Abstracts

Invited speakers

Rafi Blumenfeld, National University of Defense Technology, Changsha, and Imperial College London
Granular statistical mechanics – Status report and future directions
Sam Edwards’s formulation of statistical mechanics for granular media not only made science of particulates more rigorous but also opened a new branch of soft matter physics. In this presentation I review briefly the Edwards formalism, which has been extended to cellular and porous materials, and some of its assumptions. A certain problem with the original formalism is discussed and a new way to overcome it is proposed. Finally, a derivation of an equation of state with the new formalism is presented, which relates the volume, the boundary stress and measures of the structural and stress fluctuations.

Nicholas Buchler, Duke University
Marching to the cell cycle drum beat: Entrainment of a genetic oscillator in yeast
We have built a synthetic, two-gene oscillator in yeast using a transcriptional activator and inhibitor pair. We measured gene dynamics in single cells using time-lapse fluorescence microscopy. Our synthetic oscillator mostly exhibited 1:1 mode locking with the yeast cell cycle in fast growth conditions. To help distinguish whether our synthetic oscillator was driven or entrained by the cell cycle, we blocked the yeast cell cycle. Gene oscillation persisted with an intrinsic period similar to the cell cycle, which suggests that our synthetic oscillator is autonomous and also explains the 1:1 mode locking. Last, we demonstrate sub-harmonic entrainment of our synthetic oscillation during a slower cell cycle in poor growth conditions. The point of entrainment of our synthetic oscillator occurs during entry into the early cell cycle.

Dmitri Chklovskii, Simons Center for Data Analysis
Derivation of neural circuits from the similarity matching principle
To make sense of the world our brains must analyze high-dimensional datasets streamed by our sensory organs. Such analysis requires solving complex computational tasks such as dimensionality reduction, sparse feature discovery and clustering. To model these tasks we pursue an approach similar to the principle of least action in physics and propose a new family of objective functions based on the principle of similarity matching. From these objective functions we derive online distributed algorithms that can be implemented by biological neural networks. We also formulate minimax optimization problems from which we derive online algorithms with two classes of neurons identified with principal neurons and interneurons in biological circuits. In addition to modeling neural computation our algorithms can be used for Big Data applications.

Jim Crutchfield, University of California, Davis
Demon Dynamics: Deterministic Chaos, the Szilard Map, and the Intelligence of Thermodynamic Systems
We introduce a deterministic chaotic system—the Szilard Map—that encapsulates the measurement, control, and erasure protocol by which Maxwellian Demons extract work from a heat reservoir. Implementing the Demon’s control function in a dynamical embodiment, our construction symmetrizes Demon and thermodynamic system, allowing one to explore their functionality and recover the fundamental trade-off between the thermodynamic costs of dissipation due to measurement and due to erasure. The map’s degree of chaos—captured by the Kolmogorov-Sinai entropy—is the rate of energy extraction from the heat bath. Moreover, an engine’s statistical complexity quantifies the minimum necessary system memory for it to function. In this way, dynamical instability in the control protocol plays an essential and constructive role in intelligent thermodynamic systems. Joint work with Alexander B. Boyd.

Lucas Goehring, Max Planck Institute for Dynamics and Self-Organization
Watching paint dry: the dynamics of charged colloidal particles 
Colloidal dispersions are the basis of many paints, inks, coatings, ceramics and composites. When dried, these dispersions can show a surprising variety of patterning mechanisms. They may buckle or crack into spirals or parallel lines for example, while their dried shapes can vary from a coffee-ring to textured surfaces controlled by evaporation-lithography. This abundance of patterns suggests diverse means for the directed self-assembly of micro-structured materials, if the underlying dynamics can be understood and controlled.  We have looked at the origins of these patterns in the collective dynamics of individual colloidal particles, and their microscopic structure. Here, I will present some of our group’s recent work on slowly drying colloidal latex and silica, including features such as regular shear banding, a controllable structural anisotropy and birefringence, and an unexpectedly rich phase diagram of colloidal crystals arising from poly-disperse particles.  In other words, I will guide you through what can be learned by a careful watching of the drying of a drop of latex paint, and the many instabilities and structures that can result from it.

Rudy Horne, Morehouse University
Parity-Time (PT) Symmetric systems: An analysis of dimer and trimer models
In the late 1990s, a novel idea came to fruition concerning the study of the fundamentals of quantum mechanics. In the works of Bender and Boettcher [1] and Bender et. al [1, 2], it was proposed that Hamiltonians that respect the physical symmetries of the dynamics, namely parity (P) and time-reversal (T), could have real eigenvalues even if said Hamiltonian were non-Hermitian. In this talk, we will discuss the cases of two-site (dimer) and three-site (trimer) PT- symmetric models respecting the parity-time (PT) symmetry with linear and nonlinear gain/loss profiles. We also examine solutions arising from these dimer and trimer mod- els and their regions of stability. We will present both analytical and numerical results concerning this work. In addition, we will examine both dimer and trimer models with a rapidly-varying gain/loss profile. In this part of the work, we discuss the derivation a set of averaged equations and examine its solutions and stability regions. We also compare these results with those obtained by direct simulation of the full set of nonlinear equations.
[1] C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)
[2] C.M. Bender, S. Boettcher and P.N. Meisinger, J. Math. Phys. 40, 2201 (1999).

Sui Huang, Institute for Systems Biology (Seattle)
Cell fate commitment as high-dimensional critical state transition revealed by single-cell resolution gene expression analysis in cell populations
How does a mammalian cell transition from one stable attractor state into another one during differentiation –a drastic phenotypic cell state switch involving the alteration of expression of thousands of genes? A biological meaningful stable cell state, such as a ‘cell type’, is represented by a high-dimensional attractor x* that defines the expression level of thousands of genes. Such attractor states are very stable and spontaneous exit are extremely rare. Thus, a differentiation process likely involves destabilization of the high-dimensional attractor through some kind of bifurcation until at a critical point the attractor disappears, allowing the cells to exit and “flow” into a nearby (“lower”) attractor state that is now accessible, and represent the differentiated state. This simple picture has biological consequences that are could be observable. While critical transitions are well described for low-dimensional systems, and typically modelled as a fold-bifurcation with the associated “early warning signs”, we now show experimentally the case for a high-dimensional system: the commitment of a progenitor cell to a particular differentiated lineage. The critical slowing down and increase of autocorrelation of the fluctuations of the state variable as the hallmarks for an approach to the critical “tipping point” as the attractor “flattens”, cannot be measured in cells due to the impracticality of continuous monitoring. Instead, we exploit the fact that we have an entire cell population which represents a statistical ensemble of systems that ergodically “map out” the state space structure. Then with just a few snapshot measurements of the population distribution in the high-dimensional one can detect signs of an imminent tipping point without knowledge of the system equations dx/dt=F(x, μ), let alone of any bifurcation parameter μ. For this purpose we introduce an empirical quantity, IC computed from experimentally determined single-cell resolution, high-dimensional molecular profiles of differentiating cell populations and show that it can predict an impending critical transition, that is, a fate commitment in multipotent blood precursor cells. Consequences of such dynamics for state destabilization in development and in cancer progression are discussed.

Eleni Katifori, University of Pennsylvania
Emerging hierarchies in biological distribution networks
Biological transport webs, such as the blood circulatory system in the brain and other animal organs, or the slime mold Physarum polycephalum, are frequently dominated by dense sets of nested cycles. The architecture of these networks, as defined by the topology and edge weights, determines how efficiently the networks perform their function. We present some general models regarding the emergence and extraction of hierarchical nestedness in 2-D and 3-D biological transport networks. We show how a simple, local adaptive rule can create dense sets of hierarchically nested loops in a dynamically evolving network. We demonstrate how nestedness can be measured in a complex graph, and show that it is a distinct phenotypic trait, akin to a fingerprint, that characterizes vascular systems.

Daphne Klotsa, University of North Carolina, Chapel Hill
Spheres form strings and a swimmer from a spring
Rigid spherical particles in oscillating fluid flows form interesting patterns as a result of fluid mediated interactions. Here, through both experiments and simulations, we show that two spheres under horizontal vibration align themselves at right angles to the oscillation and sit with a gap between them, which scales in a non-classical way with the boundary layer thickness. A large number of spherical particles form strings perpendicular to the direction of oscillation. Investigating the details of the interactions we find that the driving force is the nonlinear hydrodynamic effect of steady streaming. We then design a simple swimmer (two-spheres-and-a-spring) that utilizes steady streaming in order to propel itself and discuss the nature of the transition at the onset of swimming as the Reynolds number gradually increases.

Herbie Levine, Rice University
Phenotypic transitions en route to metastatic cancer
In order to spread from the primary tumor to distant sites and resist immune suppression, cancer cells can undergo a coordinated change in their phenotypic properties referred to as the “epithelial-to-mesenchymal” transition.  We have studied the nonlinear genetic circuits that are responsible for this cellular decision-making progress and propose that the transition actually goes through a series of intermediate states. At the same time, we have formulated motility models which allow for the correlation of state of this network and the cell’s biophysical capabilities. Hopefully, these efforts will help us better understand the transition to metastatic disease and possible treatments thereof.

Xiaoming Mao, University of Michigan
Mechanical instabilities at finite temperature
Mechanical instability is an important concept in many fields of research ranging from civil engineering to physics and biology. Central to the phenomena of mechanical instability is the emergence of “floppy modes”, which are modes of deformation that cost little energy. These floppy modes not only governs the yield of granular matter but also have been used to design machines and robots because they help directing actuation to the right parts. Zero temperature mechanical instability and floppy modes have been extensively studied, revealing many interesting phenomena, including multiple universality classes. However, how thermal fluctuations change these mechanical instabilities remain largely unexplored. In this talk, I will discuss our recent theoretical studies on finite temperature mechanical instability. In particular, we show that in both ordered and disordered lattice models, a large number of floppy modes can occur at the same time, corresponding to the breaking of one unique ground state to exponentially many, and the competition between these floppy modes leads to an interesting first order rigidity transition. Our theory shares intriguing similarities with Brazovskii’s theory on finite wavelength instability which has been applied to understand pattern formation problems.

Adilson Motter, Northwestern University
Optimization of Network Dynamics: Attributes and Artifacts
The recent interest in complex networks has been largely driven by the prospect that network optimization will help us understand the workings of evolution in natural systems and the principles of efficient design in engineered systems. In this talk, I will reflect on unanticipated properties observed in three classes of network optimization problems. First, I will discuss implications of optimization for the metabolic activity of living cells and its role in giving rise to the recently discovered phenomenon of synthetic rescues. I will then comment on the problem of controlling network dynamics and show that theoretical results on optimizing the number of driver nodes often only offer a conservative lower bound to the number actually needed in practice. Finally, I will discuss the sensitive dependence of network dynamics on network structure that emerges in the optimization of network topology for dynamical processes governed by eigenvalue spectra, such as synchronization and consensus processes. It follows that optimization is a double-edged sword for which desired and adverse effects can be exacerbated in complex network systems due to the high dimensionality of their phase spaces.

Katie Newhall, University of North Carolina, Chapel Hill
The Causes of Metastability and Their Effects on Transition Times
Many experimental systems can spend extended periods of time relative to their natural time scale in localized regions of phase space, transiting infrequently between them. This display of metastability can arise in stochastically driven systems due to the presence of large energy barriers, or in deterministic systems due to the presence of narrow passages in phase space. To investigate metastability in these different cases, we take the Langevin equation and determine the effects of small damping, small noise, and dimensionality on the dynamics and mean transition time.

Lou Pecora, U.S. Naval Research Laboratory
Finding and Forming Synchronized Clusters in Complex Networks of Oscillators Using Symmetries
Many networks are observed to produce patterns of synchronized clusters, but it has been difficult to predict these clusters in general or understand the conditions for their formation.  We show the intimate connection between network symmetry and cluster synchronization.  We apply computational group theory to reveal the clusters and determine their stability. In complex networks the symmetries can number in the millions, billions, and more.  The connection between symmetry and cluster synchronization is experimentally explored using an electro-optic network.  In networks with Laplacian coupling clusters are possible which do not directly result from symmetries; however, it is possible to construct all possible synchronized clusters starting from the symmetry clusters. We show how to do this using the computational group theory as an aid and how to derive the variational equations for all the clusters.

Shane Ross, Virginia Tech
Escape from potential wells in multi-dimensional systems: experiments and partial control
Predicting the escape from a potential energy well is a universal exercise, governing myriad engineering and natural systems, e.g., buckling phenomena, ship capsize, and human balance.  Criteria and routes of escape have previously been determined for 1 degree of freedom (DOF) mechanical systems with time-varying forcing, with reasonable agreement with experiments. When there are 2 or more DOF, the situation becomes more complicated, and the theory of tube dynamics provides the criteria for which phase space states will escape. We report a verification of tube dynamics for a 2 DOF experiment of a ball rolling on a surface. Furthermore, we consider the partial control problem where the goal is to avoid escape in the presence of unknown bounded disturbances which are greater in magnitude than the controls, by adopting a safe set sculpting algorithm which explicitly incorporates tube dynamics.

Mark Shattuck, City College of New York
Statistics of frictional families of particles: experiment and simulation
Using a previously developed experimental method to reduce friction in mechanically stable packings of disks and simulation of hard disks, we find that frictional packings form tree-like structures of geometrical families that lie on reduced dimensional manifolds in configuration space. Each branch of the tree begins at a point in configuration space with an isostatic number of contacts and spreads out to sequentially higher dimensional manifolds as the number of contacts are reduced. We find that gravitational deposition of disks produces an initially under-coordinated packing stabilized by friction on a high-dimensional manifold. In experiments, using short vibration bursts to reduce friction, we compact the system through many stable configurations with increasing contact number and decreasing dimensionality until the system reaches an isostatic frictionless state. We find that this progression can be understood as the system moving through the null-space of the rigidity matrix defined by the interparticle contact network in the direction of the gravitational force. We can predict the statistics of these states using a theory developed for mechanically stable frictional packings in terms of the difference between the total number of contacts required for isostatic packings of frictionless disks and the number of contacts in frictional packings, m. The saddle order m represents the number of unconstrained degrees of freedom that a static packing would possess if friction were removed. Using a novel numerical method that allows us to enumerate disk packings for each m , we show that the probability to obtain a packing with saddle order m at a given static friction coefficient μ , P m (μ) , can be expressed as a power series in μ . Using this form for P m (μ) , we quantitatively describe the dependence of the average contact number on the friction coefficient for static disk packings obtained from direct simulations of the Cundall-Strack model for all μ and N. When we apply this model to the data obtained from the deposition and compaction experiments, we find that it can recapitulate the average number of contacts for different numbers of vibration bursts.

Mary Silber, Northwestern University
Pattern Formation in the Drylands: Self Organization in Semi-Arid Ecosystems
Much of our understanding of spontaneous pattern formation in spatially extended systems was developed in the “wetlands” of fluid mechanics. That setting allowed well-controlled table-top laboratory experiments; it came with fundamental equations governing the system; it benefitted from a back-and-forth between theory and experiment. These investigations identified robust mechanisms for spontaneous pattern formation, and inspired the development of equivariant bifurcation theory. Recently, these pattern formation perspectives have been applied to modeling the vegetation in dryland ecosystems, where satellite images have revealed strikingly regular spatial patterns on large scales. Ecologists have proposed that characteristics of vegetation pattern formation in these water-limited ecosystems may serve as an early warning sign of impending desertification. We use the framework of equivariant bifurcation theory to investigate the mathematical robustness of this approach to probing an ecosystem’s robustness. Additionally, we identify new applied pattern formation research directions in this far-from-pristine setting, where there are no fundamental equations and no controlled laboratory experiments.

Contributed speakers

Dane Taylor, University of North Carolina
Analyzing data with dynamics: Phase transitions in random walks, community detection and network centrality
The study of dynamics on networks is a general framework that can be used to explore patterns in empirical networks and arbitrary multivariate data. In particular, the analysis of random walks has important applications ranging from Google’s algorithm for ranking webpages to community detection algorithms that seek to identify multiscale, and often hierarchical, representations of a network. Identifying such low-dimensional representations has numerous applications including dimension reduction for network-coupled dynamical systems and the clustering of data sets. By studying random walks on networks, we explore the fundamental limitations on when communities are detectable. We find phase transitions in detectability to coincide with phase transitions in dynamics behavior, which we analyze using random matrix theory for matrices that encode networks. We focus our study on multilayer networks: a family of networks in which edges appear in multiple layers that encode multiple types of connections. Remarkably, we find that if the network layers are strongly coupled, then the phase transitions vanish with increasing number of layers. This disappearance marks an enhancement of community detectability due to an increase in the influence of community structure on random walks.

William Ditto, North Carolina State University
Evidence for strange non-chaotic stars
The unprecedented light curves of the Kepler space telescope document how the brightness of some stars pulsates at primary and secondary frequencies whose ratios are near the golden mean, the most irrational number. A nonlinear dynamical system driven by an irrational ratio of frequencies generically exhibits a strange but non-chaotic attractor. For Kepler’s “golden” stars, we present evidence of the first observation of strange non-chaotic dynamics in nature outside the laboratory. This discovery could aid the classification and detailed modeling of variable stars.

James Yorke, University of Maryland
Quantitative Quasiperiodicity
Chaos and quasiperiodicity are the two complicated nonlinear dynamical behaviors seen in typical systems. The Birkhoff Ergodic Theorem is about the average of a function f(x_n) along a trajectory x_n of a dynamical system that is either chaotic and quasiperiodic. It says that as the number N of iterates used goes to infinity, the average _{n = 0,..,N} converges to the space average of f(x) as N → ∞. But convergence is so Extremely slow that it is practically useless as a computational tool. For quasiperiodic systems, we have turned this into a power tool with extremely fast convergence – just by changing the weights in the averages so that the early and late terms, i.e., f(x_n) for n near 0 and near N are given very small weights in the average. Two of the examples we investigate are the standard map and the restricted three body problem. This work is highly computational and pictorial so I recommend viewing http://arxiv.org/abs/1508.00062. Beyond that paper, we have found this method is an effective tool for distinguishing between chaos and quasiperiodicity where this can be difficult in situations where both coexist. We find that our “weighted” Birkhoff average allows us to compute rotation numbers and changes of variable (using Fourier series) generally with 30-digit precision.

Sarah Rajtmajer, Pennsylvania State University
A coordination-game model for information cascades in social networks
As social media applications become increasingly ubiquitous, the nature of content sharing has begun to change. Some user behaviors move quickly and easily through peer networks in information cascades. For instance, during the summer of 2014, more than 1.2 million videos of the Ice Bucket Challenge were shared on Facebook in less than three months. What are the dynamics of these processes? Is there a “tipping point” at which widespread adoption of a behavior can be expected? We present a model for the spread of rapidly-adopted, short-lived behaviors in social networks. Our approach is more closely aligned with models designed to appraise the emergence of consensus than with classic information cascade models. Specifically, we describe the everyday practices of user-nodes through the framework of a coordination game, in which each player’s payoff is summed over each nearest-neighbor pairwise coordination game in the social network graph. We model the short-term disruption of an attractive new trend as an additional available strategy with dynamic payoff structure, and quantify regimes for the success of a trend as it depends on rate of adoption and retention of interest over time. We find there can be long-term effects of these perturbations on habitual user behavior, even after the trend has disappeared entirely from the network. This is joint work with Christopher Griffin (US Naval Academy) and Andrew Belmonte (Penn State).

Christopher Marcotte, Georgia Institute of Technology
A dynamical mechanism for sustained atrial fibrillation: insights from unstable solutions in a simple model of cardiac dynamic
Fibrillation is characterized by the uncoordinated contraction of cardiac muscle, which reduces the effectiveness of pumping blood, due to the breakup of electrochemical excitation waves into self-sustaining spiral chaos. We present a deterministic description of sustained spiral chaos informed by the instabilities of spiral waves and their interactions with boundaries or other spiral waves in a simplified model of atrial fibrillation. In particular, we describe the dependence of the spiral frequency and drift on the proximity of the spiral core to the boundaries and the size of the domain. Analysis of sequences of single-spiral solutions yields a simple map which describes the drift of the spiral core under evolution, and whose roots then correspond to structurally stable configurations of spiral waves. Finally, we investigate the shape of the adjoint eigenfunctions of time-dependent single-spiral solutions and relate these to the structure of chaotic multi-spiral configurations.

Dapeng Bi, Rockefeller University
Rigidity and glassiness in dense biological tissues
Cells must move through tissues in many important biological processes, including embryonic development, cancer metastasis, and wound healing. Often these tissues are dense and a cell’s motion is strongly constrained by its neighbors, leading to glassy dynamics. Although there is a density-driven glass transition in self-propelled particle (SPP) models for active matter, these cannot explain liquid-to-solid transitions in confluent tissues, where there are no gaps between cells and the packing fraction remains fixed and equal to unity. I will demonstrate the existence of a new type of rigidity transition that occurs in confluent tissue monolayers at constant density. The onset of rigidity is governed by a model parameter that encodes single-cell properties such as cell-cell adhesion and cortical tension. I will also introduce a new model that simultaneously captures polarized cell motility and multicellular interactions in a confluent tissue and identify a glassy transition line that originates at the critical point of the rigidity transition. This work suggests an experimentally accessible structural order parameter that specifies the entire transition surface separating fluid tissues and solid tissues.

Erin Rericha, Vanderbilt University
How do cells in a tumor mix
?
Reaction-diffusion based continuum equations are an attractive method to describe tumor changes in mass and volume in the absence or presence of drug therapy. Here we consider the growth and drug response of tumors that are heterogeneous – containing cells that respond differently to the applied therapy. We recast common continuum tumor models to account for tumors of two different cell types, one sensitive and one resistant to the treatment. In particular we consider whether the diffusion of these two cell types proceeds independently or is constrained by the other cell type. To test the proposed model variants, we generate an experimental tumor analogue composed of non-small cell lung cancer cell lines – the PC9-BR1 cell line, which continues to proliferate in the presence of the targeted therapy erlotinib (resistant) and the PC9-DS9 sub-line which exponentially decays in cell number in response to erlotinib therapy (sensitive). The cell types are labeled with different fluorescent markers, seeded into separate spatial regions then allowed to mix. With the experimentally validated model, we consider how early in the course of therapy the presence of a small number of resistant cells might be detected from magnetic resonance images of the tumor.

Yen Ting Lin, The University of Manchester
Effects of bursting noise in gene regulation networks
Including short-lived mRNA populations in models of gene regulation networks introduces both the transcriptional bursting (i.e., transcription occurs at random times) and translational bursting (i.e., random amounts of proteins are translated by each mRNA). In this talk, I will present a coarse-graining method to construct mesoscopic models for such type of dynamical systems, which fully accounts for the bursting noise. We systematically compare different levels of modeling, ranging from individual-molecule-based models including mRNA populations, over protein-only individual-based models to mesoscopic models such as diffusion-type models and our proposed model. We show that the proposed mesoscopic model outperforms conventional diffusion-type models. In a one-dimensional autoregulated network, we present closed-form analytic solutions for both the stationary distribution of protein expression as well as first-passage times of the dynamical system. We present numerical solutions for higher-dimensional gene regulation networks, in which case we also carry out analysis in the weak-noise limit. The implications of the study are multi-faceted. Bursting noise is a ubiquitous feature of many biological systems. From a modeling perspective, our proposed method provides an alternative way to analyze dynamical systems with bursting noise. Biologically, the study presents quantitative evidences, the first to our knowledge, showing that bursting noise is the predominant form of intrinsic noise in gene regulation networks. Finally, from a mathematical point of view, the proposed model belongs to a class of stochastic processes named piecewise deterministic Markov processes, and our analysis on first-passage times and weak-noise limit of the process may inspire more rigorous analytic investigations in the future. References: The talk is based on the following preprints:  arXiv:1508.02945 [q-bio.MN];  arXiv:1508.00608 [q-bio.MN]

James Hanna, Virginia Tech
A conserved quantity in thin body dynamic
We use an example from textile processing to illustrate the utility of a conserved quantity associated with metric symmetry in a thin body. This quantity, when combined with the usual linear and angular momentum currents, allows us to construct a four-parameter family of curves representing the equilibria of a rotating, flowing string. To achieve this, we introduce a non-material action of mixed Lagrangian-Eulerian type, applicable to fixed windows of axially-moving systems. We will point out intriguing similarities with Bernoulli’s equation, discuss the effects of axial flow on rotating conservative systems, and make connections with 19th- and 20th-century results on the dynamics of cables.

Cédric Barroo, Université Libre de Bruxelles
Experimental study of nonlinear dynamics on a single nanoparticle
Non-equilibrium reactions are observed in a variety of reactive systems, including biological cells, micelles and on the surface of nanoparticles of catalysts. Probing the reactions and their dynamics at the scale of a single nanoparticle remains challenging, due to the scarcity of high-resolution techniques. Field emission microscopy (FEM) is a powerful technique for studying the dynamics of catalytic reaction taking place at the surface of a nanosized metal tip which acts as a catalyst. The dynamics is probed in real-time and during ongoing reactive processes. The microscope is usually run as an open nanoreactor. Accordingly, the chemical system is kept far from its thermodynamic equilibrium, a necessary condition to observe nonlinear spatiotemporal behaviors. This study reports on the observation and analysis of various nonlinear behaviors during the catalytic hydrogenation of NO2 on a single nanoparticle of Pt. By exploiting the nanoscale resolution capabilities of the FEM, the reaction can be probed down to 10 nm2. By changing a control parameter, it is possible to observe different behaviors (monostability, aperiodic oscillations, self-sustained periodic oscillations and more complex oscillations) and to determine the type of bifurcation leading to the emergence of the oscillations. For the specific case of periodic oscillations, Fourier transform analyses and temporal autocorrelation functions are used to characterize the dynamics and to quantify the robustness of the kinetic oscillations. At such small scale, molecular fluctuations are expected to endanger the regularity of the oscillations. Here we show that the correlation time and the variance of the period of oscillations are connected by a universal constraint, as predicted theoretically for systems subject to a phenomenon called phase diffusion. We also show how the dynamical attractor and the phase space dynamics can be reconstructed from experimental time series, and prove that the periodic oscillations can be linked to a limit cycle. Experiments at high temporal resolutions show the presence of propagating processes at the surface of the catalyst. This occurrence proves that the consecutive ignition of the active facets is due to a coupling via surface diffusion. The propagation of chemical waves on a single facet of the nanocrystal is also observed. These waves take the form of target patterns, which are observed for the first time at the nanoscale. The velocity of the observed waves is of the order of a few μm/s, which is in good agreement with previous studies of catalytic reactions at the mesoscale. Field emission microscopy is a powerful technique allowing for a better understanding of catalytic systems at the molecular level. The experiments prove the robustness of dissipative, nonlinear behaviors down to the nanoscale, without a significant loss of correlation due to fluctuations inherent to small chemical systems. These results shed a new light on the conditions under which collective order can emerge at the nanoscale.

Thomas Witelski, Duke University
Experimental study of regular and chaotic transients in a non-smooth system
This project focuses on transient behavior in the dynamics of a non-smooth mechanical system. We consider a harmonically forced rigid-arm pendulum impacting an inclined barrier. Prior to settling down onto a long-term forced response, such as a periodic steady-state or chaos, this system exhibits a variety of transient behavior involving hysteresis, attraction to isolated remote solutions, and chaotic transients. Some of this behavior is closely linked to grazing bifurcations – a subtle feature common to non-smooth systems. Thorough experimental and simulation studies were carried out with respect to dependence on initial conditions. This helped to identify the presence of super-persistent chaotic transients lasting much longer than the settling time for the first time in an experimental mechanical system.

Amir Bozorg Magham, University of Maryland
Causality Analysis: Identifying the Leading Element in a Coupled Dynamical System
Coupled dynamical systems with time-varying coupling are abundant in nature. While the detailed mechanism of interactions and the governing equations of these systems are typically unknown, there is often interest in determining the dominant sub-system that has the strongest coupling effect. Detecting dominant sub-systems enables us to have a deeper understanding of the directional causal interactions and ultimately to control the dynamical behavior of the system. Convergent Cross Mapping (CCM) is a new approach to causality analysis in deterministic dynamical systems. According to this methodology, if an element of a dynamical system has a causal influence on another element of the same system, then the reconstructed phase space of the response variable has signatures of the cause such that one can use that information to estimate the corresponding phase space of the driver. This method has been recently employed in ecological systems, exhibiting weak and constant coupling coefficients between variables, to determine a causal relationship and the dominant component. Our study is motivated by the need to identify the dominant constituent in asymptotic synchronized systems that are speculated to have time-varying interconnections with switching between the dominant elements in different periods. For example, a challenging application is to identify the dominant variables of the global climate system from geophysical records of greenhouse gases concentrations and temperature proxy. For this purpose, we consider coupled systems that (i) have variable coupling parameters in sequential periods, and (ii) the larger coupling coefficient is not fixed for a specific sub-system, and switching between the dominant sub-systems is possible in consecutive periods. We investigate whether CCM can identify the leading sub-system in such a coupled system under different conditions. We perform numerical experiments with different [linear] coupling schemes applied to the Lorenz-63 system with asymptotic synchronization. Three cases are considered: periodic-constant, normally distributed, and mixed normally/non-normally distributed coupling coefficients. We also investigate the CCM results in the presence of temporal uncertainties and additive noise. Our results indicate that CCM can identify the dominant sub-system in the coupled asymptotic synchronized Lorenz-63 systems, except when the average coupling coefficients are approximately equal, i.e., when the leading sub-system is not well-defined. These numerical experiments lay the groundwork for further causality analysis that can practically assist in analyzing coupled dynamical systems with time-varying coupling parameters and asymptotic synchronization.

Richard Lueptow, Northwestern University
Cutting and Shuffling: A Dynamical Systems Paradigm for Mixing
While chaotic advection has long been studied, mixing by cutting and shuffling is not well explored or understood. Unlike the stretching and folding characteristic of chaotic advection, cutting and shuffling maps do not stretch the material, possess no positive Lyapunov exponents, and exhibit no chaotic behavior in the usual sense – yet they can mix quite efficiently under certain conditions. In fact, cutting and shuffling, which is mathematically formulated as Piecewise Isometries (PWIs), can lead to complex dynamics and mixing in the absence of chaos. A PWI transformation divides an object into a finite number of pieces and rearranges them into the object’s original shape. The simplest example is cutting and shuffling a line segment. Certain rearrangement permutations mix well in a short time while others do not. Small, random variations in the location of cuts can lead to better mixing when the ratio of subsegment lengths is rational, but makes little difference when the subsegment lengths are closer to irrational. Similar heuristics should govern cutting and shuffling a two-dimensional square, but this remains to be fully explored. In three dimensions, a physical model of cutting and shuffling is a spherical tumbler that is half-filled with a granular material undergoing a bi-axial rotation protocol – a rotation about one axis followed by a rotation about another axis for each iteration. This form of cutting and shuffling exhibits rich dynamics including non-mixing regions along curves of elliptic points within KAM tubes and mixing regions along curves of hyperbolic points. Experiments using flowing granular material in a half-filled spherical tumbler confirm elliptic regions in the physical system. The system can also be studied using PWIs on a hemispherical shell. Computationally recording the cut locations from the PWI on the hemispherical shell yields complicated patterns after many iterations of the protocol that depend on the rotation angles. The trajectory of the initial cut in the limit of infinite iterations forms the “exceptional set” for the biaxial hemispherical cut and shuffle protocol, akin to a stroboscopic map of the system. The complement of the exceptional set comprises non-mixing regions, whose measure is related directly to the degree of mixing. However, the exceptional set alone yields no information about the rate of mixing or the rate at which the structure of the dynamics is achieved, so additional insights are necessary. Nevertheless, the merging of the mathematics of PWIs, dynamical systems approaches, and physical applications is leading toward a unique paradigm for understanding and predicting mixing in physical systems based on cutting and shuffling.

Eve Armstrong, University of California, San Diego
From the complex nonlinear behavior of a single neuron to the robust pattern of a network
We are developing a biophysical model of neural circuitry in the sensory-motor junction (HVC) of the songbird vocalization network. In this effort, we have created methods addressing the problem of sparse measurements that pervades analysis of functional circuits in neuroscience and complex systems in general. A neuron is a nonlinear system that, in isolation, exhibits highly variable and often chaotic behavior. When placed within a functional circuit, however, there often emerges patterned behavior with high coherence over thousands of participating cells, and which demonstrates robustness to noise and mild trauma. The connectivity involved in this transformation is largely unknown, mainly because most of the variables governing the system dynamics are unmeasurable. We are adapting an approach to data assimilation that has demonstrated success in chaotic systems such as meteorology, and has been applied rarely to biological neural networks. These methods are designed to maximally utilize the information in sparse data sets. Data assimilation is likely to make a transformative impact in neuroscience, where sparse measurements present a great obstacle, and where the community remains largely uninformed that methods exist to address that problem. Our close collaborations with experimental neurobiologists offers us an unusual platform for demonstrating and disseminating our results. We are working with the laboratories of Daniel Margoliash at the University of Chicago and Michael Long at New York University Medical School, to first characterize each cell type in terms of distinct biophysical cellular properties, and secondly to design and assist in the analysis of experiments to ascertain the functional connectivity of the HVC network. Our approach involves a systematic probe of the question: What is the role of the complex nonlinear properties of individual cells in producing the robust patterns of a network?

Alex Arenas, Universitat Rovira i Virgili
Spontaneous synchronization driven by energy transport in multiplex networks
We presentment and stylized model consisting of a two-layer multiplex network. Energy flow is modeled as a continuum of biased random walkers on one layer (energy transport layer ), while synchronization is modeled by Kuramoto phase oscillators on the other layer (synchronization layer). The dynamics of the two process are intertwined as follows. The transition probability of the random walk on the energy transport layer depends on the the degree k_j on the synchronization layer of the destination node j, i.e. the probability P_ji of jumping from node i to its neighbor j in the energy layer is P_ji \prop k _j^\alpha . Conversely, the frequency \omega_i of oscillator i on the synchronization layer is proportional to the fraction of random walkers at node i in the energy layer f_i.

Katherine Copenhagen, University of California, Merced
Self-organized sorting limits behavioral variability in swarms
Swarming is a phenomenon that spans many length and time scales in nature and is manifested by collective, coherent and global motion arising from simple local interactions between individuals. However, the nature of the local interactions may vary depending on differences between individual swarm agents. Here, we investigate the effects of variability in behavior among the agents in finite swarms with both alignment and cohesive interactions. At the simplest level, we consider the presence of a fraction of non-aligners among the population that ignore or are unable to process directional cues from their neighbors. We show that there is a critical non-aligner fraction above which the group can no longer swarm, the maximum non-aligner carrying capacity, which has a simple relation to the length and interaction strength scales that characterize the swarm. We show that swarms can increase their non-aligner carrying capacity by manipulating these parameters, allowing for adaptation to increased non-aligner loads “on the fly” and suggesting optimal parameters for “non-aligner tolerant” artificial swarms. When the cohesive interactions are weakened, however, swarms can dynamically reorganize and sort out excess non-aligners to maintain an average fraction of non-aligners close to a critical value for that swarm, suggesting a simple, robust and efficient mechanism that allows heterogeneously mixed populations to naturally regulate their composition and remain in a collective swarming state.

Kevin Mitchell, University of California, Merced
Topological dynamics in three-dimensional fluid mixing
Symbolic dynamics, and the associated topological entropy, are well developed tools for analysing two-dimensional dynamics, such as arise in the chaotic mixing of 2D fluids. For example, topological entropy has been useful in quantifying the mixing of fluids stirred by periodically braiding rods. However, at present no analogous symbolic techniques exist for extracting topological dynamics from chaotic flows in three-dimensions. We address this challenge here, using the topology of intersecting codimension-one stable and unstable manifolds. This leads to a symbolic dynamics of material surfaces in the fluid. This symbolic dynamics can be understood as resulting from the stirring by loops that undergo a kind of 3D braiding. The resulting theory provides a rigorous lower bound on the growth rates of both two-dimensional material surfaces and one-dimensional material curves. We illustrate our theory with a mathematical model of a chaotic ring vortex.

Eldad Afik, Weizmann Institute of Science
A Lagrangian approach to elastic turbulence: Pair dispersion in a dissipative chaotic flow reveals the role of the memory of initial velocity
Pair dispersion is the basis for understanding transport phenomena in flows. Its study at the small scales is of wide interest mainly for two reasons: (i) while a large body of mixing and transport in Biology and Chemistry is due to turbulent mixing, much of the dynamics takes place at the scales dominated by viscosity, typically smaller than a millimetre; (ii) microfluidic devices are playing an important role in research as well as industrial technologies, often including complex fluids and flows, whose dynamics still lack a universal description. Recent reports on particle dispersion in dissipative chaotic flows follow the paradigm of exponential separation in the long run. We have conducted a microfluidic experiment generating `elastic turbulence’ [1,2], a flow characterised in the literature as smooth in space and random in time. We have implemented a novel 3d particle tracking method [3] to quantitatively validate a theoretical prediction in this respect. In sharp contrast to previous works, we found no signature for the exponential separation [4]; we expect this behaviour to be the common one for tracer particles, given their finite size. We provide conclusive experimental evidence that the short time dynamics, which follows a power- law quadratic in time, is dominant over a significant time. This scaling, also known as the ballistic pair separation, is universal yet it has been overlooked so far in this context. Moreover, we realized that the exponential pair separation prediction [5] relies on restrictive assumptions which are unlikely to be realizable for tracer particles in bounded natural and experimental scenarios known to us. Our finding provides a quantitative prediction for particle dispersion in viscous chaotic flows based on the initial velocity [4], and questions the applicability of the currently leading paradigm.
[1] A. Groisman and V. Steinberg, Nature 405:53-54 (2000).
[2] A. Groisman and V. Steinberg, Nature 410:905-908 (2001).
[3] E. Afik, ArXiv e-prints 1310.1371 (2013). submitted.
[4] E. Afik and V. Steinberg, ArXiv e-prints 1502.02818 (2015). submitted.
[5] G. K. Batchelor, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 213 (1114):349-366 (1952).

Jonathan Kollmer, Friedrich-Alexander-Universität Erlangen-Nürnberg
Vertically migrating shear zones in horizontally driven granular matter
In dense, sheared granular matter the strain is often localised in shear bands. In most cases, these shear bands extent in the vertical direction and are stationary. We introduce a setup that creates horizontal shear bands. This type of shear bands does migrate upwards trough the sample. Using X-Ray radiography we show that this effect is caused by the dilatancy, the reduce in volume fraction occurring in dense sheared granular media. We further show that these migrating shear bands are responsible for the periodic time evolution of the surface height.

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