Handling editor: Phanikrishna Thota

Dynamical Systems at North Carolina State

written by
Steve Schecter, North Carolina State University

Some members of the NC State dynamical systems group. From left: John Franke (faculty), Vahagn Manukian (postdoc), David Long (graduate student), Ming Jiang (graduate student), Dmitry Zenkov (faculty), Anna Ghazaryan (postdoc).

North Carolina State University (NC State), with some 31,000 students, is in the state capital, Raleigh, a short drive from two other major universities, Duke University in Durham and the University of North Carolina at Chapel Hill. At the center of the triangle formed by the three universities is Research Triangle Park, the largest research park in the world, home to over 39,000 employees working for more than 150 organizations. The three-city area, known since the 1950's as the Research Triangle, has become one of the major high-tech and biotechnology centers of the United States.

Some readers may wonder why North Carolina has both a "North Carolina State University" and a "University of North Carolina." Such pairs exist in many other states: Michigan State University and the University of Michigan, and Florida State University and the University of Florida, are two examples. In some states the names are different: Purdue University and Indiana University, for example. In each case the second is a classic arts-and-sciences university on the European model. The first is an American innovation that traces back to the Morrill Act of 1862. That law provided federal land grants to fund institutions that "without excluding other scientific and classical studies," would "teach such branches of learning as are related to agriculture and the mechanic arts, ... in order to promote the liberal and practical education of the industrial classes." Today the land-grand universities typically include agriculture and engineering schools; NC State also has schools of textiles, veterinary medicine, and natural resources (previously forestry). NC State students have been known to embrace their heritage by driving to the annual NC State-North Carolina football game on a mud-covered tractor.

The presence of two other major universities and many other research enterprises, and NC State's technical orientation, have influenced the development of dynamical systems at the University. Faculty members, the year they joined the department, and their interests are described below.

Raimond Struble (1958) and Anthony Danby (1965)

Struble and Danby are the fathers of dynamical systems at NC State. In the 1960's Struble and his students, such as Joe Marlin, who joined the faculty in 1964, wrote a series of papers on periodic and almost periodic solutions of the pendulum and Duffing equations and other nonlinear oscillators. Struble's textbook on differential equations [1], was praised by J. P. LaSalle in Math Reviews [2] as "a useful step from the out-of-date undergraduate texts still being used to the excellent advanced books that have been written in recent years." Tony Danby was an exhuberant practitioner of celestial mechanics. Of his textbook [3], R. G. Langebartel wrote in Math Reviews [2]: "The book is set apart by virtue of its plentiful supply of illustrations taken from items of current interest."

Robert H. Martin Jr. (1970)

Martin helped develop the theory of differential equations in Banach spaces. According to V. Lakshmikantham in Math Reviews [2], Martin's textbook [4], "gives an excellent account of the current progress in the field of abstract differential equations and it is warmly recommended."

John Franke and Jim Selgrade (both 1973)

Franke was a student of Bob Williams at Northwestern, hence part of the "Smale school" of dynamical systems; Selgrade studied under Charles Conley at Wisconsin, hence was a charter member of the "Conley school." The Smale and Conley groups had been developing their approaches to dynamical systems since the mid 60's with minimal contact. (Conley's interest in the general theory of dynamical systems grew out of his earlier work in celestial mechanics; the second and third papers he published, both on celestial mechanics, were reviewed in Math Reviews [2] by Danby.) Franke and Selgrade worked on bringing together the ideas of the two groups. See for example [5] and [6], bringing together the concepts of hyperbolicity, which was fundamental to the approach of the Smale school; basic sets which were defined by Smale; and chain-recurrence, a central notion of Conley.

Motivated by problems in chemistry and biology, Martin and Selgrade began in the 1970's to work on dynamical systems that preserve a partial order. See [7] and [8].

Steve Schecter (1975)

I was a student of Smale at Berkeley during his mathematical economics period. During my first few years in the department, Jim Selgrade, John Franke and I drove every week to Chapel Hill to participate in Sheldon Newhouse's seminar. In 1983 I read the book of Guckenheimer and Holmes and was hooked: who knew that you could calculate an integral and prove the existence of a horseshoe? I wrote a paper on how to calculate Melnikov integrals when an equilibrium has a zero eigenvalue [9]. A year or two later, Michael Shearer, whose field is partial differential equations and who had joined the Mathematics Department in 1985, asked me to look at a problem about traveling waves that was bothering him. Traveling waves for PDEs are represented by solutions of ODEs that connect equilibria. The bothersome aspect of Shearer's problem turned out to be that in order to see how the connections broke as parameters varied, you had to calculate a Melnikov integral at an equilibrium with a zero eigenvalue. Thanks to Shearer's question, my subsequent career has centered around traveling waves and shock waves.

Moody Chu (1982)

Moody Chu studied with T. Y. Li at Michigan State. He is a numerical analyst much of whose work uses insights and methods from dynamical system. One of his first papers gave a novel homotopy method for computing eigenvalues [10] Another early paper investigated the relationship between the QR algorithm and the Toda flow, and used a center manifold for the Toda flow to understand convergence the QR algorithm [11]. In the review articles [12] and [13] Chu gives a general framework for using matrix differential equations to understand iterative methods in numerical linear algebra.

Xiao-Biao Lin (1988)

Lin studied with Jack Hale at Brown and was a postdoc with Shui-Nee Chow at Michigan State. Lin's work at the time he came to NC State combined several ideas in a compelling way. Hale and Chow had shown how to view homoclinic and heteroclinic solutions in a function-space setting, using Lyapunov-Schmidt reduction and a reinterpretation of the Melnikov integral. Ken Palmer added the idea of systematically using exponential dichotomies. Palmer also constructed interesting solutions near homoclinic and heteroclinic solutions by concatenating pieces of the known solutions to approximate them, and then eliminating the jumps between pieces by means of a shadowing lemma. Lin combined and extended these ideas to produce a powerful approach to studying dynamics near a chain of heteroclinic solutions. See [14]. Lin's approach has since been popularized by Bernold Fiedler and Bjorn Sandstede under the name "Lin's method."

Around the same time Lin gave a beautiful treatment of singularly perturbed boundary value problems using similar ideas [15]. Lin views the truncation of an asymptotic solution to a singular perturbation problem as an approximate solution with jumps. If some hyperbolicity is present, a true solution can be produced nearby using a shadowing lemma. Lin's subsequent work has extended these ideas to partial differential equations.

Biomathematics group

Since the 1960's, NC State has had a Biomathematics Graduate Program, with faculty drawn from many departments. Steve Ellner (1986-2000), who studied with Si Levin at Cornell, was a mainstay of the Biomathematics Program until he left to take Levin's old job as Professor of Ecology and Evolutionary Biology at Cornell. Franke and Selgrade have both been increasingly involved with this program and with work in mathematical biology.

Franke's work has involved discrete dynamical systems that arise in population dynamics, ecology, and epidemiology. One of his students, Abdul-Aziz Yakubu, is also well-known in this area, and is presently chairman of the Mathematics Department at Howard University.

Selgrade has worked on problems of population genetics, starting in collaboration with Gene Namkoong, a forest geneticist at NC State. In a recent paper [16], he and James H. Roberds of the U.S. Forest Service investigate a model for gene transfer from a genetically modified population into a natural population. They determine the positions of attractors and the rate of approach to attractors, and conclude that certain transgenes will maintain themselves in the natural population. In work that started as a collaboration with a researcher at the Chemical Industry Institute for Toxicology in Research Triangle Park, Selgrade has also studied delay differential equations that model hormonal control of the menstrual cycle [17].

Three other Mathematics Department faculty who have major roles in the Biomathematics Graduate Program are Sharon Lubkin (1997), Mette Olufsen (2001), and Alun Lloyd (2003), the program's current director. All have substantial dynamical systems backgrounds. Lubkin studied with Richard Rand at Cornell and was a postdoc with J. D. Murray at the University of Washington. Olufsen was a postdoc with Nancy Kopell at Boston University. Lloyd studied with Robert May at Oxford and was a postdoc with Martin Nowak at the Institute for Advanced Study.

Lubkin's work focuses on mechanical problems in the dynamics of tissues as they grow, change shape, and differentiate. Olufsen studies the dynamics of the cardiovascular system using fluid dynamics and system-level models. Some of her recent work deals with changes in blood flow as one goes from sitting to standing. Lloyd studies the dynamics of infectious diseases. He uses both deterministic and stochastic models to understand the behavior of diseases like measles and dengue. In particular, he uses coupled ODE models to mimic city-to-city spread, and network models to study the importance of spatial and social structure.

Newest faculty

Three recent additions to the loose group of dynamical systems researchers at NC State represent new directions.

Dmitry Zenkov (2001) studies geometric mechanics, especially nonholonomic systems and discrete syststems. Readers may be familiar with his expository article with his advisor Tony Bloch and Jerry Marsden [18].

Karen Daniels (Physics, 2005) who did her postoc with Bob Behringer at Duke, is an experimental nonlinear dynamicist. Her current experiments address granular materials, surfactant-driven gel fracture, instabilities in thin fluid flows, and modeling of geophysical processes such as earthquakes and meteor impacts. The March 31, 2006, issue of Science [19] includes an interview with Daniels about an experiment done with Behringer in which vigorous shaking caused a can of plastic beads to freeze into a crystal-like order. She holds an NSF CAREER award.

Jie Yu (Department of Civil, Construction, and Environmental Engineering, 2006) was a postdoc with Chris Jones at the University of North Carolina. She studies near-shore ocean phenomena such as sand bars, rip currents, and wave-current interactions.

Dynamics seminars

Two long-term seminars deal with dynamical systems. The Differential Equations Seminar in the Mathematics Departments covers dynamical systems and partial differential equations. The seminar of the Biomathematics Program alternates student speakers one semester and invited speakers the next. Many of the topics revolve around dynamical systems, including disease models, dynamics of ecological systems, neural dynamics, and dynamic aspects of biomechanics.

Since Fall 2005, Chris Jones of the University of North Carolina at Chapel Hill, Xiao-Biao Lin, and I have run a working seminar on traveling waves and geometric singular perturbation theory. The impetus was the arrival of two postdocs, Vahagn Manukian and Anna Ghazaryan, who had just finished Ph.D.'s with Bjorn Sandstede at Ohio State; Manukian is at NC State, Ghazaryan at both UNC and NC State. The seminar has evolved into the go-to place for your traveling wave needs. For example, Shearer presented work he has been doing with his former student Rachel Levy, now at Harvey Mudd College, and Tom Witelski of Duke on traveling waves in thin film flows with a surfactant. After his presentation, Manukian and I developed a reinterpretation and extension of their work based on geometric singular perturbation theory. Karen Daniels has begun doing experiments on these flows in her lab in the Physics Department.

References

[1]		R.A. Struble, Nonlinear Differential Equations, McGraw-Hill, New York 1961.
[2]		MathSciNet, Mathematical Reviews on the Web, http://www.ams.org/mathscinet/, American Mathematical Society, 2008.
[3]		J.M.A. Danby, Fundamentals of Celestial Mechanics, The Macmillan Company, New York, 1962.
[4]		R.H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley, New York, 1976.
[5]		J.E. Franke and J.F. Selgrade,"Abstract omega-limit sets, chain recurrent sets, and basic sets for flows," Proc. Amer. Math. Soc. 60 (1976), 309-316.
[6]		J.E. Franke and J.F. Selgrade, "Hyperbolicity and chain recurrence," J. Differential Equations 26 (1977), 27-36.
[7]		R. H. Martin, "Asymptotic stability and critical points for nonlinear quasimonotone parabolic systems," J. Differential Equations 30 (1978), 391-423.
[8]		J. F. Selgrade, "Asymptotic behavior of solutions to multiple loop positive feedback systems," SIAM J. Math. Anal. 12 (1981), 669-678.
[9]		S. Schecter, "The saddle-node separatrix-loop bifurcation," SIAM J. Math. Anal. 18 (1987), 1142-1156.
[10]		M.T. Chu, "A simple application of the homotopy method to symmetric eigenvalue problems," Linear Algebra Appl. 59 (1984), 85-90.
[11]		M.T. Chu, "The generalized Toda flow, the QR algorithm and the center manifold theory," SIAM J. Algebraic Discrete Methods 5 (1984), 187-201.
[12]		M.T. Chu, "Matrix differential equations: a continuous realization process for linear algebra problems, Nonlinear Anal. 18 (1992), 1125-1146.
[13]		M.T. Chu, "Linear algebra algorithms as dynamical systems," Acta Numerica, to appear.
[14]		X.-B. Lin, "Using Melnikov's method to solve Silnikov's problems," Proc. Royal Soc. Edinburgh 116A (1990), 295-325.
[15]		X.-B. Lin, "Heteroclinic bifurcation and singularly perturbed boundary value problems," J. Diff. Equations 84 (1990), 319-382.
[16]		J.F. Selgrade and J.H. Roberds, "Global attractors for a discrete selection model with periodic immigration," J. Difference Equ. Appl. 13 (2007), 275-287.
[17]		L. Harris Clark, P.M. Schlosser and J.F. Selgrade, "Multiple stable periodic solutions in a model for hormonal control of the menstrual cycle," Bulletin of Math. Biology 65 (2003), 157-173.
[18]		A.M. Bloch, J.E. Marsden, and D.V. Zenkov,"Nonholonomic Mechanics," Notices of the AMS 52 (2005), 324-333.
[19]		A. Cho, "American Physical Society Meeting: In a Jumble of Grains, a Good Hard Shake Restores Order," Science 311, no. 5769 (2006) 1860 - 1861.

A birthday party invitation There have been several dynamical systems conferences at NC State, most notably a 1997 CBMS conference on Dynamic Systems in Structured Population Dynamics with Jim Cushing of the University of Arizona as the main speaker. The next is a conference, The Geometry and Analysis of Dynamical Systems, on February 22-23, 2008. The excuse is the sixtieth birthdays of Xiao-Biao Lin and myself, which happen to be a week apart. We hope to see you there!

Xiao-Biao Lin and author Steve Schecter.