## Kyoung Jin Lee : A scary, yet interesting, scenario to the fibrillating heart

- Nonlinear and Complex Systems ( 210 Views )Alternans, a beat-to-beat temporal alternation in the sequence of heart beats, is a known precursor of the development of cardiac fibrillation, leading to sudden cardiac death. The equally important precursor of cardiac arrhythmias is the rotating spiral wave of electro-mechanical activity, or reentry, on the heart tissue. In this talk, I will show that these two seemingly different phenomena can have a remarkable relationship: In well controlled in-vitro tissue cultures, isotropic populations of rat ventricular myocytes sustaining a temporal rhythm of alternans can support period-2 oscillatory re-entries, and vice versa. These re-entries bear `line defects' across which the phase of local excitation slips rather abruptly by $2\pi$, when a full period-2 cycle of alternans completes in $4\pi$. In other words, the cells belonging to the line defects are period-1 oscillatory whereas all the others in the bulk medium are period-2 oscillatory. We also find that a slowly rotating line defect results in a quasi-periodic like oscillation in the bulk medium. Some key features of these phenomena can be well reproduced in computer simulations of a nonlinear reaction-diffusion model.

## Eric Vanden-Eijnden : Transition Pathways of Rare Events

- Nonlinear and Complex Systems ( 200 Views )Many processes in nature occur in the form of rare but important events. Well known examples of such events include conformation changes of biomolecules, chemical reactions, and nucleation events during phase transformation. Rare events do not happen very often on the internal clock of the system (which makes their simulation very challenging), but this clock can be very fast and this leaves plenty of room for the appearance of rare events in our daily life. I will review classical theories for the description of rare events, recent theoretical developments such as Transition Path Theory, concept such as reaction coordinate or free energy of a reaction and I will discuss how to compute the pathway and rate of rare events efficiently using the String Method. As illustrations, I will discuss the hydrophobic collapse of a polymeric chain, phase transitions in the Ising model, and a genetic toggle switch.

## Roberto Camassa : Spinning rods, microfluidics, and mucus propulsion by cilia in the lung

- Nonlinear and Complex Systems ( 177 Views )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.

## Doug Durian : Growth of dynamical heterogeneity in dense granular materials on approach to jamming

- Nonlinear and Complex Systems ( 175 Views )The physics of granular flow is of widespread practical and fundamental interest, and is also important in geology and astrophysics. One challenge to understanding and controlling behavior is that the mechanical response is nonlinear, with a forcing threshold below which the medium is static and above which it flows freely. Furthermore, just above threshold the response may be intermittent even though the forcing is steady. Two familiar examples are avalanches on a heap and clogging in a silo. Another example is dynamical heterogeneities for systems brought close to jamming, where intermediate-time motion is correlated in the form of intermitted string-like swirls. This will be reviewed in the context of glassy liquids and colloids, and more deeply illustrated with experiments on three different granular systems. This includes air-fluidized beads, where jamming is approached by density and airspeed; granular heap flow, where jamming is approached by depth from the free surface; and dense suspensions of NIPA beads, where jamming is approached by both density and shear rate. Emphasis will be given to measurement and analysis methods for quantifying heterogeneities, as well as the scaling of the size of heterogeneities with distance to jamming.

## Daniel Gauthier : Nonlinear stability analysis of a time-delay opto-electronic oscillator

- Nonlinear and Complex Systems ( 169 Views )I will describe some recent work on the dynamics of an optoelectronic time-delay oscillator that displays high-speed chaotic behavior with a flat, broad power spectrum. The chaotic state coexists with a linearly stable fixed point, which, when subjected to a finite-amplitude perturbation, loses stability initially via a periodic train of ultrafast pulses. A nonlinear stability analysis is required to understand the device dynamics. Through such an analysis, an approximate mapping is derived that does an excellent job of capturing the observed instability. The oscillator provides a simple device for fundamental studies of time-delay dynamical systems and can be used as a building block for ultrawide-band sensor networks. The results of this study recently appeared in print and can be found here: PRL The work is the part of Kristine Callan's PhD dissertation research and was in collaboration with Zheng Gao, Lucas Illing, and Eckehard Schoell.

## Zohar Nussinov : The detection of hidden structures in glasses and complex systems by multi-scale clustering

- Nonlinear and Complex Systems ( 169 Views )We will discuss the application of multi-scale graph theory based methods to the detection of general structures in networks, lattices, and amorphous physical systems. These methods enable the detection of the "natural" system structures on all scales. We specifically analyze lattices and spin systems with defects and various glass formers (including an analysis based on experimental data) to ascertain dominant structures at different temperatures. We will discuss general features of the phase diagram related to this analysis.

## Amilcare Porporato : Random Jumps in Eco-Hydrology: Non-Gaussian Forcing in the Nonlinear Soil-Plant-Atmosphere System

- Nonlinear and Complex Systems ( 169 Views )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., DOdorico P., Phase transitions driven by state-dependent Poisson noise, Phys. Rev. Lett. 92(11), 110601, 2004.

DOdorico 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., DOdorico 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.

## Chris Wiggins : Learning Networks from Biology, Learning Biology from Networks

- Nonlinear and Complex Systems ( 168 Views )Both the 'reverse engineering' of biological networks (for example, by integrating sequence data and expression data) and the analysis of their underlying design (by revealing the evolutionary mechanisms responsible for the resulting topologies) can be re-cast as problems in machine learning: learning an accurate prediction function from high-dimensional data. In the case of inferring biological networks, predicting up- or down- regulation of genes allows us to learn ab intio the transcription factor binding sites (or `motifs') and to generate a predictive model of transcriptional regulation. In the case of inferring evolutionary designs, quantitative, unambiguous model validation can be performed, clarifying which of several possible theoretical models of how biological networks evolve might best (or worst) describe real-world networks. In either case, by taking a machine learning approach, we statistically validate the models both on held-out data and via randomizations of the original dataset to assess statistical significance. By allowing the data to reveal which features are the most important (based on predictive power rather than overabundance relative to an assumed null model) we learn models which are both statically validated and biologically interpretable.

## Holger Stark : Active motion: Understanding the nonequilibrium

- Nonlinear and Complex Systems ( 167 Views )Active motion of microorganisms or artificial microswimmers in a fluid at low Reynolds number is an appealing subject which has attracted much attention recently. Since these swimmers move constantly in nonequilibrium, they give rise to novel phenomena which, in particular, occur when external fields are applied or when they move collectively.

The talk reviews three situations where active motion manifests itself. First, a swimmer under Poiseuille flow shows nonlinear dynamics reminiscent of the nonlinear pendulum. Bounding walls introduce "dissipation" [1] and an elliptical crosssection of the microchannel leads to chaotic motion. Secondly, I discuss the collective motion of model swimmers, so-called squirmers, in a quasi 2D geometry by means of multi-particle collision dynamics. This is a particle based method to solve the Navier-Stokes equations and helps to elucidate the role of hydrodynamics in collective phenomena. Indeed, we find gas-like and cluster phases as well as phase separation which is strongly influenced by hydrodynamic near-field interactions and the swimmer type. Thirdly, I discuss dynamic clustering of active or self-propelling colloids that interact by diffusiophoresis reminiscent of chemotaxis in bacterial systems.

[1] A. Zoettl and H. Stark, Phys. Rev. Lett. 108, 218104 (2012).

## Peter J. Mucha : Stochastic Dynamics in Near-Wall Velocimetry

- Nonlinear and Complex Systems ( 160 Views )The tracking of small, colloidal particles is a common technique for measuring fluid velocities, highly successful at the micro-scale and recently extended to measurements at nano-scales. The Brownian fluctuations of the colloidal tracers are typically isotropic in the bulk; but in the near-wall region, these fluctuations are strongly affected by the hydrodynamic interaction with the wall and by the no-flux condition imposed there. Such wall effects can, under appropriate conditions, bias particle image velocimetry (PIV) measurements based on colloidal tracers, potentially leading to significant overestimation of near-wall velocities. The quantification of the resulting bias is presented in terms of the size of the imaged region and the measurement interval between PIV images. The effect of the steady state particle distribution is additionally explored, and implications for near-wall velocimetry measurements are briefly discussed.

This talk represents collaborative work with Christel Hohenegger, Minami Yoda, Reza Sadr, and Haifeng Li.

## Cecilia Clementi : Multi-resolution protein modeling by combining theory and experiment

- Nonlinear and Complex Systems ( 160 Views )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.

## Michael W. Deem : Antigenic Distance, Glassy Dynamics, and Localization in the Immune System

- Nonlinear and Complex Systems ( 160 Views )The immune system normally protects the human host against death by infection. I will introduce a hierarchical spin glass model of the evolutionary dynamics that occurs in the antibody-mediated and T cell-mediated immune responses. The theory will be used to provide a mechanism for original antigenic sin, wherein an initial exposure to antigen degrades the response of the immune system upon subsequent exposure to related, but different, antigens. A new order parameter to characterize antigenic distance will be introduced from the theory. This order parameter predicts effectiveness of the influenza vaccine more reliably than do results from animal model studies currently used by world health authorities. This order parameter would seem to be a valuable new tool for making vaccine-related public health policy decisions. Next, I will note that while the immune system normally protects the human host against death by infection, the method used by the immune system to search sequence space is rather slow --- interestingly there exist biological mechanisms that can find antibodies with higher affinity and also find them more quickly. Thus, one would think that these more powerful evolutionary mechanisms would give an immune system that responds faster and more effectively against disease. So, why didn't we evolve that kind of adaptive response? I will show that the slow glassy dynamics of the immune system serves a functional role of inhibiting the autoimmune diseases that these more powerful searching mechanisms would induce. I will suggest that the controversy related to the correlation between chronic infection and autoimmune disease might be addressed by searching for the broad distribution of onset times for autoimmune disease predicted from the theory.

## Sreekanth Pannala : Multiscale/Multiphysics simulation strategy for gas-solids flow reactors

- Nonlinear and Complex Systems ( 153 Views )Gas-solids chemically reacting flows are omnipresent in many multiphase flow reactors in various industries like Chemical, Fossil and Nuclear. The challenging aspect of modeling these reacting flows are the wide range of both temporal and spatial scales encountered in these systems. The challenge is to accurately account and bridge (as seamlessly as possible) the length and time scales involved in the problem. First, the problem is introduced using biomass gasifier/pyrolyser and nuclear fuel coater with sample results as examples and provide an overview of the various models currently used at the different scales. In particular, the critical role of the granular dynamics in the overall performance of the reactors will be highlighted. The ongoing development of a multiphysics and multiscale mathematics framework for coupling various modeling methods over a range of scales will be presented. The development of a general wavelet-based multiscale methodology called compound wavelet matrix (CWM) for bridging spatial and temporal scales will be reported. Finally, the steps needed to generalize the current methodology for arbitrary heterogeneous chemically reacting flows or other applications involving multiscale/multiphysics coupling will be elucidated. The challenges and opportunities of employing these models for rapid deployment of clean energy solutions based on multiphase flow reactors to the market place will be discussed.

## Daniella E. Raveh : Nonlinear Dynamics of Aeroelastic Airfoil Systems in Buffeting Flows

- Nonlinear and Complex Systems ( 151 Views )Transonic flows over airfoils at certain combinations of Mach numbers and steady mean angle of attack exhibit buffet; a phenomenon of large shock-wave oscillations due to flow separation and vortex shedding at a characteristic flow frequency. Buffet may occur even when the airfoil does not move. The seminar will present two recent studies of numerical simulations of an airfoil that a) undergoes prescribed harmonic oscillations, and b) is suspended by a spring in transonic buffeting flows. Both studies focus on the nonlinear interaction between the two oscillatory systems, namely the buffeting flow and the oscillating airfoil. Flow simulations of prescribed airfoil motions (using a Navier-Stokes turbulent flow solver) reveal a lock-in phenomenon. Certain combinations of amplitude and frequency of a prescribed airfoil oscillatory motion caused the buffet flow oscillations to lock into the prescribed frequency. The combinations of prescribed frequencies and amplitudes that cause lock-in present an .Arnold tongue. structure. There is a broad analogy between this flow phenomenon and the flow field of the Von Karman vortex street found behind a cylinder with the cylinder undergoing a prescribed oscillation. Flow simulations of an airfoil that is suspended on a spring reveal three distinct response characteristics, depending on the relationship of the elastic system.s natural frequency to the buffet frequency, and on the system.s mass ratio (the structural to fluid mass ratio). Elastic systems with natural frequencies that are lower than the buffet frequency exhibit a single-frequency response, with a frequency that is shifted form the buffet frequency towards the elastic natural frequency as the mass ratio is decreased (and the magnitude of the elastic response increases). On the other hand, an elastic system with a natural frequency that is the same as the buffet frequency exhibits resonance. Finally, elastic systems with natural frequencies that are higher than the buffet frequency exhibit a response with two distinct frequencies, that of the buffet and that of the elastic natural frequency. As long as the pitch amplitudes are small, the response is mostly at the buffet frequency. As the pitch amplitudes increase there is more power in the elastic natural frequency, and less in the buffet frequency. As the pitch amplitudes further grow, the response is in the elastic natural frequency solely, and the buffet frequency vanishes. To the best of the authors. knowledge the nonlinear dynamics of elastic systems in buffeting flows has not been reported previously. The authors are interested to learn whether similar phenomena are known in other research communities.

## Clarence W. Rowley : Low-order models for control of fluids

- Nonlinear and Complex Systems ( 140 Views )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.

## Nicholas Ouellette : Multiscale Dynamics and Coherent Structures in Turbulent Flow

- Nonlinear and Complex Systems ( 129 Views )Despite an enormous range of applications and centuries of scientific study, understanding and predicting the flow of fluids remains a tremendous challenge, particularly when the flow is chaotic or turbulent. Turbulent flows tend to be characterized by violent fluctuations, enormous numbers of strongly coupled degrees of freedom, and significant variability in space and time. But despite all this complexity, turbulence is not random. Rather, it tends to self-organize into striking but transient patterns and features that arise from nonlinear interactions. Some of these "coherent structures," such as strong vortices, are readily apparent; others are more subtle. But how much can we learn or predict about the flow from studying coherent structures? The answer may lie in the energetics of the flow, since these same nonlinearities couple dynamics on different scales and, in turbulence, drive a net transfer of energy from the scales at which it is injected into the flow to the scales at which it is dissipated. To begin to make quantitative links between the nonlinear dynamics of the flow and the spontaneous generation of spatiotemporal order, I will discuss experimental results from a quasi-two-dimensional turbulent flow. Using a filtering technique, we extract the spatially localized scale-to-scale flux of energy, and show that it is linked to suitably defined coherent structures. I will also discuss the self-organization of the turbulent stress that drives this energy transfer.

## Matthieu Wyart : Discontinuous shear thickening without inertia in dense non-Brownian suspensions

- Nonlinear and Complex Systems ( 126 Views )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.

## Ken Kamrin : Toward a predictive continuum model for dense granular flows

- Nonlinear and Complex Systems ( 124 Views )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.

## Hongqiang Wang : Non-equipartition in a binary granular system and measurement of velocity distribution in a 3D vibrated granular system

- Nonlinear and Complex Systems ( 123 Views )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.

## Thomas Peacock : Sailing on Diffusion

- Nonlinear and Complex Systems ( 120 Views )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.

## Lawrence Virgin : Identifying chaos using spectral content

- Nonlinear and Complex Systems ( 116 Views )The characterization of chaos as a random-like response from a deterministic dynamical system with an extreme sensitivity to initial conditions is well-established, and has provided a stimulus to research in nonlinear dynamical systems in general. In a formal sense, the computation of the Lyapunov Exponent (LE) spectrum establishes a quantitative measure, with
at least one positive LE (and generally bounded motion) indicating a local exponential divergence of adjacent trajectories. Other measures are associated with certain geometric features of a chaotic attractor, e.g., the fractal dimension, and broadband frequency content. However, although the extraction of LE's can be accomplished with (necessarily noisy) experimental data, this is still a relatively data-intensive, sensitive (and frustrating) endeavor.

We present here an alternative, pragmatic approach to identifying chaos as a function of system parameters, based on frequency content and extending the concept of the spectrogram. This talk will describe this approach applied to systems of increasing complexity, ranging from direct numerical simulations of familiar archetypal systems like Lorenz and the pendulum to experimental data generated from mechanical systems. The accuracy and utility of the approach, including the effect of noise, is tested relative to the standard (LE) approach.

## Yair Mau : Reversing desertification: a pattern formation approach

- Nonlinear and Complex Systems ( 116 Views )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.