Session K47: Spatio-Temporal Pattern Formation

A Detailed Thermodynamic Study of Rayleigh-Benard Cells

Systems that are out-of-equilibrium and exhibit complex patterns are difficult to characterize. These systems are thermodynamically open, thereby constantly exchanging heat and matter with the surrounding. In order to be able to systematically and rigorously characterize such a system we present an experimental study of an out-of-equilibrium thermodynamic system, the Rayleigh-Benard convection cells. When a thin layer of liquid is evenly heated from the bottom the liquid tends to self-organize into patterns of hexagonal cells or a series of rolls. The cells or rolls are an outcome of an upward flow of the hot layer of the liquid from the bottom, and a downward flow of the cool layer of the liquid from the top along with the competing effects of viscosity and thermal gradient. The goal of this presentation is to measure the flow of energy through the system and be able to quantify it, along with steady state measurements of the temperature fluctuation, entropy and internal work calculation by infrared imaging as a function of fluid viscosity and thickness over a wide range of heat input. The resulting thermal patterns will be interpreted in terms of a balance between internal entropy and work.   

Velocity waves in large-scale human crowds

I will discuss the propagation of velocity waves in large-scale human crowds. Firstly, I will present a set of measurements performed on a number of different pedestrian crowds. I will demonstrate and quantitatively characterise longitudinal and non dispersive velocity waves propagating upstream the average pedestrian flow and show that transverse velocity fluctuations are strongly over damped. Secondly, I will show how to infer a predictive hydrodynamic description of some human crowds from their velocity wave spectra.   

Multistable Dynamical Network of Diffusively Coupled Chemical Oscillators

Coupled-oscillators networks are an important class of dynamical systems found in both the natural and engineered worlds. The spatio-temporal patterns exhibited by such networks depend heavily on inter-nodal interactions and topology. Using the Belousov-Zhabotinksy chemical reaction as our oscillator and a microfluidic chip to control topology, we examine the dynamics of a 4-ring network of inhibitor-coupled wells. While the network is small, it possesses multiple stable attractor states each possessing distinct dynamics. As such, it is a model system for understanding how to design controllers for complex systems with multiple fixed points. In this talk, experimental results will be compared to numerical models. Our goal is to create minimal models that predict the possible attractor states of the system and their basins of attraction in order to guide control schemes for switching between attractor states.   

Symmetries as Building Blocks for Synchronization Clusters: Theory and Experiment

Networks of coupled oscillators are sometimes observed to produce patterns of synchronized clusters where all the oscillators in each cluster have identical trajectories, but not the same as oscillators in other clusters. We show the intimate connection between network symmetry and cluster synchronization using computational group theory to reveal the symmetry clusters (SC) and determine their stability. Other synchronization clusters such as equitable partitions clusters (EC) or Laplacian coupling clusters (LC) also exist. We show that EC's and LC's can be constructed by the merging of appropriate SC's. We show that these types of dynamical behaviors can exist experimentally in a set of fiber ring lasers from which we form complex networks. We also demonstrate that the stability theory for such clusters applies to the experiment.   

Pattern Guiding via “Structured Noise”

Far-from equilibrium systems, such as reaction diffusion systems, exhibits the unique feature of pattern formation [1]. Nonlinear variations of these systems further enrich observed patterns and underlying dynamics [2,3]. Here, we show that, in recently developed Nonlinear Laser Lithography (NLL) [4] and Swift-Hohenberg systems, it is possible to control the emerging patterns employing engineered defects in the form of "structured noise". In NLL, "structured noise" enables us to tile the material surface with all possible periodic structures of isotropic unit cells, namely Bravais lattices. The nanopatterning is demonstrated both in simulations and experiments. Using similar approach with the Swift-Hohenberg system, we numerically show that pattern selection and direction can be achieved with initial defects. As a characteristic feature of such systems that are responsive to noise, we demonstrate stochastic resonance behavior in both systems. The control capability arises due to thresholding behavior of pattern transitions based on nonlinear feedback mechanisms.

[1] Turing, A. M., Phil. Trans. R. Soc. B, 237,32-72 (1952).
[2] Ilday, S. et al., Nat. Comm. 8,(2017).
[3] Tokel, O. et al., Nat. Photonics 11,639-645 (2017).
[4] Oktem, B. et al., Nat. Photonics 7,897-901 (2013).   

Transport and Feedback in Models of Self-Organizing Vegetation Patterns in Dryland Ecosystems: Some Comparisons with Satellite Images

Bands of vegetation alternating periodically with bare soil have been observed in many dryland environments since their discovery in the Horn of Africa in the 1950s. Mathematical modeling efforts over the past two decades have sought to account for these bands via a self-organizing interaction between vegetation and water resources. Understanding the processes underlying vegetation patterns in arid and semi-arid regions is important to predict desertification risk under increasing anthropogenic pressure. Various modeling frameworks have been proposed that are capable of generating similar patterns through self-organizing mechanisms which stem from key assumptions regarding plant feedbacks on surface/subsurface water transport. We discuss a hierarchy of hydrology-vegetation models for the coupled dynamics of surface water, soil moisture, and vegetation biomass on a hillslope. We identify distinguishing features and trends for the periodic traveling wave solutions when there is an imposed idealized topography and make some comparisons to satellite images of large-scale banded vegetation patterns in drylands of Africa, Australia and North America.   

The Saturation Bifurcation in Coupled Oscillators

We examine examples of weakly nonlinear systems whose steady states undergo a bifurcation with increasing forcing, such that a forced subsystem abruptly ceases to absorb additional energy, instead diverting it into an initially quiescent, unforced subsystem. We derive and numerically verify analytical predictions for the existence and behavior of such saturated states for a class of oscillator pairs. We also examine related phenomena, including zero-frequency response to periodic forcing, Hopf bifurcations to quasiperiodicity, and bifurcations to periodic behavior with multiple frequencies.   

An interacting random walk model produces Turing-like filamentation and a new phase transition in time

We show a new percolation-like phase transition in time, viz., sudden disappearance of long-range spatial connectivity, in a system of interacting random walkers on a square lattice. Our model consists of walkers that are placed randomly or uniformly at sites of a square lattice at time t=0. The dynamics of the walk is simulated by a simple interaction model. At time t, let n be the number of walkers on a lattice site and m the sum of the number of walkers on the neighboring sites. If n≥m the walker hops to one of the neighboring sites with equal probability and if n<m it does a lazy walk, i.e., hops only with probability exp(m-n). We declare a site active if there is at least one walker on it and inactive otherwise. Two neighboring sites have an edge if both are active. We compute the wrapping probability of the clusters formed at each time step, averaging over many simulations. We observe a sharp phase transition in the wrapping probability at a time threshold. A numerical finite-size scaling analysis shows the universality class to be the same as that of standard site percolation. In addition, we observe Turing-like patterns in the clusters as we go through the phase transition, much like laser filamentation in a strong pulse propagating through a self-Kerr ionizing medium.   

Dynamic Phases, Stratification, Laning, and Pattern Formation for Driven Bidisperse Disk Systems in the Presence of Quenched Disorder

Using numerical simulations, we examine the dynamics of driven two-dimensional bidisperse disks flowing over quenched disorder. The system exhibits a series of distinct dynamical phases as a function of applied driving force and packing fraction such as a phase separated state and a smectic state with liquidlike or polycrystalline features depending on the global disk density. At low driving forces, the system exhibits a clogged phase with an isotropic density distribution, while at intermediate driving forces the disks separate into bands of high and low density, where the high density bands can have either liquidlike structure or polycrystalline structure. In addition to the phase separation in the overall density we find that in some cases there is a fractionation of the disk species, particularly for large size ratio differences. These species phase separated regimes are associated with a variety of patterns such as large disks separated by chains of smaller disks or other types of patterns that are affected by the disk size ratio.   

Spatiotemporal Dynamics of the Kuramoto-Sakaguchi Model with Time-dependent Connectivity

We study the dynamics of the Kuramoto-Sakaguchi synchronization model with interaction involving a time-dependent coupling range. Statistical quantities measuring coherence and phase discontinuity are used to distinguish between various collective states and we plot their phase diagrams in the planes defined by various parameters of the system. A sinusoidal variation of the coupling radius leads to a hysteretic response of the system depending on the strength of coupling. This indicates the existence of an intrinsic time-scale in the system competing with the time period of the coupling range, which is verified from the mean field Ott-Antonsen analysis as well as direct simulation. The findings may be applied to diverse situations, e.g., dynamics involving plastic neural connections or human beings gathered by some common time-dependent source of excitation; and hence can open new vistas of diverse applicability of the decades-old Kuramoto model while the latter may then widen its scope to explain natural and social phenomena arising out of collective dynamics.
Main Publication: Phys. Rev. E 94, 022213 (2016)   

A novel significant parameter in the dynamics of periodic waves in excitable media

Wave selection in excitable media is an important problem of nonlinear dynamics. Same aspects of this problem have been solved already in application to a propagation of a solitary wave. In this talk a periodic sequence of wave segments is analysed. By direct numerical simulations performed on the modified FitzHugh-Nagumo model and by application of a free-boundary approach it is shown that such sequence can be stabilized only in a restricted domain within the parameter space. An introduced parameter predetermines the propagation velocity and the shape of the wave segments and reaches a constant critical value at the whole boundary of the existence domain. Moreover, another constant value of this novel parameter corresponds to zero tension of a scroll wave filament in a 3D medium.
  

Orientation Patterns of non-spherical Particles in Turbulence

In experiments and numerical simulations we measured angles between the orientations of small spheroids in turbulence. Since turbulent strains tend to align nearby spheroids, one might think that their relative angles are quite small. We show that this intuition fails in general: the distribution of relative angles has heavy power-law tails, and the dynamics evolves to a fractal attractor despite the fact that the fluid velocity is spatially smooth at small scales. The fractal geometry depends on particle shape, and it determines the power-law exponents. This talk is based on joint work by L. Zhao, K. Gustavsson, R. Ni, S. Kramel, G. A. Voth, H. I. Anderson, and B. Mehlig (arXiv:1707.06037).