June 3 - 7, 2013
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.
Keywords of the presentation: continuous spectrum; generalized eigenvalue; Kuramoto model
The Kuramoto model is a system of ordinary differential equations for describing
synchronization phenomena defined as a coupled phase oscillators.
In this talk, an infinite dimensional Kuramoto model is considered, and
Kuramoto's conjecture on a bifurcation diagram of the system will be proved.
A linear operator obtained from
the infinite dimensional Kuramoto model has the continuous spectrum on the
imaginary axis, so that the usual spectrum does not determine the dynamics.
To handle such continuous spectra,
a new spectral theory of linear operators based on Gelfand triplets is developed.
In particular, a generalized eigenvalue (resonance) is defined.
It is proved that a generalized eigenvalue determines the stability and bifurcation of the system.
Keywords of the presentation: functionalized Cahn-Hilliard model, bi-layers, pearling & meandering
The functionalized Cahn-Hilliard (fCH) equation is a a model for interfacial energy in phase separated mixtures with an amphiphilic structure. On the one hand, it has recently attracted attention in the study of polymer-electrolyte membranes in which hydrophobic polymers are functionalized by the addition of anorganic side chains. On the other hand, it has captured the attention as a rich mathematical model that describes the intriguing formation, interactions and dynamics of localized structures of varying co-dimension: bi-layers, pores and micelles. Here, we focus on the most simple co-dimension 1 bi-layer structures in 3 space-dimensions: flat plates and spherical or cylindrical shells. The existence problem for stationary structures corresponds to constructing homoclinic solutions in a 4-dimensional (spatial) dynamical system that has the structure of a perturbed integrable Hamiltonian system. The stability of the plates and/or shells is established by a careful analysis of the 4th-order linearized operator associated to these homoclinic solutions. Two potential destabilization mechanisms well-known from observations – the meandering and the pearling instability – play a central role in the analysis. While the meandering instability is strongly linked to both the local geometry of the bi-layer and its interactions with other bi-layers, our results indicate that the pearling instability only depends on the parameters of the model and the nature of the underlying double-well potential.
Keywords of the presentation: localized solution, spot solution, curved surface
The movement of a localized pattern appearing after Turing instability is a much attractive topic. In this talk, we consider reaction-diffusion systems which possess localized solutions in two dimensional spaces and investigate the dynamics of the solutions on a two dimensional curved surface. In order to analyze them, we first assume the existence of a linearly stable localized solution in the whole space and consider the movement of the solution when the domain is deformed to a curved surface. By using the center manifold reduction, we can reduce the dynamics to ODE systems expressed by the gradient of curvatures.
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.
Keywords of the presentation: volume-constraints, free boundary problems, variational methods, droplet motion
We present on the use of minimizing movements for analyzing certain free boundary problems expressing the motion of liquid droplets and bubbles. The variational nature of our approach allows one to constrain the volume of each droplet, as well as to control the contact angles that each makes with obstacles. We will discuss the corresponding numerical methods and then show the results of these methods applied to both scalar and non-scalar motions.
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.
Keywords of the presentation: Persistent Homology, Persistent Diagrams, Protein Compressibility
Persistent homology and persistent diagrams have been developed as tools of topological data analysis. They provide a robust topological characterization of a given (discrete) geometrical data. In this talk, I will present our recent researches on applying persistent diagrams to protein structural analysis. Topological characterizations of protein compressibilities and H/D exchanges will be explained in detail. Time permitting, I will also introduce a notion of the three ladder persistence as representaions on a certain class of associative algebra, which will be useful for comparing two proteins in view of homological information.
In this talk, I will discuss some recent results concerning front propagation into unstable states in systems of reaction-diffusion equations. The system aspect will be key. The focus will be on some of the qualitative differences between wavespeed selection in systems of reaction-diffusion equations and their scalar counterparts. In particular, we will consider some examples of coupled Fisher-KPP equations.
Keywords of the presentation: microorganism, Stokes flow, boundary interaction, bifurcation
It is known that some microorganisms accumulate near a boundary, which has been recently found a hydrodynamic phenomenon.
To understand the diversity and complexity on the behavior and to give a comprehensive explanation, we have numerically investigated motion of an ellipsoidal squirmer in Stokes flow swimming near both a no-slip and free-slip infinite plane boundary. It is found that the fluid interaction due to the no-slip boundary is not only purely attractive or repulsive but rather more complicated with depending on the cell geometry and stroke pattern. Compared with the no-slip boundary, the swimmer cannot be stably confined near the boundary in presence of the free-slip boundary. As an example, swimming stability of a model sperm cell near a boundary will be also discussed.
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.
Keywords of the presentation: vortex merging, core growth model
We model various configurations of vortex merger problems for the Navier-Stokes equations using the core growth model for vorticity evolution coupled with the passive particle field and an appropriately chosen time-dependent rotating reference frame. Using the combined tools of analyzing the topology of the streamline patterns along with the careful tracking of passive fields, we highlight the key features of the stages of evolution of vortex merger, pinpointing deficiencies in the low-dimensional model with respect to similar experimental/numerical studies. The model, however, reveals a far richer and delicate sequence of topological bifurcations than has previously been discussed in the literature for this problem, and, at the same time, points the way towards a method of improving the model, which might provide further insight of the merging process, especially in how the vortex merger starts.
This is joint work with Eva Kanso, Paul K. Newton and David T. Uminsky.
Read More...
We present a family of dipole models for describing the far-field hydrodynamic interactions in populations of self-propelled bodies. Interestingly, but perhaps not surprisingly, the dipolar far-field effect is descriptive of swimming bodies (e.g., fish) at high Re numbers as well as self-propelled particles (e.g., bacteria) in confined geometries such as in Hele-Shaw cells. We argue that our models can be used in both contexts, that is, fish schooling and confined motile micro-particles. We then discuss some of the physical insights these models bring to the large-scale behavior of these systems.
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).
Keywords of the presentation: Marangoni effect, Stokes equation
A self-propelled systems have attracted broader interest. I will
introduce following two systems in which a droplet exhibits spontaneous motion induced by interfacial tension gradient. We consider mathematical model based on Stokes equation, and compare the theoretical results with experimental ones.
(i) We proposed a theoretical framework for the spontaneous motion of a droplet coupled with pattern formation inside it in a three-dimensional system. The nonlinearity of the chemical process inside the droplet induces inhomogeneous concentration profile, which leads to interfacial tension gradient at the droplet interface. The droplet is driven by the gradient through the surrounding flow due to the Marangoni effect. We also demonstrated experiments in which Belousov-Zhabotinsky (BZ) reaction droplet is moving coupled with chemical waves inside it.
(ii) We analyze spontaneous rotation of a droplet induced by the Marangoni flow in a two-dimensional system. The droplet with the small particle which supplies a surfactant at the interface is considered. We calculated flow field around the droplet using the Stokes equation and found that advective nonlinearity breaks symmetry for rotation. Theoretical calculation indicates that the droplet spontaneously rotates when the radius of the droplet is an appropriate size.
Reference:
H. Kitahata, N. Yoshinaga, K. H. Nagai, and Y. Sumino, Phys. Rev. E, 84,
015101 (2011).
H. Kitahata, N. Yoshinaga, K. H. Nagai, and Y. Sumino, Chem. Lett., 41,
1052 (2012).
K. H. Nagai, F. Takabatake, Y. Sumino, H. Kitahata, M. Ichikawa, and N. Yoshinaga, Phys. Rev. E, 87, 013009 (2013).
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.
Keywords of the presentation: > localized states, phase field crystal, conserved Swift-Hohenberg equation
The conserved Swift-Hohenberg equation with cubic nonlinearity provides the
simplest microscopic description of the thermodynamic transition from a
fluid state to a crystalline state. The resulting phase field crystal
model describes a variety of spatially localized structures, in addition
to different spatially extended periodic structures. The location of these
structures in the temperature versus mean order parameter plane is
determined using a combination of numerical continuation in one
dimension and direct numerical simulation in two and three dimensions.
Localized states are found in the region of thermodynamic coexistence
between the homogeneous and structured phases, and may lie outside of the
binodal for these states. The results are related to the phenomenon of
slanted snaking but take the form of standard homoclinic snaking when the
mean order parameter is plotted as a function of the chemical potential,
and are expected to carry over to related models with a conserved order
parameter.
This work is joint work with U Thiele, A J Archer, M J Robbins and H Gomez.
Keywords of the presentation: Persistence digrams, pattern formation
Persistence diagrams are extremely useful for describing complicated patterns in a simple but meaningful way. We will demonstrate this idea on the patterns appearing in the convection flows and granular media. This procedure allows us to transform samples from the experiment into a point cloud in the space of persistence diagrams. The space of persistence diagrams is a complete metric space. There is a whole family of matrices on this space. By choosing different metrics one can interrogate the pattern locally or globally which provides deeper insight into the dynamic of the process of pattern formation. The Morse-Conley database can be used to identify invariant sets of the dynamical system. We will concentrate on fixed points and periodic orbits.
Keywords of the presentation: Periodic orbits, Kuramoto-Sivashinsky, Rigorous numerics
In this talk, we introduce a rigorous computational method for periodic orbits of dissipative PDEs. The idea is to consider a space-time Fourier expansion of a periodic solution and to solve for its Fourier coefficients in a space of algebraically decaying sequences. The rigorous computation is based on the radii polynomials approach, which provide an efficient means of constructing a ball centered at a numerical approximation which contains a genuine periodic solution. We apply this method to show the existence of several periodic solutions in the Kuramoto-Sivashinsky equation. This is joint work with Marcio Gameiro (Sao Carlos, Brazil).
Keywords of the presentation: Magnetic Fluids, Hamiltonian Mechanics, Free-surface boundary problems
Ferro-solitons and patches of localised patterns have been observed experimentally on the surface of a magnetic fluid in the presence of a uniform vertical magnetic field by Richter and Barashenkov (PRL 2005). In this talk, I will present some recent experimental results of Richter and link these with some analysis and numerical results of the equations governing the free-surface. It turns out that the system possesses an energy formulation and for the linear magnetisation law case a Hamiltonian formulation exists. This Hamiltonian formulation is then used to do Normal form analysis for one-dimensional interfacial localised patterns. We then present numerical results for these patterns and trace-out a two-parameter bifurcation diagram depicting the existence of one-dimensional interfacial localised patterns. Finally, we present some preliminary results looking at a general nonlinear magnetisation law and two-dimensional interfacial localised patterns.
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.
Keywords of the presentation: localized states, supercritical Turing bifurcations, forced complex Ginzburg-Landau equation
Much previous work on localized states has focused on the vicinity of subcritical Turing bifurcations, which create spatially uniform equilibria and spatially periodic patterns that are both stable. Indeed supercritical Turing bifurcations create stable Turing patterns together with *unstable* equilibria, which are insufficient by themselves to form robust localized patterns. However, recently robust localized patterns are found after a supercritical Turing bifurcation in the 1:1 forced complex Ginzburg-Landau equation, in a parameter regime where the unstable equilibria connect to *stable* equilibria on an S-shaped bifurcation curve. Before the Turing bifurcation, there exist localized states that resemble bounded fronts between two stable equilibria. The bifurcation structure, spectral stability, and temporal dynamics of these localized states in both one and two dimensions are determined using numerical continuation and direct numerical simulation. In particular, the bifurcation structure of 1D steady localized patterns takes the form of defect-mediated snaking that differs sharply from standard homoclinic snaking. In 2D there exist both radially symmetric localized states including circular fronts and localized rings, and fully 2D localized states including localized hexagons bounded by circular and planar fronts.
This is joint work with E. Knobloch.
Keywords of the presentation: Convection, binary fluid, suspension, bifurcation, pattern formation
We study dynamics of localized structures in fluids. First, we present a detailed network connecting various solutions representing steady and traveling localized states in binary fluid. The structure of the network can be classified into four categories based on the local network connections. Second, we show an experimental work of bioconvection caused by Euglena suspension illuminated from below. In this convection, localized convection cells are observed as well as the binary fluid convection; But the mechanism has not been clarified. In our experiment, an annular container was used to achieve S^1 condition in the azimuthal direction, and the width of the container is a similar size of the single convection cell in three-dimensional domain. After introducing the basic mechanism of bioconvection, we show the result suggesting the bistability structure of this suspension. The topics of this talk are joint works with Dr. T. Watanabe(JAXA), E. Shoji(Hiroshima University) and Prof. Y. Nishiura(Tohoku 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.
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.
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.
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.
Keywords of the presentation: vortex, boundaries, multiply-connected, equilibria
Some brief examples of systems where the motion of vortices play an important role in the dynamics will first be discussed to motivate the remainder material to be presented. Following this, the equations of motion governing the motion of point vortices in multiply-connected domains (for inviscid, incompressible fluids) will be introduced. A method for numerically calculating equilibria in multiply connected domains, applied to a Kasper Wing configuration, will then be presented and various properties of the flow (for example, the lift on the airfoils) will be analyzed. Time permitting, the motion of finite patches of constant vorticity in multiply-connected domains will then be introduced and used to study the robustness of the Kasper wing equilibria.
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.
Keywords of the presentation: singular limit, interaction laws, fronts, pulses
We consider the asymptotic interaction laws of pulses and fronts in the so-called semi-strong regime of strongly differing diffusion lengths for reaction-diffusion systems. An asymptotic expansion and matching approach is applied in a model independent unifying framework. In contrast to the universal laws of motion arising in weak interaction, semi-strong interaction comes in different types that we refer to as first and second order. For illustration, we focus on a class of examples that includes the Gray-Scott and Schnakenberg models.
Read More...
Keywords of the presentation: pattern formation, localized structures, geometric blow-up
Many planar spatially extended systems exhibit localized standing structures such as pulses and oscillons. In particular, such structures can emerge at Turing and forced Hopf bifurcations. In this talk, I will give an overview of these mechanisms and show how geometric blow-up techniques can be used to analyze them: among the findings is the bifurcation of localized structures that have significantly larger amplitudes than expected from formal considerations. This is joint work with Kelly McQuighan, David Lloyd, and Scott McCalla.
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.
Keywords of the presentation: aortic aneurysms, wall shear stress, fluid dynamics
Thoracic endovascular aortic repair (TEVAR) has become widely accepted as an important option for treatment of thoracic aortic diseases. Many studies have proven the safety and efficacy of TEVAR with satisfactory short-term to mid-term outcomes. Nevertheless, even if the initial TEVAR treatment technically succeeds, some patients show recurrence and progression of disease many years after treatment. Based on long-term follow-up examinations, it has been inferred that such long-term morphological changes and hemodynamics interact with positive feedbacks. The wall shear stresses show different patterns depending on the original aorta morphologies. In this talk, the relationship between them will be discussed using patient-specific models of the thoracic aorta as constructed from CT scans.
Read More...
Keywords of the presentation: pulse waves, generalized FitzHugh-Nagumo system
Heterogeneity is one of the most important and ubiquitous types of external perturbations. I study a spontaneous pulse generating mechanism caused by the heterogeneity of jump type. Such a pulse generator has attracted much interest in relation to potential computational abilities of pulse waves in physiological signal processing. Exploring the global bifurcation structure of pulse generators as periodic solutions, I find firstly the conditions under which they emerge,
i.e., the onset of pulse generators, secondly a candidate for the organizing center producing a variety of pulse generators. This is joint work with M. Yadome, X. Yuan, and Y. Nishiura.
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.
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.
Keywords of the presentation: nonlocal swarm aggregation integrodifferential biology
I discuss a partial integrodifferential equation model for the the collective motion of biological groups. The model describes a population density field u(x,t) advected by a velocity field v = q * u + f. The convolution q * u represents pairwise social interactions between swarm members and f(x) models exogenous forces such as food or light. Because social interactions are difficult to measure in experiment, one challenge in aggregation modeling is to choose a social interaction kernel q(x) that produces qualitatively correct macroscopic behavior. For f = 0, we determine conditions on q for u to asymptotically spread, contract, or reach steady state. For nonzero f, we use a variational formulation to find exact solutions for swarm equilibria in one spatial dimension. Typically, these are localized solutions with jump discontinuities or delta-concentrations at the group’s edges. We apply some results to locust swarms.
Keywords of the presentation: network, synchronization, loop-searching system
Self-recovery of function is one of the remarkable properties of biological systems and its implementation in autonomous distributed systems is highly desirable. In this study focusing on designing logical connections by using dynamical systems, we propose an autonomous distributed system which is capable of searching for closed loops. A closed loop is defined as a phase synchronization of a group of oscillators belonging to the corresponding nodes.
We demonstrate that the system can find a possible loop when one of the connections in the existing loop is suddenly removed.
Keywords of the presentation: localized patterns, bifurcation, stability analysis, non-local PDEs
We present an overview of recent results on localized pattern formation in non-local PDEs that arise in swarming and self-assembly models. Much work has been done in one dimension but two dimensions and higher has been more challenging. We present a mathematical framwork which predicts the rich array of localized patterns which have been observed in two and three dimensions. In particular we compute the non-local, linear stability analysis for particles which bifurcate away from radially symmetric states such as rings and spheres. The linear theory accurately characterizes patterns in the ground states of the fully nonlinear problem. This aspect of the theory allows us to solve the inverse problem of designing specified potentials which assemble into targeted patterns. Time permitting I will talk about applications of the theory to specific self-assembly problems.
We present results for an advection-reaction-diffusion model describing malignant tumour (i.e. skin cancer) invasion. Numerical solutions indicate that both smooth and shock-fronted travelling wave solutions exist for this model.
We verify the existence of both type of these solutions using techniques from geometric singular perturbation theory and canard theory. Moreover, we provide numerical results on the stability of the waves and the actual observed wave speeds.
This is joint work with K. Harley, G. Pettet, R. Marangell and M. Wechselberger.
Keywords of the presentation: Nucleation, stochastic Cahn-Hilliard equation, stochastic Cahn-Morral system, pattern formation
Stochastic partial differential equation systems serve as basic models for several phase separation phenomena in multi-component metal alloys. In a process called nucleation, the additive noise in the system forces the formation of localized droplets formed by one or more components of the system. In this talk, I will discuss dynamical aspects of this behavior in the context of stochastic versions of the celebrated Cahn-Hilliard and Cahn-Morral models. In addition to a brief description of the theoretical background, numerical studies will be presented in the context of alloys consisting of three metallic components which give a statistical classification for the distribution of droplet types as the component structure of the alloy is varied. We relate these statistics to the equilibrium structure of the deterministic Cahn-Morral system and show that even highly unstable equilibria can be observed during the nucleation process, and in fact serve as organizing centers for the dynamics. In addition, we try to shed some light on the size of the generated droplets by considering binary systems perturbed by degenerate noise of certain wavelengths.
Keywords of the presentation: singular perturbations, localized patterns, hot spot, nonlocal eigenvalue problem
Two topics in the theory and applications of localized pattern formation are
discussed. 1) Hot-spot patterns of urban crime; 2) Localized patterns for the
Brusselator on the sphere.
We analyze localized patterns of criminal activity for some reaction-diffusion models of urban crime introduced by Short et al. Such patterns, characterized by the concentration of criminal activity in localized spatial regions, are referred to as hot-spot patterns and they occur in a parameter regime far from the Turing point associated with the bifurcation of spatially uniform solutions. Hot-spot patterns are constructed and their stability properties analyzed from a new class of nonlocal eigenvalue problems for a three-component RD system where the effect of police deterrence is included. The stability analysis of hot-spot solutions suggest optimal strategies for police enforcement. Secondly, we show that there is a parameter regime where hot-spot patterns can nucleate from a quiescent background when the inter hot-spot spacing exceeds a threshold. The bifurcation structure and dynamical behavior of hot-spot patterns in this regime is analyzed.
Next, we study the existence and stability of localized spot patterns for the Brusselator RD model on the surface of the sphere. A key difficulty with a weakly nonlinear analysis of patterns on the sphere is that the Laplacian eigenfunctions exhibit a high degree of mode degeneracy. In contrast, in the context of localized pattern, spot patterns for the Brusselator on the sphere are shown to exhibit three types of instabilities: self-replication, competition, and oscillatory instabilities of the spot amplitudes. An analysis of these instabilities is given and the results are compared with full numerical computations. The study of localized patterns on the sphere has some similarity to the well-studied problem of Eulerian point vortices on the sphere.
Keywords of the presentation: stability threshold, logrithmic expansions, nonlocal eigenvalue problem
We consider the problem of refined stability thresholds for
multiple spots in general reaction-diffusion systems (Gierer-Meinhardt,
Schnakenberg, Gray-Scott,...). The stability thresholds given by
Wei-Winter (2001, 2003) provide the leading logrithm order which
doesnot depend on the location. Here we obtain the next O(1) term in terms
of Green's function on the locations. We then combine with Floquet-Bloch
theory to study which lattice of spots in the whole space is the most
stable. (Joint work with D. Iron and M. Ward).
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.
Keywords of the presentation: Flapping Flight, Stability, Control
The stability of the forward flight of a periodically-flapping butterfly is numerically investigated. The flight is longitudinally unstable owing to the flapping motion and the flow structures, and the control of the pitching angle is essential for stable periodic flights. The control is observed to be done by the abdominal motion, and the butterflies adopt a different approach to stabilize the flight from the human-made control systems. The control mechanism of butterfly's flapping flights will be discussed by comparing to the other control systems.
This is joint work with Kei Senda, Makoto Iima and Norio Hirai.