Several geometric and probabilistic methods for studying chaotic phase space transport have been developed and fruitfully applied to diverse areas from orbital mechanics to fluid mechanics and beyond. Increasingly, systems of interest are determined not by analytically defined model systems, but by data from experiments or large-scale simulations. This emphasis on real-world systems sharpens our focus on those features of phase space transport in finite-time systems which seem robust, leading to the consideration of not only invariant manifolds and invariant manifold-like objects, but also their connection with concepts such as symbolic dynamics, chaos, coherent sets, and optimal control. We will highlight some recent applications to areas such as spacecraft trajectories, microfluidic mixing, ship capsize prediction, and biological invasions.

The terrestrial water balance is forced by highly intermittent and unpredictable pulses of rainfall. This in turn impacts several related hydrological and ecological processes, such as plant photosynthesis, soil biogeochemistry and has feedbacks on the local climate. We treat the rainfall forcing at the daily time scale as a of marked (Poisson) point processes, which is then used the main driver of the stochastic soil water balance equation. We analyze the main nonlinearities in the soil water losses and discuss the probabilistic dynamics of soil water content as a function of soil-plant and vegetation characteristics. Crossing and mean-first-passage-time properties of the stochastic soil moisture process define the statistics of plant water stress, which in turn control plant dynamics, as shown in application to tree-grass coexistence in the Kalahari transect.
In the second part of this overview, we briefly illustrate: i) the propagation of soil moisture fluctuations through the nonlinear soil carbon and nitrogen cycles, ii) the possible emergence of persistence and preferential states in rainfall occurrence due to soil moisture feedback, and iii) the impact of inter-annual rainfall variability in connection to recent theory of ‘superstatistics’.

REFERENCES:
Rodriguez-Iturbe I. and A. Porporato, Ecohydrology of water controlled ecosystems: plants and soil moisture dynamics. Cambridge University Press, Cambridge, UK. 2004.
Laio F., Porporato A., Ridolfi L., and Rodriguez-Iturbe I. (2001) Plants in water controlled ecosystems: Active role in hydrological processes and response to water stress. II. Probabilistic soil moisture dynamics. Advances in Water Research, 24, 707-723.
Porporato A., Laio F., Ridolfi L., and Rodriguez-Iturbe I. (2001) Plants in water controlled ecosystems: Active role in hydrological processes and response to water stress. III. Vegetation water stress. Advances in Water Research, 24, 725-744.
Porporato A., D’Odorico P., Phase transitions driven by state-dependent Poisson noise, Phys. Rev. Lett. 92(11), 110601, 2004.
D’Odorico P., Porporato A., Preferential states in soil moisture and climate dynamics, Proc. Nat. Acad. Sci. USA, 101(24), 8848-8851, 2004. Manzoni S., Porporato A., D’Odorico P. and I. Rodriguez-Iturbe. Soil nutrient cycles as a nonlinear dynamical system. Nonlin. Proc. in Geophys. 11, 589-598, 2004.
Porporato A., G. Vico, and P. Fay, Interannual hydroclimatic variability and Ecosystem Superstatistics. Geophys. Res. Lett., 33, L5402, 2006.
Daly, E., and A. Porporato, Inter-time jump statistics of state-dependent Poisson processes, Phys. Rev. E, 75, 011119, 2007.

In nature, insects, fish, birds and other animals flock. A simple two-dimensional model due to Vicsek et al treats them as self-propelled particles that move with constant speed and, at each time step, tend to align their velocities to an average of those of their neighbors except for an alignment noise (conformist rule). The distribution function of these active particles satisfies a kinetic equation. Flocking appears as a bifurcation from an uniform distribution of particles whose order parameter is the average of the directions of their velocities (polarization). This bifurcation is quite unusual: it is described by a system of partial differential equations that are hyperbolic on the short time scale and parabolic on a longer scale. Uniform solutions provide the usual diagram of a pitchfork bifurcation but disturbances about them obey the Klein-Gordon equation in the hyperbolic time scale. Then there are persistent oscillations with many incommensurate frequencies about the bifurcating solution, they produce a shift in the critical noise and resonate with a periodic forcing of the alignment rule. These predictions are confirmed by direct numerical simulations of the Vicsek model. In addition, if the active particles may choose with probability p at each time step to follow the conformist Vicsek rule or to align their velocity contrary or almost contrary to the average one, the bifurcations are of either period doubling or Hopf type and we find stable time dependent solutions. Numerical simulations demonstrate striking effects of alignment noise on the polarization order parameter: maximum polarization length is achieved at an optimal nonzero noise level. When contrarian compulsions are more likely than conformist ones, non-uniform polarized phases appear as the noise surpasses threshold.

The challenge of predicting velocity and stress fields in any flowing granular material has proven to be a difficult one, from both computational and theoretical perspectives. Indeed, researchers are still in search of the ``Navier-Stokes''-equivalent for flowing granular materials. Granular flows can be adequately predicted using grain-by-grain discrete element methods (DEM), but these approaches become computationally unrealistic for large bodies of material and long times. A robust continuum model, once identified, would have the practical benefit that it could be implemented at a meso-scale saving many orders of magnitude in computation time compared to DEM.

Here, we begin by synthesizing a 3D elasto-viscoplastic law for steady granular flow, merging an existing "frictional fluid" relation with a nonlinear granular elasticity relation to close the system. The flow rate vanishes within a frictional (Drucker-Prager) yield surface and the elastic response is based on a mean-field model generalizing Hertz's contact law. The resulting form is general, able to produce flow and stress predictions in any well-posed boundary value problem. We implement it using ABAQUS/Explicit finite-element package and run test simulations in multiple geometries. The solutions are shown to compare favorably against a number of experiments and DEM simulations.

While this relation appears to function well for rapid flows, experimental results can often differ from the predictions in regions of slower flows. We have been able to attribute many of these phenomena to nonlocal effects stemming from the finite-ness of the grain size. To correct this, we consider the addition of a simple nonlocal term to the rheology in a fashion similar to recent nonlocal flow models in the emulsions community. The results of this extended model are compared against many DEM steady-flow simulations in three different 2D geometries. Quantitative agreement is found for all geometries and over various geometrical/loading parameters. By natural extension, the nonlocal model is then converted to three dimensions with minimal changes, and is implemented numerically as a User-Element in the ABAQUS package. We show that a single calibration of the 3D model quantitatively predicts hundreds of experimental flows in different geometries, including, for the first time, the wide-shear zones observed in split-bottom cells, a geometry made infamous for resisting a theoretical treatment for almost a decade.

The problem of reforestation is studied by solving a vegetation model in drylands. The "shikim" water harvesting method is seen as a parametric periodic forcing of a pattern forming system, where the resulting stripes and spots patterns are 1:1 and 2:1 resonant solutions. A modified Swift-Hohenberg equation helps us understand the dynamics of collapse and expansion of patterned states. I conclude by addressing preventive measures that make the vegetation system more resilient to climatic changes, and help avoid catastrophic regime shifts.

The stiffness of cells is commonly assumed to depend on the stiffness of their surrounding: bone cells are much stiffer than neurons, and each exists in surrounding tissue that matches the cell stiffness. In this talk, I will discuss new measurements of cell stiffness, and show that that cell stiffness is strongly correlated to cell volume. This affects both the mechanics and the gene expression in the cell, and even impacts on the differentiation of stem cells.

Understanding the self-organization principles and collective dynamics of non-equilibrium matter remains a major challenge despite considerable progress over the last decade. In this talk, I will introduce a hydrodynamic analog system that allows us to investigate simultaneously the wave-mediated self-propulsion and interactions of effective spin degrees of freedom in inertial and rotating frames. Millimetric liquid droplets can walk across the surface of a vibrating fluid bath, self-propelled through a resonant interaction with their own guiding wave fields. A walking droplet, or `walker, may be trapped by a submerged circular well at the bottom of the fluid bath, leading to a clockwise or counter-clockwise angular motion centered at the well. When a collection of such wells is arranged in a 1D or 2D lattice geometry, a thin fluid layer between wells enables wave-mediated interactions between neighboring walkers. Through experiments and mathematical modeling, we demonstrate the spontaneous emergence of coherent droplet rotation dynamics for different types of lattices. For sufficiently strong pair-coupling, wave interactions between neighboring droplets may induce local spin flips leading to ferromagnetic or antiferromagnetic order. Transitions between these two forms of order can be controlled by tuning the lattice parameters or by imposing a Coriolis force mimicking an external magnetic field. More generally, our results reveal a number of surprising parallels between the collective spin dynamics of wave-driven droplets and known phases of classical condensed matter systems. This suggests that our hydrodynamic analog system can be used to explore universal aspects of active matter and wave-mediated particle interactions, including spin-wave propagation and topologically protected dynamics far from equilibrium.

As a voltage is applied on a layer of an electroactive polymer, the polymer can reduce in thickness and expand in area, giving an actuation strain over 100%. This talk will discuss the large deformation, instabilities, and energy conversion of electroactive polymers. We will particularly focus on new phenomena of electroactive polymers recently observed at Duke Soft Active Materials Laboratory. Interestingly, these phenomena are closely related to daily-life issues such as skin wrinkling and creasing, physical topics such as the Taylor-Cone instability, and engineering applications such as high-energy-density capacitors and anti-biofouling.

Cardiac cells, like toilets, are excitable: Giving a sufficiently strong push to the handle of a quiescent toilet elicits a dramatic response (flush) followed by a gradual return to the resting state. Likewise, supplying a sufficiently strong electrical stimulus to a quiescent cardiac cell elicits a prolonged elevation of the membrane potential (an action potential).

Suppose that one end of an excitable fiber of cardiac cells is paced periodically. If the period is large, the generic response is a stable periodic wave train of the sort associated with normal, coordinated contraction of heart muscle tissue. Reducing the period (think "speeding up the heart rate") can cause the onset of an instability which can have devastating physiological consequences. Echebarria and Karma (Chaos, 2002) argued that if one attempts to stabilize the periodic wave train by using feedback control to perturb the pacing period, success can be achieved only within some small radius of the stimulus site. Those authors used a special case of the ETDAS control method that Dan Gauthier and Josh Socolar devised.

Here, I will offer an explanation as to WHY algorithms like ETDAS, applied locally, cannot achieve global results in this context. Then, I'll argue that it actually IS possible to stabilize the periodic wave train if the perturbations are chosen more carefully. While these findings may seem encouraging from an experimental or clinical standpoint, I will close by describing some recent work of Flavio Fenton which I believe is even more promising.

Buoyancy-driven flows, which are fluid flows driven by spatial variations of fluid density, play many key roles in the environment. Examples include winds in valleys and over glaciers, mineral transport in rock fissures, and ocean boundary mixing. To date, however, all investigations of buoyancy-driven flow have considered flow induced by a fixed boundary that influences fluid density (e.g. by heating or cooling). We have discovered that buoyancy-driven flows provide a previously unrecognized means of propulsion for freely-floating objects, and we demonstrate this new concept to surprising effect in a series of laboratory experiments.

The detailed characterization of the overall free energy landscape associated with the folding process of a protein is the ultimate goal in protein folding studies. Modern experimental techniques provide accurate thermodynamic and kinetic measurements on restricted regions of a protein landscape. Although simplified protein models can access larger regions of the landscape, they are oftentimes built on assumptions and approximations that affect the accuracy of the results. We present new methodologies that allows to combine the complementary strengths of theory and experiment for a more complete characterization of a protein folding landscape at multiple resolutions. Recent results and possible applications will be discussed.

Upon approaching the glass transition a liquid gets extremely sluggish without obvious structural changes. Despite decades of work, the physical origin of this glassy slowdown remains controversial. A common explanation relies on the increasing roughness of the underlying free-energy landscape, but the theoretical and experimental underpinnings of this scenario are still lacking. In this talk, I will survey recent advances that let us unambiguously identify and track the growing amorphous order, a manifestation of the rarefaction of metastable states in the rugged landscape. I will further explore the crucial role this order plays in driving the glassy slowdown.

Developing descriptive and predictive models of human behavior and decision making in complex sociotechnical systems is critical for system design and evaluation. However, developing such models is difficult due to individual variability, brittle assumptions, and the need to often integrate qualitative and quantitative data. This talk will discuss various human-systems modeling techniques developed in the Humans and Autonomy Laboratory.

Understanding and modeling how human lungs function is in large part based on the hydrodynamics of the mucus fluid layers that coat lung airways. In healthy subjects, the beating of cilia is the primary method of moving mucus. With the aim of establishing a quantitative benchmark of how cilia motion propels the surrounding fluid, we study the idealized situation of one rod spinning in a fluid obeying the Stokes approximation, the appropriate limit for a Newtonian fluid with typical dimensions and time scales of cilia dynamics. New approximate -- for cylindrical rods pinned to a flat plane boundary, and exact -- for ellipsoidal rods freely spinning around their center -- solutions for the fluid motion will be presented and compared with the experimental data collected with spinning magnetic nano-rods in water. In order to assess the influence of Brownian perturbations in this micro-scale experiment, data from an experimental set-up scaled by dynamical similarity to macroscopic (table-top) dimensions will also be presented and compared to the theoretical predictions.

Brains have an impressive ability to store information about the external world on time scales that range from seconds to years. The rules of information storage in neuronal circuits are the subject of ongoing debate. Two scenarios have been proposed by theorists: In the first scenario, specific patterns of activity representing external stimuli become fixed point attractors of the dynamics of the network. In the second, the network stores sequences of patterns of network activity so that when the first pattern is presented the network retrieves the whole sequence. In both scenarios, the correct dynamics are achieved thanks to appropriate changes in network connectivity. I will describe how methods from statistical physics can be used to investigate the storage capacity of such networks, and the statistical properties of network connectivity that optimizes information storage (distribution of synaptic weights, probabilities of specific network motifs, degree distributions, etc) in both scenarios. Finally, I will compare the theoretical results with available data on cortical connectivity.

We present experiments on pattern formation and front propagation in the Belousov-Zhabotinsky (BZ) chemical reaction in flowing systems with chaotic advection. The flow is a chain of alternating vortices that oscillate and/or drift in the lateral direction. Mixing between the vortices is chaotic in this flow with either (enhanced) diffusive or superdiffusive transport. Experiments with the excitable BZ reaction are used to study the motion of reaction fronts in this system. If the vortices oscillates laterally, reaction fronts typically mode-lock to the external forcing. If the vortices drift with constant velocity, fronts typically pin to the leading vortex, remaining motionless in a reference frame that drifts with the vortices. Experiments with the oscillatory BZ reaction are used to study synchronization of a network of oscillators by chaotic mixing. We find that the system is globally-synchronized only if the long-range transport is superdiffusive, characterized by Levy flight trajectories. Time-permitting, we will also present results of experiments on chemical fronts and patterns in a two-dimensional array of vortices.

The past several decades have seen an exponential growth of computer processing speed and memory capacity. The massive, complex simulations that run on supercomputers allow exploration of fields for which physical experiments are too impractical, hazardous, and/or costly. Accurate and efficient high-fidelity simulations are critical to many energy, defense, and health applications, e.g., global climate simulations, optimal design of wind systems for power generation, combustion simulations aimed at increasing fuel efficiency and reducing carbon emissions, simulations of heart fibrillation, and many others. Unfortunately, even with the aid of massively parallel next-generation computers, high-fidelity simulations are still too expensive for real-time and multi-query applications such as uncertainty quantification, design, optimization, and control. For this reason, interest in model order reduction continues to grow. In this talk I will summarize recent advances in nonlinear model reduction for high-Reynolds-number fluid flows, structural dynamics, and computational finance.

Fluidized granular systems with inelastic inter-particle collisions exhibit distinguishing behavior from it's elastic counterpart. Two species of particles in a binary granular system typically do not have the same mean kinetic energy, in contrast to the equipartition of energy required in equilibrium. It is found that not only the mechanical properties of these two types of particles, but also the heating mechanism plays an important role in affecting the extent of nonequipartition of kinetic energy, even in the bulk of the system. An experimental measurement of the velocity distribution of a 3D vibration fluidized granular medium by spatial resolved high speed video particle tracking is also reported. It is found that the distribution is wider than a Gaussian and broadens continuously with increasing volume fraction.

A consensus is emerging that discontinuous shear thickening (DST) in dense suspensions marks a transition from a flow state where particles remain well separated by lubrication layers, to one dominated by frictional contacts. We show here that reasonable assumptions about contact proliferation predict two distinct types of DST in the absence of inertia. The first occurs at densities above the jamming point of frictional particles; here the thickened state is completely jammed and (unless particles deform) cannot flow without inhomogeneity or fracture. The second regime shows strain-rate hysteresis and arises at somewhat lower densities where the thickened phase flows smoothly. DST is predicted to arise when finite-range repulsions defer contact formation until a characteristic stress level is exceeded.

For many biological systems, the timescale at which ecological interactions occur is much shorter than the timescale at which evolutionary changes occur. For rapidly evolving pathogens such as influenza, however, this is not the case; influenza researchers therefore need to understand both the ecological interactions between the host and the pathogen and the virus?s evolutionary changes in order to ultimately control the disease in humans. Recently, a study looking at the evolutionary patterns of influenza showed that, while the virus?s genetic evolution occurred gradually, its antigenic evolution occurred in a punctuated manner. (Genetic evolution refers to how the virus?s nucleotides change over time; antigenic evolution refers to how the virus changes over time with respect to how our immune system recognizes it.) Previous research from our group hypothesized that these differences in evolutionary patterns could be explained by the presence of /neutral networks/ in the virus?s genotype space: networks of sequences that differ genetically from one another but fold into the same protein conformation and thereby share antigenic properties. Here, I will present a simple epidemiological model that implicitly incorporates these neutral networks. I show that this model can reproduce (1) the seasonal and interannual outbreak patterns of influenza, (2) the quantitative patterns of influenza?s antigenic evolution, and (3) the patterns of the virus?s genetic evolution, including its characteristic phylogenetic tree. I end with how this model may be useful in understanding patterns of viral diversity in other host species (e.g., avian and equine hosts).

Qualitatively new behavior often emerges when large numbers of similar entities are interacting at high densities, no matter how simple the individual components. One prototypical example is granular matter such as fine dry sand, where individual grains are solids. In this talk I will discuss several striking phenomena, including the formation of jets and their break-up into droplets, where large ensembles of grains behave very much like a liquid, except that they do so without apparent surface tension.

Simple models of classical particles, interacting via soft- or hard-core repulsive contact interactions, have been used to model a wide variety of granular and soft-matter materials, such as dry granular particles, foams, emulsions, non-Brownian suspensions, and colloids. Such materials display a variety of complex behaviors when in a state of steady shear driven flow. These include (i) Jamming: where the system transitions from a flowing liquid to a rigid but disordered solid as the particle packing increases; (ii) Shear Banding: where the system becomes spatially inhomogeneous, separating into distinct bands flowing at different sh ear strain rates; (iii) Discontinuous Shear Thickening: where the shear stress jumps discontinuously as the shear strain rate is increased. In this talk we will consider a simple numerical model of athermal soft-core interacting frictionless disks in steady state shear flow. We will show that the mechanism by which energy is dissipated plays a key role in determining the rheology of the system. For a model with a tangential viscous collisional dissipation, but no elastic friction, we will show that as the particle packing increases there is a sharp first order phase transition from a region of Bagnoldian rheology (stress ~ strain-rate^2) to a region of Newtonian rheology (stress ~ strain-rate), that takes place below the jamming transition. In a phase diagram of varying strain-rate and packing fraction (or strain-rate and pressure) this first order rheological phase transition manifests itself as a coexistence region, consisting of coexisting bands of Bagnoldian and Newtonian rheology in mechanical equilibrium with each other. Crossing this coexistence region by increasing the strain-rate at fixed packing, we find that discontinuous shear thickening can result if the strain-rate is varied too rapidly for the system to relax to the true shear-banded steady state. We thus demonstrate that the rheology of simply interacting sheared disks can be considerably more complex than previously realized, and our model suggests a simple mechanism for both the phenomena of shear banding and discontinuous shear thickening in spatially homogeneous systems, without the need to introduce elastic friction.

The ability to effectively control a fluid would enable many exciting technological advances, such as the design of quieter, more efficient aircraft. Model-based feedback control is a particularly attractive approach, but the equations governing the fluid, although known, are typically too complex to apply standard tools for dynamical systems analysis or control synthesis. This talk addresses model reduction techniques, which are used to simplify existing models, to obtain low-order models tractable enough to be used for analysis and control, while retaining the essential physics. In particular, we will discuss two techniques: balanced truncation and Koopman modes. Balanced truncation is a well-known technique for model reduction of linear systems, with provable error bounds, but it is not computationally tractable for very large systems. We present an approximate version, called Balanced POD, that is computationally tractable, and produces much better models than traditional Proper Orthogonal Decomposition (POD), at least for the examples studied. Koopman modes are based on spectral analysis of the Koopman operator, an infinite-dimensional linear operator that describes the full nonlinear dynamics of a nonlinear system, and we show how the associated modes can elucidate coherent structures in examples including a jet in crossflow and the wake of a flat plate.

We apply super-resolution imaging and single-molecule tracking to gain insight into how proteins assemble to form organized structures in cells. We describe several new tools that were developed to study diverse systems, from viruses to chromatin. The HIV structural protein Gag assembles to form spherical particles of radius ~70 nm. During the assembly process, the number of Gag proteins increases over several orders of magnitude, from a few at nucleation to thousands at completion. We demonstrated an approach that permits quantitative morphological and molecular counting analysis of hundreds of HIV-Gag clusters at the cellular plasma membrane, thus elucidating how different fluorescent labels can change the assembly of virions. Higher-order chromatin structure determines the degree of local DNA condensation, which in turn influences gene accessibility and therefore the expression of particular genes. We present two complementary approaches to address this limitation: super-resolution imaging of directly labeled DNA, and singlemolecule high density tracking of proteins participating in DNA packaging. For STORM imaging of DNA, we stained cells with the DNA-specific dye Picogreen, and obtained a ~5-fold improvement in resolution, resolving the sub-diffraction organization of chromatin structures in living cells. For single molecule tracking (sptPALM), we used small chemical tags to target synthetic dyes to specific protein targets, and visualized their dynamics3. The combination of DNA and protein superresolution imaging and single particle tracking will allow us to study chromatin organization in living cells, and rearrangements in response to exogenous signals.