Boston University/Keio University Workshop 2014

Dynamical Systems

September 15-19, 2014

Abstracts of Talks and Lecture Notes



Table of Contents


  • Margaret Beck: Stability of patterns in reaction-diffusion equations

  • Paul Blanchard: Checkerboard Julia sets

  • Bernard Chan: Dynamics of coupled gyroscopes

  • Osmand Chowdhary: Center Manifolds and Taylor Dispersion

  • Zhengyu Chen: On the metric theory of Diophantine approximations for complex numbers

  • Sara Clifton: A mathematical model for the sexual selection of extravagant and costly mating displays

  • Bob Devaney: Cantor and Sierpinski, Julia and Fatou: crazy topology in complex dynamics (powerpoint file)

  • David Fried: Symbolic dynamics and dynamical zeta functions

  • Dick Hall: Variational methods and open problems in celestial mechanics

  • Takuya Ikeda: Central limit theorem for a class of nonsingular transformations with quasi-compact Perron-Frobenius operator

  • Gabriella Jaramillo: Patterms and inhomogeneities in spatially extended systems

  • Eiko Kin: The magic 3-manifold, horseshoe braids and minimal dilatation problem

  • Kelly McQuighan: Using numerical continuation to study "large amplitude" solutions to the 2:1 forced complex Ginzburg-Landau equation

  • Yasushi Nagai: Existence of finite sequences and non-existence of infinite sequence in tilings

  • Ahmet Özer : Modeling and controllability issues for piezoelectric structures

  • Kengo Shimomura: The Hausdorff dimension of the overlapping Cantor set

  • Mitsuhiro Shishikura: Dynamical charts for irrationally indifferent fixed points

  • Hiroki Sumi: Negativity of Lyapunov Exponents of Generic Random Dynamical Systems of Complex Polynomials. Lecture notes; Discussion of open problems

  • Kiyotaka Suzaki: An SDE approach to leafwise diffusions on foliated spaces and its application to a limit theorem

  • Shintaro Suzuki: The Artin-Mazur zeta functions of generalized beta-transformations

  • Hiroki Takahasi: Lyapunov optimizing measures for Henon-like maps at the first bifurcation

  • Yosuke Tokunaga: Measures with maximum total exponent of C^1 diffeomorphisms with basic sets

  • Theo Vo: Canard- and Hopf-induced bursting in pituitary cells

  • Gene Wayne: Dynamical systems and the asymptotics of the Navier-Stokes equations Survey article; Lectures (click "Download Chapter")

  • Qiliang Wu: Existence of pearled interfaces in the 2-D functionalized Cahn-Hilliard equation: Lecture notes; Lecture slides.

  • Stability of patterns in reaction-diffusion equations


    Margaret Beck

    Boston University

    Reaction-diffusion equations model a wide variety of chemical and biological processes. Such systems are well known for exhibiting patterns, such as traveling waves and spatially- and/or temporally-periodic structures. One important property of such solutions is whether or not they are stable, which is important because it is typically only the stable solutions that are observed in real world settings. In this talk, I will discus the difference between spectral, linear, and nonlinear stability, and highlight some key methods for analyzing stability.

    Lecture Notes with Exercises


    Checkerboard Julia Sets


    Paul Blanchard

    Boston University

    We discuss work done at Boston University during the last few years on the families of rational maps

    z^n + lambda/z^d

    where lambda is a nonzero complex parameter. The integer n is always at least 2, and the positive integer d is usually greater than 1.

    Maps in this family are interesting because they behave like the map z^n far from the origin but have a pole at the origin. This dual behavior yields unusual Julia sets, both from a topological and a dynamical point of view. We were motivated to study these families by McMullen's examples whose Julia sets are Cantor sets of quasicircles.

    Lecture Notes


    Dynamics of Coupled Gyroscopes


    Bernard Chan

    San Diego State University

    Gyroscopes are mechanical devices for measuring and maintaining orientation. While commonly used in navigation, these purely mechanical sensors are limited in application due to their size and friction wear. Advances in manufacturing techniques in microelectromechanical systems (MEMS) allow for mass manufacturing of low- cost and miniaturized vibratory gyroscopes. Given their size and price, these MEMS gyroscopes are commonly found in a broad range of handheld electronics. At the same time, these smaller gyroscopes are more prone to external perturbations. Small disturbances, such as thermo interference, can increase phase drifts in the oscillatory signal and give inaccurate results.

    To remedy the aforementioned problem, researchers are considering networks of coupled MEMS gyroscopes. Experimental and numerical studies have shown that networked MEMS gyroscopes can increase the sensitivity while minimizing phase drift. In this work, we use analytic methods to study a network of symmetrically coupled gyroscopes in a Hamiltonian setting. We first investigate the effects of coupling topology on the gyroscopic array. Normal form techniques are used to obtain the equations of motion of the reduced system. Synchronization behavior from bifurcation and numerical analysis of the system are presented. Other issues of coupled Hamiltonian systems are also discussed.

    Lecture notes


    Center Manifolds and Taylor Dispersion


    Osman Chaudhary

    Boston University

    Taylor dispersion, first described by G.I. Taylor in 1953, occurs when the diffusion of a solute in a pipe or channel is enhanced by a background flow, in that the solute asymptotically approaches a form that solves a different diffusion equation (with larger diffusion coefficient, in an appropriate moving frame). By introducing scaling variables, I'll show how one can obtain this asymptotic form in a very natural way using a center manifold reduction. This is joint work with Margaret Beck and Gene Wayne.


    On the metric theory of Diophantine approximations for complex numbers


    Zhengyu Chen

    Keio University

    In 1941, R. J. Duffin and A. C. Schaeffer conjectured that for an inequality of the form |x - m/n| < psi(n)/n with g.c.d.(m, n) = 1, there are infinitely many solutions of positive integers m and n for almost all real number x if and only if the sum over n of phi(n)psi(n)/n diverges. As a partial result, in 1978 J. D. Vaaler proved this conjecture under the additional condition psi(n) = O(1/n). I have done some research about the metric theory of Diophantine approximation over an imaginary quadratic field with a square-free d < 0, and proved that a Vaaler-type theorem holds in this case.


    A mathematical model for the sexual selection of extravagant and costly mating displays


    Sara Clifton

    Northwestern University

    The evolution of extravagant and costly ornaments on animals has intrigued biologists since Darwin suggested that female preference for exaggerated courtship displays drives the sexual selection of these ornaments. We propose a minimal mathematical model which incorporates two components of ornament evolution: an intrinsic cost of ornamentation to an individual, and a social benefit of relatively large ornaments within a population. Using bifurcation analysis and perturbation theory, we show that animals will split into two niches, one with large ornaments and one with small, a phenomenon observed in many species.

    Lecture Slides


    Cantor and Sierpinski, Julia and Fatou: Crazy Topology in Complex Dynamics


    Bob Devaney

    Boston University

    In this talk, we shall describe some of the rich topological structures that arise as Julia sets of certain complex functions including the exponential and rational maps. These objects include Cantor bouquets, indecomposable continua, and Sierpinski curves.


    Symbolic Dynamics and Dynamical Zeta Functions


    David Fried

    Boston University

    We will explain how symbolic dynamics is used to study dynamical zeta functions for hyperbolic systems. We will describe some explicit symbolic dynamics for geodesic flows on negatively curved compact orbifolds.


    Variational Methods and Open Problems in Celestial Mechanics


    Dick Hall

    Boston University

    Over the last 15 years, variational techniques have lead to significant discoveries in the Newtonian N-body problem. We discuss the application of these techniques with particular emphasis on their relationship to open problems including Saari's conjecture (on solutions with constant moment of inertia), relative equilibrium solutions, the existence of families of periodic orbits in the three body problem, and others.

    Lecture Notes


    Central limit theorem for a class of nonsingular transformations with quasi-compact Perron-Frobenius operator


    Takuya Ikeda

    Osaka University

    We consider a class of nonsingular transformations, which contains piecewise C^2 expanding maps on the interval. Given a transformation T in such a class and real valued function f satisfying an appropriate condition, we can show a central limit theorem of mixed type with nice convergence rate for the sum \sum_{k=0}^{n-1}f \circ T^k as n \rightarrow \infty provided that the limiting variance is nondegenerate. This is a joint work with Takehiko Morita in Osaka University.

    Lecture Notes


    Patterms and inhomogeneities in spatially extended systems


    Gabriela Jaramillo

    University of Minnesota

    We look at the effects of inhomogeneities on pattern formation as a perturbation problem. We present a few examples of spatially extended pattern forming systems and explain why regular perturbation theory fails due to critical continuous spectrum. We show how Kondratiev spaces can help alleviate this difficulty: the linearization at periodic patterns becomes a Fredholm operator, albeit with negative index. Together with far-field matching procedures, we obtain deformed stripe patterns or pacemakers using implicit function type continuation.

    Lecture Notes


    The magic 3-manifold, horseshoe braids and minimal dilatation problem


    Eiko Kin

    Osaka University

    We consider the mapping class groups on the oriented surfaces possibly with punctures. If we fix a surface S and we take a number C>1, then the set of the dilatations, which are smaller than C, of pseudo-Anosovs on S is finite. The minimal dilatation problem asks which number is the minimal pseudo-Anosov dilatation on S, and which pseudo-Anosov mapping class realizes the minimum. This problem is wide open for most surfaces. In this talk, I will explain how a single hyperbolic 3-manifold, so called the magic manifold, appears in the study of the problem and some roles the magic manifold plays. I will also explain a relation between the minimal dilatation problem on the punctured disk and the horseshoe braids which are determined by the periodic orbits of the Smale horseshoe map.

    Lecture Notes


    Using numerical continuation to study "large amplitude" solutions to the 2:1 forced complex Ginzburg-Landau equation


    Kelly McQuighan

    Boston 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, which are modeled mathematically with the 2:1 forced complex Ginzburg-Landau equation. We perform a numerical continuation study of localized solutions to the planar forced Ginzburg-Landau equation using Matlab and AUTO.

    Lecture Notes


    Existence of finite sequences and non-existence of infinite sequence in tilings


    Yasushi Nagai

    Keio University

    A tiling is a cover of R^d by tiles (often polygons) that overlap only on their borders. A patch is a configuration consisting of finitely many tiles that appears in a tiling. Given a tiling, we can construct a dynamical system that encodes information of the original tiling. We investigate a relation between distribution of patches in tilings and properties of the corresponding dynamical systems. In particular we show the existence of finite sequences of patches and non-existence of infinite sequences of patches in certain tilings from a property of the corresponding dynamical systems.

    Lecture Notes


    Modeling and controllability issues for piezoelectric structures


    Ahmet Özer

    University of Nevada-Reno

    The control theory (controllability/stabilization) of the basic partial differential equations (PDEs), i.e. wave equation, heat equation or ShrodingerĘs equation, are well-understood in the last four decades. An interesting route in the last decade has been investigating the control theory of the coupled system of PDEs. A motivating example to be presented in this talk is the control of piezoelectric (smart) structures which have a unique characteristic of converting mechanical energy to electrical and magnetic energy, and vice versa. More precisely, when voltage is supplied at the electrodes of the beam, an electric field is created between the electrodes, and therefore the beam either shrinks or extends. Mathematically speaking, mechanical equations are strongly coupled to the electric and magnetic equations. Even though these types of PDEs are practically having the same nature, a different methodology is needed to design a feedback controller.

    Derivation of the PDEs, i.e. modeling of the physical phenomenon, also plays an important role for both to understand the physics of the problem and to accurately control the dynamics. Accurate modeling is necessary since some physical effects, which are considered to be minor in comparison to other major effects, may be ignored and they may cause instabilities in the system. In this talk, derivation of the well- posed linear PDEs models for piezoelectric structures (for different types of actuation, i.e. voltage, charge, or current) by a variational approach is also briefly mentioned. Finally, the controllability/stabilization problems are considered.


    The Hausdorff dimension of the overlapping Cantor set


    Kengo Shimomura

    Osaka University

    We consider the Hausdorff dimension of the limit set of some overlapping iterated function system(IFS). We know the Hausdorff dimension of a IFS when it satisfies an open set condition. In this talk, we generate the Cantor set by a IFS and change a scaling factor of it with one parameter. If it is over 0.5, the IFS is overlapping and it does not satisfy an open set condition. However we can calculate the Hausdorff dimension of the Cantor set when its scaling factor takes some values. I talk about it and my general evaluation of the Hausdorff dimension of the overlapping Cantor set.


    Dynamical charts for irrationally indifferent fixed points


    Mitsuhiro Shishikura

    Kyoto University

    We introduce a notion of dynamical charts for holomorphic dynamics around indifferent fixed points. In the charts, the dynamics has a simple and standard form and important information will be encoded in the gluing of the charts. For quadratic polynomials with an irrationally indifferent fixed point of high type rotation number, such charts are constructed via near-parabolic renormalization. The original dynamics is recovered by gluing the canonical maps by gluing maps according to a rotation combinatorics. As an application, we show that the hedgehog consists of hairs and hairs can be indexed by a Denjoy odometer.

    Lecture Notes


    Negativity of Lyapunov Exponents of Generic Random Dynamical Systems of Complex Polynomials


    Hiroki Sumi

    Osaka University

    In this talk, we consider random dynamical systems of complex polynomial maps on the Riemann sphere. Dynamical systems can be describe many things in nature, and since nature has a lot of random terms, it is natural to consider random dynamical systems. Also, one of the most popular and interesting topics of dynamical systems is the polynomial dynamics. Thus, it is very important to consider random dynamical systems of complex polynomials on the Riemann sphere.

    It is well-known that for each rational map $f$ on the Riemann sphere with $\deg (f)\geq 2$,

    (a) the Julia set $J(f)$ is a non-empty perfect compact subset of the Riemann sphere (thus $J(f)$ contains uncountably many points),
    (b) the dynamics of $f: J(f)\rightarrow J(f)$ is chaotic (at least in the sense of Devaney), and
    (c) the Hausdorff dimension of the set of points $z$ in the Riemann sphere for which the Lyapunov exponent of the dynamics of $f$ is positive, is positive.

    However, we show that for generic i.i.d. random dynamical systems of complex polynomials, all of the following (1) and (2) holds.

    (1) For all points $x$ in the Riemann sphere, the orbit of the Dirac measure $\delta _{x}$ at $x$ under the dual of the transition operator of the system converges to a periodic cycle of probability measures on the Riemann sphere.
    (2) For all but countably many points $x$ in the Riemann sphere, for almost every sequence $\gamma =(\gamma _{1}, \gamma _{2}, \gamma _{3},\cdots )$ of polynomials, the Lyapunov exponent along $\gamma $ starting with $x$ is negative.

    Note that each of (1) and (2) cannot hold in the usual iteration dynamics of a single rational map $f$ with $\deg (f)\geq 2.$ Therefore the picture of the random complex dynamics is completely different from that of the usual complex dynamics. We remark that even if the chaos of the random system disappears in the $C^{0}$ sense, the chaos of the system may remain in the $C^{1}$ sense, and we have to consider the ``gradation between chaos and order''.

    References:
    [1] H. Sumi, Random complex dynamics and semigroups of holomorphic maps, Proc. London. Math. Soc. (2011), 102 (1), 50--112.
    [2] H. Sumi, Cooperation principle, stability and bifurcation in random complex dynamics, Advances in Mathematics 245 (2013) 137--181.

    Lecture notes
    Discussion of open problems


    An SDE approach to leafwise diffusions on foliated spaces and its application to a limit theorem


    Kiyotaka Suzaki

    Osaka University

    As is well-known, foliations, and consequently foliated spaces can be regarded as generalizations of dynamical systems. In this talk, we construct leafwise diffusions on foliated spaces via an SDE approach. As an application, we state a central limit theorem for a class of additive functionals of the leafwise diffusion starting at almost every point with respect to any harmonic measure.

    Lecture Notes


    The Artin-Mazur zeta functions of generalized beta-transformations


    Shintaro Suzuki

    Osaka University

    In this talk, we consider the generalized beta-transformation, introduced by Pawl Gora in 2007 and study the analytic properties of its Artin-Mazur zeta function. We state that it can be extended to a meromorphic function which is expressed by using the f-expansion of 1. As an application, we relate the analytic properties of its poles to the ergodic properties of the corresponding the generalized beta-transformation.


    Lyapunov optimizing measures for Henon-like maps at the first bifurcation


    Hiroki Takahasi

    Keio University

    We study the dynamics of a strongly dissipative Henon-like map at the first bifurcation parameter at which the uniform hyperbolicity is destroyed by the formation of homoclinic or heteroclinic tangencies inside the limit set. Under an open condition on the unstable eigenvalues of the two fixed saddles, we show the existence of an invariant Borel probability measure which minimizes the unstable Lyapunov exponent.

    Lecture Notes

    Measures with maximum total exponent of C^1 diffeomorphisms with basic sets


    Yusuke Tokunaga

    Osaka University

    Measures with maximum total exponent can be defined for a diffeomorphism on a compact manifold. We prove that for any C^1-diffeomorphism with a basic set, there exists a C^1-neighborhood satisfying the following properties. A generic element in the neighborhood has a unique measure with maximum total exponent which is of zero entropy and fully supported on the continuation of the basic set. To the contrary, we show that for r >= 2 any C^r-diffeomorphism with a basic set does not have a C^r-neighborhood satisfying the above properties.


    Canard- and Hopf-Induced Bursting In Pituitary Cells


    Theo Vo

    Boston University

    It has been shown that large conductance potassium (BK) current tends to promote bursting in pituitary cells. This requires fast activation of the BK current, otherwise it is inhibitory to bursting. In this work we combine theoretical (geometric singular perturbation theory), experimental (dynamic clamp) and numerical (AUTO) methods to understand why the BK activation must be fast in order to promote bursting.

    Lecture Notes


    Dynamical systems and the asymptotics of the Navier-Stokes equations


    Gene Wayne

    Boston University

    In this lecture I will describe how one can use ideas of dynamical systems theory to give a quite complete picture of the long time asymptotics of solutions of the two-dimensional Navier-Stokes equation. I will discuss the existence and properties of invariant manifolds for dynamical systems defined on Banach spaces and review the theory of Lyapunov functions, again concentrating on the aspects of the theory most relevant to infinite dimensional dynamics. I will then explain how one can apply both of these techniques to the two-dimensional Navier-Stokes equation to prove that any solution with integrable initial vorticity will be asymptotic to a single, explicitly computable solution known as an Oseen vortex equations.

    Survey article
    Lectures (click "Download Chapter")


    Existence of Pearled Interfaces in the 2-D functionalized Cahn-Hilliard equation


    Qiliang Wu

    Michigan State University

    The functionalized Cahn-Hilliard equation is a continuum model for the rich morphologies observed in blends of amphiphilic polymer and solvent. A central feature captured by this model is the energetic competition among morphologies of distinct co-dimension, whose transitions are typically achieved via the ``pearling bifurcation'': a high-frequency sideband instability which breaks strings into beads (or pearls). Moreover, the bifurcation can be supercritical: we demonstrate the existence of pearled and graduated pearled equilbria as they bifurcate from flat and circular co-dimension one interfaces immersed in $R^2$. Using spatial dynamics and center manifold reduction to reduce the PDE into an ODE system, we obtain the rich 1:1 resonant normal form and show the existence of these patterns within the truncated normal from up to cubic order, as well as their persistence as equilibria of the full problem.

    Lecture Notes
    Lecture Slides