MSRI Dynamical Systems Abstracts

Abstract: I will present a model of fish behavior which I will be using to describe the annual migration of the Capelin around Iceland. I will describe the model which my research group is currently analyzing, how it evolved, and the relevant biology behind it. I will then talk about the associated system of ODEs and the solutions which we have found. I have movies which I made in MATLAB of many of the solutions, and I will be showing these. This presentation is very accessible.

Margaret A. Beck (U Surrey)

A Geometric Construction of Traveling Waves in a Bioremediation Model

Abstract: Bioremediation is a promising technique for cleaning contaminated groundwater and soil. We study a bioremediation model involving a substrate (contaminant to be removed), electron acceptor (added nutrient), and microorganisms in a 1-D soil column. Using geometric singular perturbation theory, we construct traveling waves (TW) corresponding to motion of a Biologically Active Zone, in which the microorganisms consume both substrate and acceptor. For certain values of the parameters, the traveling waves exist on a 3-dimensional slow manifold within the 5-dimensional phase space. In order to prove persistence of the slow manifold under perturbation, we control the nonlinearity via a change of coordinates and construct the wave in the transverse intersection of appropriate stable and unstable manifolds. We study how the TW depends on the half saturation constants and other parameters.

Nitsan Ben-Gal (Brown)

Bifurcation and stability properties of periodic solutions to two nonlinear spring-mass systems

Abstract: Ever since the Tacoma Narrows Bridge Disaster in 1940 there has been interest in modeling the nonlinear behavior arising from a periodically forced spring-mass system. In my talk I will discuss the nature of periodic solutions to two nonlinear spring-mass models. The first is the well-known Lazer-McKenna Bridge Oscillation Model, while the second uses a smooth nonlinear restoring force to better adhere to the physical reality of the problem. I will be contrasting the multiplicity, bifurcation, and stability of periodic solutions arising from these two models, as well as discussing chaotic behavior arising from unstable solutions. This is work with K.S. Moore of the University of Michigan and P.J. McKenna of the University of Connecticut.

Lorena Bociu (U Virginia)

Existence, uniqueness and blow-up of solutions to wave equations with supercritical boundary sources and boundary damping

Abstract: This poster summarizes the wellposedness and blow-up results obtained for the wave equation driven by boundary sources with critical and supercritical exponents. Taking advantage of the boundary damping, local and global existence of finite energy solutions are obtained, while in the absence of boundary "overdamping", finite time blow-up of weak solutions is exhibited. The results extend those obtained recently in the literature, where boundary sources and dampings of polynomial structures with subcritical exponents were considered. In addition, the local and global existence theory presented is sharp, since the blow-up phenomenon is exhibited in the complimentary (to global existence) region.

Lauren Childs (Cornell)

The dynamics of macrophage activation

Abstract: The invasion of antigen into the body during an infection initiates an immune response with both broad and specific effects that involves many cells. Macrophages, a particular type of immune cell, are activated not only during the immune response - by clearing antigens - but also in the aftermath of the immune response- helping tissue heal. During an immune response, macrophage activation involves competition for one substrate, L-arginine, by two enzymes, Nitric Oxide Synthase and Arginase. The propensity towards either the clearing-of-antigen activation state versus the wound-healing activation state depends not only on the amount of each enzyme present in the system but also on each enzyme’s ability to encounter sufficient substrate. The substrate-enzyme dynamics are further complicated by interactions between some intermediates and the competing enzymes. All macrophages have the facility to become activated to clear antigen or to heal wounds but some macrophages may also be able to modify their activation state later in life. The ability of a macrophage to convert between the two distinct activation states is an unresolved question about macrophage activation which may have important impact on the efficiency and effectiveness of an immune response.

Gerda de Vries (U Alberta)

Modelling the spatio-temporal dynamics of nuclear proteins

Abstract: The combination of fluorescence microscopy techniques and mathematical modelling facilitates the study of the spatio-temporal dynamics of nuclear proteins in living cells. In this talk, I will highlight ongoing modelling work on two related aspects, namely (1) the quantitative assessment of the mobility of the proteins, and (2) their spatial organization within the nucleus. The work is the result of a collaboration between applied mathematicians and an experimental cell biologist. (1) The mobility of proteins is characterized experimentally by fluorescence recovery data. In the past, diffusion models were fit to the data to extract an effective diffusion coefficient for the proteins of interest. I will show that simple reaction-diffusion models and compartmental models can provide a much better fit to the data, suggesting that such models are more appropriate to describe the behaviour of diffusive proteins undergoing binding events. A perturbation analysis of the models leads to an elegant explanation of two important limiting types of behaviour exhibited by fluorescence recovery data, namely reduced diffusive and biphasic behaviour. (2) The spatial organization of splicing factors (proteins which play an important role in RNA processing) within the nucleus can be described by a fourth-order aggregation-diffusion model, derived from a random walk analysis. A linear stability analysis of this model reveals the emergence of spatial patterns under conditions that are in accordance with experimental observations.

Ana Christina Freitas (U Porto)

Extremal properties of quadratic maps

Abstract: We consider the logistic family of maps on the Misiurewicz parameters for which there is an invariant probability measure that is absolutely continuous with respect to Lebesgue. For each of these chaotic dynamical systems, we study the extreme value distribution of certain stationary stochastic processes associated to the time evolution of the systems. Using the techniques developed by Benedicks and Carleson, we show that the limiting distribution of M(n), which is the maximum of the first n random variables of the process, is the same as that which would apply if the random variables of the process were independent. This result allows us to obtain that the asymptotic distribution of M(n) is of Type III (Weibull).

See the full PDF version of the abstract for additional information.

Anna Ghazaryan (U North Carolina)

Traveling waves in porous media combustion: uniqueness of waves for small thermal diffusivity

Abstract: We study traveling wave solutions arising in Sivashinsky's model of subsonic detonation which describes combustion processes in inert porous media. Subsonic (shockless) detonation waves tend to assume the form of reaction front propagating with a well defined speed. It is known that traveling waves exist for any value of thermal diffusivity. Moreover, it has been shown that, when the thermal diffusivity is neglected, the traveling wave is unique. The question of whether the wave is unique in the presence of thermal diffusivity has remained open. We analytically resolve the issue of uniqueness of the wave in the presence of non-zero diffusivity through applying geometric singular perturbation theory.
Co-authors: C.K.R.T. Jones (UNC-Chapel Hill), and P. Gordon (NJIT).

Jenny Harrison (UC Berkeley)

Chainlet theory and dynamics

Abstract: In this lecture we describe a class of objects called 'chainlets', a particularly well-behaved subspace of de Rham currents. Chainlets are a generalization of $k$-surfaces in $n$-space, but with algebraic information encoded both locally and globally. Equipped with a new, 'natural' norm, we discard many cumbersome methods used in geometric measure theory, instead building an elegant calculus from the ground up on domains that include, but are not limited to, discrete and continuous domains, soap films, fractals, vector fields and charged particle. Applications arise in a vast array of applications, from analysis to differential and algebraic topology, calculus of variations, PDEs, dynamical systems, and physics.

Kathleen Hoffman (U Maryland, Baltimore County)

Stability results for elastic rods with electrostatic self-repulsion

Abstract: Conjugate points, attributed to Jacobi, have been a part of the classical calculus of variations literature for over a century, however, the classical theory pertains only to the standard calculus of variations problems. In this talk, I will outline the classical methods of conjugate points in the standard setting, and generalize those results to calculus of variations problems with integral constraints. I will then present a general theory of conjugate points for variational problems satisfying generic assumptions. The motivation for this work is to determine the stability of an elastic rod with an electrostatic self-repulsion. The singular, non-local repulsive potential makes the problem remarkably different from the standard calculus of variations problem, yet a theory of conjugate points can still be used to identify minima, or stable equilibria. Results for the two-dimensional elastic strut will be presented.

Aimee Johnson (Swarthmore)

The Relative Growth of Information in Two-Dimensional Partitions

Abstract: In a 1964 paper, Lochs considered how much knowledge the first n digits in the decimal expansion of a number from the unit interval would give about the digits in its continued fraction expansion, and showed that for a.e. x, the limiting behavior of these two numbers of digits is always the same. In this talk, we will recast this problem in terms of measurable dynamical systems and see how using partitions and entropy lead to the same result and in fact, let us put the result in a more general framework. Finally we will consider how the dynamics and geometry change when we consider points from the unit square.

Alice Jukes (Imperial College)

Symmetric homoclinic bifurcation

Abstract: We consider heteroclinic networks with a transitive group action. Suppose the equilibria in the network have real leading eigenvalues. For generic codimension one equivariant networks of this type we show that the non-wandering dynamics is conjugate to a topological Markov chain. The situation is more complicated in codimension two and we demonstrate a novel example to illustrate the difficulties and interesting phenomena that occur. This example is of a resonant homoclinic loop to an equilibrium with D_3 symmetry.

Maryam Kamgarpour (UCB)

Survey of Control of Dynamical Systems over a Communication Channel

Abstract: Classical control theory has been widely studied and applied everywhere in daily lives, such as aircraft control, chemical process and coordination of group of agents. Similarly, classical information theory has been a cutting-edge research with applications in computer networks, aircraft, and satellite communications. However, until recently, information theoretical issues are decoupled from decision and control problems. With the advent of technology, it has become possible to consider control of dynamic systems over a communication network. This has opened up many challenging and interesting questions for both control and communication community. For example, the perfect information and synchronization assumptions made in traditional control theory are no longer valid. This presentation surveys and summarizes some research that has been done in integrating these two areas.

Yun Kang (Arizona State)

A model on interaction between plant and herbivore: modeling on gypsy moth

Abstract: The interaction between a forest pest(gypsy moth) and the tree cover is moderated by availability and recycling nutrients in the soil. A non-overlapping model is developed that determines the dynamics of this interaction as a function of the total available nutrients and the nutrient uptake rate. The most interesting phenomena observed involve bistability between chaotic and stationary dynamics. We also found the existence of 8-periodical cycle is matching the fact that gypsy moth has outbreak every 8-10 years. Our purpose is to discuss the qualitative properties and biological meaning of our model. From our bifurcation analysis, we can predict the outbreak of gypsy moth and through controlling the key parameters, we can regulate the population of pest under threshold of outbreak.

See the full PDF version of the abstract for additional information.

Lina Kim (UC Santa Barbara)

Transient growth for the linearized Navier-Stokes equations

Abstract: We analytically solve the linearized Navier-Stokes equations for streamwise-invariant sinusoidal shear flow. This is accomplished by studying the infinite dimensional system of ordinary differential equations derived via Galerkin projection onto Fourier modes, a representation which allows one to interpret the dynamics in terms of interactions between streaks and streamwise vortices. We characterize transient energy growth for this system, a mechanism which may trigger nonlinear effects that lead to sustained turbulence. This includes numerically calculating perturbations which give optimal initial and total energy growth for large enough truncations to capture the behavior of the full system. We also numerically determine Reynolds number scalings and find optimal wavenumbers for maximum transient energy growth.

See the full PDF version of the poster for additional information.

Kay Kirkpatrick (UC Berkeley)

The linear Landau equation in the weak coupling limit

Abstract: Part of Hilbert's sixth problem is to derive macroscopic descriptions of particle systems (such as gases) from the microscopic Newtonian interactions. Doing this for the hard-sphere model of a dilute gas leads to a Boltzmann equation, in what's called the Boltzmann-Grad limit. By contrast, the analogous soft-sphere model of a plasma leads to a Landau equation, in the weak coupling limit (a.k.a. the limit of grazing collisions). The harder problem is to better approximate the real-life situation where particles interact according to the Coulomb potential, which has infinite range, unlike the compactly supported potentials in the hard- and soft-sphere models.

Allison Kolpas (UC Santa Barbara)

Coarse analysis of stochasticity-induced switching in a schooling model

Abstract: Many organisms display ordered collective motion, such as geese flying in formation, locusts swarming in sub-Saharan Africa, and fish schooling. Thousands to millions of organisms can be involved, with all individuals responding rapidly to their neighbors to maintain the collective motion. Individual based models are often used when modeling aggregation since they can incorporate detailed descriptions of the interactions among agents. However, for realistic numbers of individuals, the models can become computationally and analytically intractable. I will highlight an individual-based stochastic model for fish schooling which exhibits noise induced transitions between two metastable collective states. I will then describe a coarse, "equation-free" computational framework that efficiently allows the construction of an effective potential, enabling coarse bifurcation analysis and the estimation of mean residence times in each state. This framework provides a new approach to the analysis of emergent phenomena in individual-based aggregation models.

Rachel Kuske (U British Columbia)

Multi-scale dynamics and noise sensitivity

Abstract: Small noise can change both quantitative and qualitative behavior in many systems. However, when both multi-scale behavior and noise are present it is often difficult to separate the deterministic and stochastic effects. In addition, the sensitivity often makes simulation expensive. This talk will focus on new asymptotic approaches for analyzing these types of behavior, illustrated through models of neural dynamics, meta-stable interfaces, and dynamical models with memory. Although these systems are intrisically very different, they have many similarities from the viewpoint of stochastic dynamics. Results include analytical approximation for measuring noise-sensitivity as well as reduced models for fast simulation.

Florence Lin (U of Southern California)

Applications of geometric phases and Hamiltonian dynamical systems in classical molecular dynamics

Abstract: The classical molecular dynamics of polyatomic systems are treated as N-body Hamiltonian dynamical systems. Applying techniques of geometric mechanics previously applied to two-body molecular dynamics [1,2] and to the dynamics of a triatomic molecule [3], the dynamics of atom-diatomic molecule systems are described. Just as a geometric phase arises in three-body classical mechanics and is described by the holonomy of a mechanical connection [4], a geometric phase also arises in three-body classical molecular dynamics and will be described by the holonomy of an analogous molecular connection [5,6]. Physical consequences of this geometric phase include a contribution to overall rotation in the center-of-mass frame due to "internal" motions, such as bends and rotations, to (i) the rotation of a generalized Eckart frame in weakly-bound atom-diatomic molecule complexes, (ii) the recoil angle of an atom departing from a dissociating triatomic molecule, (iii) the scattering angle of the atom in atom-diatomic molecule collisions, and (iv) the rotation of a generalized Eckart frame in metal trimers. Observations of this classical geometric phase in differential geometric, computational, and experimental settings are discussed.

References [1] F. J. Lin and J. E. Marsden, J. Math. Phys. 33, 1281 (1992). [2] S. Smale, Invent. Math. 10, 305 (1970); 11, 45 (1970). [3] F. J. Lin, Phys. Lett. A 234, 291 (1997). [4] J. E. Marsden, R. Montgomery, and T. Ratiu, Mem. Am. Math. Soc. 88, No. 436, (American Mathematical Society, Providence, RI, 1990). [5] F. J. Lin, Molecular rotation due to internal motions, 2006. [6] F. J. Lin, Rotation due to internal motions in internuclear distances in three-body molecular dynamics, 2006.

Liyan Liu (U North Carolina at Chapel Hill)

Assimilation of Lagrangian data into Layer ocean model

Abstract: The physical state variables in ocean are usually the velocity components, pressure, density, temperature and salinity. Data acquisition in the ocean is too difficult to make field estimations by direct measurements. Since much surface ocean data is Lagrangian in nature, its assimilation into ocean models is a key element of an ocean forecasting system. We assess the feasibility and effectiveness of Lagrangian data assimilation of ocean models. We advanced our research to more complicated ocean model, such as a two and a half layer shallow water double gyre with wind forcing. By observing only drifter positions in the top layer of the system, we are able to correct the errors in the entire flow region by assimilating these Lagrangian observations.

Eva Pokojska (Czech Academy of Sciences), Rachid Ouifki

Delay dynamics of nuclear transcription autoregulation

Abstract: One of many critical processes in early development is a rhythmic segment formation, regulated by intracellular control of transient oscillatory expression in key nuclear transcription regulators. We generalize the mathematical model of such transcription autoregulation represented by a 2-dimensional monotone negative cyclic feedback system incorporating delay. We focus on analyzing the role of delay on stability, bifurcation direction, periodic solutions, and the global attractor structure in particular phase space regions determined by reaction nonlinearities, and production/decay rates. Existence of two functionally different classes of dynamic transcription regulators is conjectured based on the theoretical results.

Lea Popovic (Cornell)

Degenerate Diffusion Limits in Gene Duplication

Abstract: The mechanisms responsible for the preservation of duplicate genes have been debated for decades. Recently a new explanation has been proposed according to which after duplication two gene copies specialize to perform complementary functions. The quantities of interest are the probability that the separation of functionality occurs, and the amount of time after duplication that it takes for this occur.

Mathematically this problem can be described by a diffusion in six dimensional space in the simplest example when the duplicated gene is in charge of two functions. The diffusion describes the stochastic behavior of genotypic frequencies in the population. Simulations show that this diffusion spends most of its time near a particular one dimensional curve in the six dimensional space. This degenerate diffusion limit is an example of a solution to a stochastic differential equation that is forced to a lower dimensional manifold by a very srong drift. One can show that if the drift is a vector field whose deterministic flow has an asymptotically stable manifold of fixed points, then this drift term forces the stochastic process to stay close to and any limitting process must actually stay on the stable manifold.

We show that the dynamical system in the gene duplication model is asymptotically stable, and use the limitting diffusion process to give results for the quantities of interest.

Claire Postlethwaite (Northwestern)

Controlling travelling waves of the complex Ginzburg-Landau equation with spatial feedback

Abstract: Previous work has shown that Benjamin-Feir unstable travelling waves of the complex Ginzburg-Landau equation (CGLE) in two spatial dimensions cannot be stabilised using a particular time-delayed feedback control mechanism known as `time-delay autosynchronisation'. In this paper, we show that the addition of similar spatial feedback terms can be used to stabilise such waves. This type of feedback is a generalization of the time-delay method of Pyragus (Phys. Letts. A 170, 1992) and has been previously used to tabilize waves in the one-dimensional CGLE by Montgomery and Silber (Nonlinearity 17, 2004). We consider two cases in which the feedback contains either one or two spatial terms. We give a numerical example to demonstrate our linear stability results.

Ami Radunskaya (Pomona College)

Stochastic perturbations of growth models

Abstract: Many biological and physiological processes involve self-regulating mechanisms that prevent too much growth while ensuring against extinction. Here we are interested in a class of processes whose rate of growth is not entirely deterministic, but depends on the current state of the system. In this talk we will look at some simple examples of dynamcial systems in this class, and discuss some analytical results along with their implications.

Mary Silber (Northwestern)

Controlling pattern formation

Abstract: Faraday waves, of startling beauty and complexity, may form on the surface of a fluid layer when it is shaken up and down. The spatial symmetries of these intricate wave patterns depend on the frequency content of the forcing function in subtle ways that we have tried to illuminate. This in turn suggests ways to control the pattern formation process by an appropriate design of the forcing function. Our analysis is based in equivariant bifurcation theory, while the problems are motivated by laboratory experiments; both will be described.

Ana Dias (University of Porto)

Coupled cell networks: ODE-equivalence, minimality and quotients

Abstract: Many important real world networks can be modelled by dynamical systems on a graph and therefore coupled cell networks. It is known that non-isomorphic coupled cell networks can correspond to equivalent dynamics and these are called ODE-equivalent. In this talk we describe some recent results in the theory of coupled cell networks related with ODE-equivalence, minimality and quotients of coupled cell networks.

Daniel Toundykov (U Virginia)

Finite-dimensionality and smoothness of the global attractor for a semilinear wave equation with localized dissipation, and a source of critical exponent

Abstract: This work addresses long-term behavior of solutions to a semilinear damped wave equation with a source term. The dissipation is nonlinear and affects only a small sub-domain adjacent to a portion of the boundary; the source is modeled by the Nemytski operator associated to a nonlinear map whose order may include (in dimensions above 2) the corresponding critical Sobolev exponent (for the embedding H^1 to L^2). It is known that existence of attractors is strictly linked with asymptotic compactness of evolution trajectories, while at the level of critical exponents the compactness of Sobolev embeddings is lost and the extensive machinery developed for sub-critical settings no longer applies. The study of asymptotic properties under critical damping and sources is a challenging problem, and a comprehensive treatment of such models appeared only recently in the literature -- in the work of I. Lasiecka and I. Chueshov. The results so far, however, have dealt primarily with full interior or full boundary damping. In this case we investigate the issue of critical exponents combined with geometrically restricted dissipation. We prove that such a system possesses a smooth (bounded in H^2 X H^1) global compact attractor of a finite fractal dimension.

Paul Wright (Courant)

Some rigorous results for the periodic oscillation of an adiabatic piston

Abstract: A simple model of an adiabatic piston consists of a heavy piston of mass M that separates finitely many ideal, unit mass gas particles moving inside two gas containers. Averaging techniques, used to study the motion of the slow-moving piston in the limit where M tends to infinity, suggest that the piston should oscillate periodically. For one-dimensional chambers, the effects of the gas particles can be essentially decoupled, and I showed that we recover a strong law of large numbers that is characteristic of classical averaging over just one fast variable: the deviation of the piston from its averaged behavior is no more than O(M^{-1/2}) on a time scale O(M^{1/2}). I also showed that for a very general gas chamber in higher dimensions, the actual motions of the piston converge in probability to the averaged behavior on that time scale, although a strong law is no longer possible. I learned about this problem from the papers of Neishtadt and Sinai, who derived the averaged equations and pointed out that an averaging theorem due to Anosov could be extended to this case.

Na Yu (U of British Columbia)

Noise-induced mixed mode in a pair of weakly coupled neuronal oscillators

Abstract: Synaptically coupled neurons show in-phase or anti-phase synchrony depending on the chemical and dynamical nature of the synapse. Deterministic theory helps predict the phase difference between the two phase-locked oscillators when the coupling is week. In the presence of noise, however, deterministic theory faces difficulty when the coexistence of multiple oscillatory solutions occur. We analyzed the solution structure of two coupled neuronal oscillators near a subcritical Hopf bifurcation point and a saddle point of the periodic solution branch which revealed a rich variety of co-existing and multi-mode solutions. The normal of such a bifurcation scenario is derived using a simpler model. We show that noise causes important changes in the phase dynamics of coupled oscillators when distinct co-existing solutions are randomly visited.