that switches at a given rate between a set of possible graphs. The existence of any graph edge is probabilistic and independent from the existence of any other edge. We further allow each edge to be weighted differently. Even if the network is always instantaneously not connected, we show that sufficient information is propagated through the network to allow almost sure local synchronization as long as the expected value of the network is connected, and that the switching rate is sufficiently fast.

Hybrid chaos synchronization and its application in information processing

In this paper, through numerical studies, we explore a new methodology for chaos synchronization via a hybrid (generalized plus identical) synchronization. An arbitrary signal, generated by an unknown dynamical system, can be synchronized by the hybrid chaotic system. The signal can then be stored for future application such as password and message identification. Each finite-length signal can, in principle, be labelled and stored by a unique number, provided that the key hybrid system parameter used for the purpose is suitably chosen within a one-to-one mapping range. The new methodology enables us to encode an arbitrary signal accurately and efficiently. Sufficient numerical simulations are shown to verify the proposed design. Potential applications of the developed hybrid chaos synchronization system include information storage, message identification, and certain types of secure signal and image communication.

Master stability functions for coupled nearly identical dynamical systems

We derive a master stability function (MSF) for synchronization in networks of coupled dynamical systems with small but arbitrary parametric variations. Analogous to the MSF for identical systems, our generalized MSF simultaneously solves the linear-stability problem for near-synchronous states (NSS) for all possible connectivity structures. We also derive a general sufficient condition for stable near-synchronization and show that the synchronization error scales linearly with the magnitude of parameter variations. Our analysis underlines the significant role played by the Laplacian eigenvectors in the study of network synchronization of near-identical systems.

Targeting chaotic orbits to the Moon through recurrence

Abstract Transport times for a chaotic system are highly sensitive to initial conditions and parameter values. In a previous paper, we presented a technique to find rough orbits (epsilon chains) that achieve a desired transport rapidly. The strategy is to build the epsilon chain from segments of a long orbit — the point is that long orbits have recurrences in neighborhoods where faster orbits must also pass. If a local hyperbolicity condition is satisfied, then a nearby shadow orbit may be constructed with significantly smaller errors. In this paper, we modify the technique to find real orbits, in configuration space, of the restricted three body problem. We find a chaotic Earth-Moon transfer orbit that achieves ballistic capture and that requires 38% less total velocity boost than a comparable Hohmann transfer orbit.

High Resolution MEMS Accelerometers to Estimate VO2 and Compare Running Mechanics between Highly Trained Inter-Collegiate and Untrained Runners

Background The purposes of this study were to determine the validity and reliability of high resolution accelerometers (HRA) relative to VO2 and speed, and compare putative differences in HRA signal between trained (T) and untrained (UT) runners during treadmill locomotion. Methodology Runners performed 2 incremental VO2max trials while wearing HRA. RMS of high frequency signal from three axes (VT, ML, AP) and the Euclidean resultant (RES) were compared to VO2 to determine validity and reliability. Additionally, axial rms relative to speed, and ratio of axial accelerations to RES were compared between T and UT to determine if differences in running mechanics could be identified between the two groups. Principal Findings Regression of RES was strongly related to VO2, but T was different than UT (r = 0.96 vs 0.92; p<.001) for walking and running. During walking, only the ratio of ML and AP to RES were different between groups. For running, nearly all acceleration parameters were lower for T than UT, the exception being ratio of VT to RES, which was higher in T than UT. All of these differences during running were despite higher VO2, O2 cost, and lower RER in T vs UT, which resulted in no significant difference in energy expenditure between groups. Conclusions/Signficance These results indicate that HRA can accurately and reliably estimate VO2 during treadmill locomotion, but differences exist between T and UT that should be considered when estimating energy expenditure. Differences in running mechanics between T and UT were identified, yet the importance of these differences remains to be determined.

What is special about diffusion on scale-free nets?

We study diffusion (random walks) on recursive scale-free graphs and contrast the results to similar studies in other analytically soluble media. This allows us to identify ways in which diffusion in scale-free graphs is special. Most notably, scale-free architecture results in a faster transit time between existing nodes when the network grows in size; and walks emanating from the most connected nodes are recurrent, despite the networks infinite dimension. We also find that other attributes of the graph, besides its scale-free distribution, have a strong influence on the nature of diffusion.

REVIEW OF CHAOS COMMUNICATION BY FEEDBACK CONTROL OF SYMBOLIC DYNAMICS

This paper is meant to serve as a tutorial describing the link between symbolic dynamics as a description of a chaotic attractor, and how to use control of chaos to manipulate the corresponding symbolic dynamics to transmit an information bearing signal. We use the Lorenz attractor, in the form of the discrete successive maxima map of the z-variable time-series, as our main example. For the first time, here, we use this oscillator as a chaotic signal carrier. We review the many previously developed issues necessary to create a working control of symbol dynamics system. These include a brief review of the theory of symbol dynamics, and how they arise from the flow of a differential equation. We also discuss the role of the (symbol dynamics) generating partition, the difficulty of finding such partitions, which is an open problem for most dynamical systems, and a newly developed algorithm to find the generating partition which relies just on knowing a large set of periodic orbits. We also discuss the importance of using a generating partition in terms of considering the possibility of using some other arbitrary partition, with discussion of consequences both generally to characterizing the system, and also specifically to communicating on chaotic signal carriers. Also, of practical importance, we review the necessary feedback-control issues to force the flow of a chaotic differential equation to carry a desired message.

Causation entropy identifies indirect influences, dominance of neighbors and anticipatory couplings

Abstract Inference of causality is central in nonlinear time series analysis and science in general. A popular approach to infer causality between two processes is to measure the information flow between them in terms of transfer entropy. Using dynamics of coupled oscillator networks, we show that although transfer entropy can successfully detect information flow in two processes, it often results in erroneous identification of network connections under the presence of indirect interactions, dominance of neighbors, or anticipatory couplings. Such effects are found to be profound for time-dependent networks. To overcome these limitations, we develop a measure called causation entropy and show that its application can lead to reliable identification of true couplings.