Ph.D. Recipients and their Thesis Abstracts

Differential Equations / Dynamical Systems

Algebra, Analysis, Combinatorics, Differential Equations / Dynamical Systems, Differential Geometry, Geometry, Interdisciplinary, Lie Groups, Logic, Mathematical Physics, PDE / Numerical Analysis, Probability, Statistics, Topology

On the Numerical Construction of Hyperbolic Structures for Complex Dynamical Systems

Abstract: Our main interest is using a computer to rigorously study ε-pseudo orbits for polynomial diffeomorphisms of C². Periodic ε-pseudo orbits form the ε-chain recurrent set, R_ε. The intersection ∩_{ε > 0}R_ε is the chain recurrent set, R. This set is of fundamental importance in dynamical systems.

Due to the theoretical and practical difficulties involved in the study of C², computers will presumably play a role in such efforts. Our aim is to use computers not only for inspiration, but to perform rigorous mathematical proofs.

In this dissertation, we develop a computer program, called Hypatia, which locates R_ε, sorts points into components according to their ε-dynamics, and investigates the property of hyperbolicity on R_ε. The output is either "yes", in which case the computation proves hyperbolicity, or "not for this ε", in which case information is provided on numerical or dynamical obstructions.

A diffeomorphism f is hyperbolic on a set X if for each x there is a splitting of the tangent bundle of x into an unstable and a stable direction, with the unstable (stable) direction expanded by f (f^--1). A diffeomorphism is hyperbolic if it is hyperbolic on its chain recurrent set.

Hyperbolicity is an interesting property for several reasons. Hyperbolic diffeomorphisms exhibit shadowing on R, i.e., ε-pseudo orbits are delta-close to true orbits. Thus they can be understood using combinatorial models. Shadowing also implies structural stablity, i.e., in a neighborhood in parameter space the behavior is constant. These properties make hyperbolic diffeomorphisms amenable to computer investigation via ε-pseudo orbits.

We first discuss Hypatia for polynomial maps of C. We then extend to polynomial diffeomorphisms of C². In particular, we examine the class of Hénon diffeomorphisms, given by

This is a large class of diffeomorphisms which provide a good starting point for understanding polynomial diffeomorphisms of C². However, basic questions about the complex Hénon family remain unanswered.

In this work, we describe some Hénon diffeomorphisms for which Hypatia verifies hyperbolicity, and the obstructions found in testing hyperbolicity of other examples.

Using Spider Theory to Explore Parameter Spaces

Abstract: For a fixed integer $d\ge2$, consider the family of polynomials $P_{d,\lambda}(z)=\lambda(1+z/d)^d$, where $\lambda$ is a complex parameter. In this work, we study the location of parameters $\lambda$ for which $P_{d,\lambda}$ has an attracting cycle of a given length, multiplier, and combinatorial type.

Two main tools are used in determining an algorithm for finding these parameters: the well-established theories of external rays in the dynamical and parameter planes and Teichmüller theory. External rays are used to specify hyperbolic components in parameter space of the polynomials and study the combinatorics of the attracting cycle. A properly normalized space of univalent mappings is then employed to determine a linearizing neighborhood of the attracting cycle.

Since the image of a univalent mapping completely determines the mapping, we visualize these maps concretely on the Riemann sphere; with discs for feet and curves as legs connected at infinity, these maps conjure a picture of fat-footed spiders. Isotopy classes of these spiders form a Teichmüller space, and the tools found in Teichmüller theory prove useful in understanding the Spider Space. By defining a contracting holomorphic mapping on this spider space, we can iterate this mapping to a fixed point in Teichmüller space which in turn determines the parameter we seek.

Finally, we extend the results about these polynomial families to the exponential family $E_\lambda(z)=\lambda e^z$. Here, we are able to constructively prove the existence and location of hyperbolic components in the parameter space of $E_\lambda$.

Construction of Markov Partitions for Linear and Nonlinear Automorphisms of Tori

Abstract: We construct and investigate Markov partitions for hyperbolic linear and nonlinear maps of the torus. Our approach is based upon the arithmetic sofic partitions constructed by Kenyon and Vershik. We improve their construction by showing how to produce an irreducible sofic system in every case. In addition, we describe a method for determining the boundaries of the rectangles, and use this to calculate their Hausdorff dimension in simple cases. We also investigate the construction of Markov partitions for nonlinear maps of the torus. Our method involves computing an approximation to a conjugacy with a linear map via the Shadowing Lemma, and using this conjugacy to define a Markov partition for the nonlinear map as the image of one for the linear map.

On the Combinatorics of External Rays in the Dynamics of the Complex Hénon Map

Abstract: We present combinatorial models that describe quotients of the solenoid arising from the dynamics of the complex Hénon map

f_{a,c}: C^2 \to C^2, (x,y) \mapsto (x^2 + c– ay, x).

These models encode identifications of external rays for specific mappings in the Hénon family. We investigate the structure of a region of parameter space in R^2 empirically, using computational tools we developed for this study.

We give a combinatorial description of bifurcations arising from changes in the set of identifications of external rays.

Our techniques enable us to detect, predict, and locate bifurcation curves in parameter space.

We describe a specific family of bifurcations in a region of real parameter space for which the mappings were expected to have simple dynamics. We compute the first few bifurcation curves in this family and label them combinatorially.

Our computer experiments also indicate the existence of gaps within the region of real parameter space where Hénon family f_{a,c} has connected Julia set J_{a,c}. We show why the verification of this gap would imply the existence of values of a for which the level-a Mandelbrot set, M_a= {c\in C: J_{a,c} is connected}, is not connected.

Computing Thom-Boardman Singularities

Abstract: Thom-Boardman singularities describe the hierarchical degeneracy of smooth mappings between smooth manifolds. This thesis presents a numerical method to compute a regular set of defining equations for Thom-Boardman singularities. Using an automatic differentiation package, we have written C++ and Matlab programs to test the algorithm developed in the thesis. As a byproduct of the development of the algorithm, we present algorithmic proofs of some of the theoretical results proved in Boardman [3]. Using bordering matrix techniques, we have also derived a simpler algorithm that computes non-regular defining equations. Numerical experiments show that the algorithm that computes regular equations is much more robust than the algorithm based on bordering matrices.

Non-normal Dynamics and Hydrodynamic Stability

Abstract: This thesis explores the interaction of non-normality and nonlinearity in continuous dynamical systems. A solution beginning near a linearly stable fixed point may grow large by a linear mechanism, if the linearization is non-normal, until it is swept away by nonlinearities resulting in a much smaller basin of attraction than could possibly be predicted by the spectrum of the linearization. Exactly this situation occurs in certain linearly stable shear flows, where the linearization about the laminar flow may be highly non-normal leading to the transient growth of certain small disturbances by factors which scale with the Reynolds number.

These issues are brought into focus in Chapter 1 through the study of a two-dimensional model system of ordinary differential equations proposed by Trefethen, et. al. [Science261, 1993].

In Chapter 2, two theorems are proved which show that the basin of attraction of a stable fixed point, in systems of differential equations combining a non-normal linear term with quadratic nonlinearities, can decrease rapidly as the degree of non-normality is increased, often faster than inverse linearly.

Several different low-dimensional models of transition to turbulence are examined in Chapter 3. These models were proposed by more than a dozen authors for a wide variety of reasons, but they all incorporate non-normal linear terms and quadratic nonlinearities. Surprisingly, in most cases, the basin of attraction of the laminar flow shrinks much faster than the inverse Reynolds number.

Transition to turbulence from optimally growing linear disturbances, streamwise vortices, is investigated in plane Poiseuille and plane Couette flows in Chapter 4. An explanation is given for why smaller streamwise vortices can lead to turbulence in plane Poiseuille flow. In plane Poiseuille flow, the transient linear growth of streamwise streaks caused by non-normality leads directly to a secondary instability.

Certain unbounded operators are so non-normal that the evolution of infinitesimal perturbations to the fixed point is entirely unrelated to the spectrum, even as t\to\infty. Two examples of this phenomenon are presented in Chapter 5.

A Computer Generated Proof for the Existence of Periodic Orbits for Three-Dimensional Vector Fields

Abstract: In this dissertation we develop a program for proving the existence of a closed orbit for vector fields in R^3. We assume that the existence of an attracting limit cycle has been observed through numerical methods, and one point in the basin of attraction is among the program's inputs. The program constructs a positively invariant torus around the limit cycle. Interval analysis is used to prove rigorously that the constructed torus and all its normal sections are transverse to the vector field. We prove that under such conditions, the torus will contain a periodic orbit of the flow. We report the results of using the program to prove the existence of closed orbits for certain parameter ranges of the Lorenz system. A C++ implementation of the key parts of the algorithm is included in Appendix A.

Combinatorics and Holomorphic Dynamics: Captures, Matings and Newton's Method

Abstract: This thesis studies three interrelated topics of one dimensional holomorphic dynamics: captures, matings and Newton's method. The main ingredients are Thurston's characterization theorem of critically finite rational maps and Yoccoz's puzzle arguments. The latter works in many cases in which critical finiteness is not assumed. Both employ combinatorics heavily and essentially.

We prove a conjecture on captures for quadratic polynomials. The result has a significant impact on the study of parameter spaces for many quadratic rational map families. For example, it is used in the study of matings involving Yoccoz polynomials in this thesis, although we mainly relied on Yoccoz's puzzle argument to do the proof.

Newton's method has been a favorite application topic for holomorphic dynamics. We study the Newton maps for a class of higher degree polynomials which includes all polynomials of form z^d+AZ+b (d>=3 and a,b are arbitrary parameters).

Regularized Models of Phase Transformation in One-Dimensional Nonlinear Elasticity

Abstract: We investigate the asymptotic behavior of two models of solid-solid phase transitions in one-dimensional nonlinear elasticity. These models illustrate two different methods of regularizing the continuum mechanical equations of elasticity for a multiphase material. Both systems are studied in the framework of infinite-dimensional gradient dynamical systems theory.

First, we consider a viscoelastic model with capillarity. We show that this system possesses strong, regular solutions and a global attractor which can be characterized as the unstable set of the equilibrium states. We study the local bifurcation of equilibria, the structure of the attractor, and the asymptotic behavior of solutions.

The second model introduces an order parameter to determine the phase. This phase-field system is composed of two coupled evolution equations: one hyperbolic and the other parabolic. We establish the existence of classical solutions and show that all solutions approach the equilibrium set in the weak topology, while the phase-field stabilizes strongly. We also present numerical simulations.

Internal Addresses in the Mandelbrot Set and Irreducibility of Polynomials

Abstract: For a complex parameter c, consider the quadratic polynomial p_c= z^2 + c. Its periodic points of period n solve the polynomial equation P_n(z)= p_c^{\circ n}(z) – z. Dividing out periodic points of lower period, we arrive at polynomials Q_n(z) which are recursively defined by P_n(z)= \prod_{k| n} Q_k(z). The first main result is the following: the polynomials Q_n are irreducible overC(c). We even determine its Galois group and show that it consists of all permutations of periodic points of exact period n which commute with the dynamics of the iteration.

The principal tool, and second main result, is a combinatorial description of the Mandelbrot set using internal addresses. Moreover, they give the Mandelbrot set an interpretation as the parameter space of kneading sequences, and they answer the question: How can you tell where a rational external ray lands at the Mandelbrot set, without having Adrien Douady at your side?

Both the irreducibility result and the internal addresses are generalized to the polynomials z^d + c, and applications to the exponential family \lambda e^z are briefly discussed. In addition, we give a new, much simplified proof of a classical result of Douady and Hubbard: every external ray of the Mandelbrot set at a rational angle lands at a well-defined point c, such that there is a precisely described relation between the angle of the ray and the dynamics of the polynomial z^2 + c.

Quasisymmetric Conjugacy of Degree N Critical Circle Map

Abstract: Circle diffeomorphism has been studied since the time of Poincaré. Through the works of Arnold, Herman and Yoccoz, one has a very deep understanding of circle diffeomorphism. However the techniques employed by the above people fail if the map is merely a circle homeomorphism and has critical points. To overcome this difficulty, van Strien and independently \'Swiatek introduced a new technique called distortion of cross-ratio. People have since tried to analyze this technique and create in the process a new principle to study one dimensional real dynamics.

On the other hand, much less is known about covering circle map of degree n> 1. Even though such a map is topologically semi-conjugate to the map z\to z^n, we may not have a topological conjugacy because the map may have attracting periodic orbits. Using the new techniques developed by Sullivan, de Melo and van Strien, this work gives sufficient conditions for a degree n critical covering circle map to be quasisymmetrically conjugate to the map z\to z^n.

As an application of this theorem to complex dynamical systems, I make use of Douady's and Ghy's construction which gives sufficient conditions for a quadratic polynomial to have a Siegel disk which is a quasi-disk. We consider a one parameter family of cubic polynomials with a superattractive fixed point at the origin, the family being parametrized by the other free critical point. I am able to give sufficient conditions for the immediate basin of the origin to be bounded by a quasi-circle.

Local Connectivity in a Family of Cubic Polynomials

Abstract: In this work we consider the one parameter family of cubic polynomials with a critical fixed point, the family being parameterized by the other free critical point. Because these polynomials have a single free critical point we are able to use many of the techniques developed to study quadratics (which also have a single free critical point). In particular we will be using some relatively recent constructions due to J. C. Yoccoz which he introduced in his work on the Mandelbrot set.

Our first results are in the dynamical plane. We are able to show that if the polynomial is not renormalizable then its Julia set is locally connected. Furthermore we show that the boundary of the immediate basin of attraction of the critical fixed point is locally connected in all cases.

In the last two chapters we turn to the parameter plane and study the connectedness locus for this family of cubics. We show that it contains infinitely many embedded copies of the Mandelbrot set and is locally connected everywhere else.

Mathematical Theory of the Barotropic Model in Geophysical Fluid Dynamics

Abstract: We use the geometric method to study the Hamiltonian structures in the mechanical system of ideal 2-D fluid moving on a uniformly rotating sphere. Stability of some relative equilibria include modons and discontinuous vortices are discussed by the energy-momentum method.