# Reception and Poster Session

Tuesday, June 4, 2013 - 4:00pm - 6:00pm

Lind 400

**Poster: An Agent-Based Model for Stripe Formation in Zebra Fish**

Bjorn Sandstede (Brown University)

Zebra fish develop stripe patterns composed of pigmented cells during their early development. Experimental work by Kondo and his collaborators has shed much light on how stripes form and how they are regenerated when pigmented cells are removed. Theoretical work has focused on reaction-diffusion models. Here, we use an agent-based model for cell birth and movement to gain further insight into the processes and scales involved in stripe formation and regeneration.**Poster: Adiabatic Stability Under Semi-strong Interactions: The Weakly Damped Regime**

Thomas Bellsky (Arizona State University)

We rigorously derive multi-pulse interaction laws for semi-strong interactions in a family of singularly-perturbed and weakly-damped reaction-diffusion systems in one space dimension. Most significantly, we show the existence of a manifold of quasi-steady N-pulse solutions and identify a normal-hyperbolicity condition which balances the asymptotic weakness of the linear damping against the algebraic evolution rate of the multi-pulses. Our main result is the adiabatic stability of the manifolds subject to this normal hyperbolicity condition. More specifically, the spectrum of the linearization about a fixed N-pulse configuration contains essential spectrum that is asymptotically close to the origin as well as semi-strong eigenvalues which move at leading order as the pulse positions evolve. We characterize the semi-strong eigenvalues in terms of the spectrum of an explicit NxN matrix, and rigorously bound the error between the N-pulse manifold and the evolution of the full system, in a polynomially weighted space, so long as the semi-strong spectrum remains strictly in the left-half complex plane, and the essential spectrum is not too close to the origin.**Poster: Localized Perturbation of Striped Patterns: Recovering Fredholm Properties via Kondratiev Spaces**

Gabriela Jaramillo (University of Minnesota, Twin Cities)

We study the effects of adding a local perturbation in a pattern forming system, taking as example the Ginzburg-Landau equation with a small localized inhomogeneity in two dimensions. Linearizing at a periodic pattern, $A_*=sqrt{1-k^2}e^{ik cdot x}$, one finds an unbounded linear operator that is not Fredholm in typical translation invariant or weighted spaces. We show that Kondratiev spaces provide an effective means to circumvent this difficulty. These spaces consist of functions with algebraical localization that increases with each derivative. We establish Fredholm properties in these spaces and use the result to construct deformed periodic patterns using the Implicit Function Theorem. This is joint work with Arnd Scheel.**Poster: Triggered Fronts in Complex Ginzburg Landau**

Ryan Goh (University of Minnesota, Twin Cities)

We study patterns that arise in the wake of an externally triggered, spatially propagating instability in the complex Ginzburg-Landau equation. We model the trigger by a spatial inhomogeneity moving with constant speed. In the comoving frame, the trivial state is unstable to the left of the trigger and stable to the right. At the trigger location, spatio-temporally periodic wavetrains nucleate. Our results show existence of coherent, “heteroclinic” profiles when the speed of the trigger is slightly below the speed of a free front in the unstable medium. Our re- sults also give expansions for the wavenumber of wavetrains selected by these coherent fronts. A numerical comparison yields very good agreement with observations, even for moderate trigger speeds. Technically, our results provide a heteroclinic bifurcation study involving an equilibrium with an algebraically double pair of complex eigenvalues. We use geometric desingularization and invariant foliations to describe the unfolding. Leading order terms are determined by a condition of oscillations in a projectivized flow, which can be found by intersecting absolute spectra with the imaginary axis.

This is joint work with A. Scheel.**Poster: Localised Crime Hotspots and Police Deterrence**

David Lloyd (University of Surrey)

In this poster, I will present some results on hotspots of an urban crime model. I will show that the crime model gives rise to localised 1D states that undergo a process known as homoclinic snaking and and can form multi-pulses. We path-follow these localised states into a singular limit region where some more detail analysis can be done. We then discuss what happens to two-dimensional localised hotspots and relate the results to the crime application. Finally, we will discuss the effect of a Police deterrent on the crime hotspots.**Poster: Diffusive Stability of Turing Patterns via Normal Forms**

Qiliang Wu (University of Minnesota, Twin Cities)

We investigate dynamics near Turing patterns in reaction-diffusion systems posed on the

real line. Linear analysis predicts diffusive decay of small perturbations. We construct a

“normal form” coordinate system near such Turing patterns which exhibits an approximate

discrete conservation law. The key ingredients to the normal form is a conjugation of the

reaction-diffusion system on the real line to a lattice dynamical system. At each lattice site,

we decompose perturbations into neutral phase shifts and normal decaying components. As

an application of our normal form construction, we prove nonlinear stability of Turing patterns

with respect to small localized perturbations, with sharp rates.**Poster: Dynamics of Two Interfaces in a Hybrid System with Jump-type Heterogeneity**

Takashi Teramoto (Asahikawa Medical College)

We consider the dynamics of an oscillatory pulse (standing breather,SB)

of front-back type, in which the motion of two interfaces that interact

through a continuous field is described by a mixed ODE-PDE system.

We carry out a center manifold reduction around the Hopf singularity

of a stationary pulse solution, which provides us insight into the underlying mechanism

for a sliding motion of SB in a jump-type spatial heterogeneous medium.

This is joint work with K. Nishi and Y. Nishiura.**Poster: Coherent Structures in a Model for Mussel-algae Interaction**

Vahagn Manukian (Miami University)

We consider a model for formation of mussel beds on soft sediments. The model consists of coupled nonlinear pdes that describe the interaction of mussel biomass on the sediment with algae in the water layer overlying the mussel bed. Both the diﬀusion and the advection matrices in the system are singular. We use Geometric Singular Perturbation Theory to capture nonlinear mechanisms of pattern and wave formation in this system.**Poster: Oscillons near Hopf Bifurcations of Planar Reaction Diffusion Equations**

Kelly McQuighan (Brown University)

Oscillons are planar, spatially localized, temporally oscillating, radially symmetric structures. They have been observed in various experimental contexts including fluid systems, granular systems, and chemical systems. Oscillons often arise near forced Hopf bifurcations of periodically forced diffusive systems. Such systems are modeled mathematically with the forced complex Ginsburg-Landau equation (FCGLE). We study the two dimensional FCGLE.

We present a proof of the existence of oscillon solutions to the FCGLE in one of the parameter regions. Our proof relies on a geometric blow-up analysis of the far field. Our choice of blow-up coordinates capture the desired far-field exponential behavior. Our analysis is complemented by a numerical continuation study of oscillon solutions to the FCLGE using AUTO. Numerical stability of these solutions was then computed using Matlab.**Poster: Predicting the Bifurcation Structure of Localized Snaking Patterns**

Elizabeth Makrides (Brown University)

We expand upon a general framework for studying the bifurcation diagrams of localized spatially oscillatory structures, and demonstrate how this approach can be used to predict the bifurcation diagrams of localized structures upon certain perturbations. Building on work by Beck et al., we provide an analytical explanation for the numerical results of Houghton and Knobloch on symmetry breaking in systems with one spatial dimension, and make predictions on the effects of symmetry breaking in more general settings, including planar systems. In particular, we predict analytically, and subsequently confirm numerically, the formation of isolas upon particular symmetry breaking perturbations.**Poster: Data Series Analysis Using SCC Decomposition Algorithm**

Ippei Obayashi (Kyoto University)

In this poster, we present a new method for time series data

analysis using a directed graph and a graph algorithm,

known as strongly connected components (SCC) decomposition.

An SCC is a recurrent object in a directed graph.

We construct a directed graph from time series data and

its SCCs are considered to be approximated recurrent sets

of the dynamical systems behind the data.

The formalization of the method, some mathematical discussions,

and numerical experiments are shown.**Poster: Steady-State Mode Interactions of Radially Symmetric Modes for the**

Lugiato-Lefever Equation on a Disk

Tomoyuki Miyaji (Kyoto University)

We consider the bifurcation of stationary solution for the nonlinear

Schrodinger equation with damping and spatially homogeneous forcing

terms.

It was proposed by Lugiato and Lefever as a model equation for pattern

formation in the ring cavity with the Kerr medium.

By numerical simulations, it is known that a localized solution for

two-dimensional Lugiato-Lefever equation undergoes the Hopf bifurcation,

and the resulting limit cycle undergoes the homoclinic bifurcation.

We consider the equation in a disk and study the steady-state mode

interactions between two radial modes.

We analyze numerically the vector field on the center manifold,

and show that the Hopf and homoclinic bifurcations can occur as a result

of mode interactions.**Poster: Spontaneous Motion of an Elliptic Camphor Particle Driven**

by Surface Tension Gradient

Hiroyuki Kitahata (Chiba University)

The coupling between deformation and motion in a self-propelled system has attracted broader interest. In the present study, we consider the motion of an elliptic camphor particle in order to investigate the effect of particle shape on spontaneous motion. It is concluded that the symmetric spatial distribution of camphor molecules at the water surface becomes unstable first in the direction of a short axis, which induces the camphor disk motion in the short-axis direction. Experimental results also support the theoretical analysis. From the present study, we suggest that when an elliptic particle supplies surface-active molecules to the water surface, the particle can exhibit translational motion only in the short-axis direction.

Reference: H. Kitahata, K. Iida, and N. Nagayama, Phys. Rev. E, 87, 010901 (2013).**Poster: What is the Origin of Rotational Motion in Dissipative Systems?**

Takashi Teramoto (Asahikawa Medical College)

The drift instability causes a head-tail asymmetry in spot shape, and the peanut one implies a deformation

from circular to peanut shape. Rotational motion of spots can be produced by combining these instabilities

in a class of three-component reaction-diffusion systems. Partial differential equations dynamics can be reduced

to a finite dimensional one by projecting it to slow modes for spatially localized spots near codimension 2 singularity.

Such a reduction clarifies the bifurcational origin of rotational motion of traveling spots in two dimensions in close

analogy to the normal form of 1:2 mode interactions. This is joint work with K. Suzuki and Y. Nishiura.**Poster: A Signed-Distance Approach to the Vector-Type BMO Algorithm**

Elliott Ginder (Hokkaido University)

We construct an algorithm, based on the method of Bence-Merriman-Osher

(BMO), for computing multiphase, volume-constrained, curvature-driven

motions. Our approach uses signed distance functions to alleviate

restrictions on the time and grid spacings used in its implementation,

and we show how our algorithm allows for additional contact energies.

These energies can be used to control phase contact angles and we will

show the numerical results of the algorithm. We will also discuss how

the signed distance approach allows one to obtain uniform estimates

for the minimizing movement, which is a feature that is not available

to the original BMO.**Poster: Spatial Localization in Non-Homogeneously Forced Systems**

Edgar Knobloch (University of California, Berkeley)

The effects of spatial inhomogeneities on homoclinic snaking of spatially localized structures in the quadratic-cubic and the cubic-quintic Swift-Hohenberg equations are studied using a combination of numerical and analytical techniques. Spatial inhomogeneities employed include periodic spatial forcing and forcing in the form of a Gaussian bump or dip. Spatial forcing selects the location of the localized structures and is responsible for several distinct transitions in the snaking scenario.