List of researchers in CAMS working on problems related to Dynamical Systems: Aubry, Blackmore, Bose, Elmer, Goldman, Golowasch, Jiang, Kapraff, Kondic, Kriegsmann, Matveev, Miura, Moore, Nadim, Papageorgiou, Siegel, Tao, Tavantzis, Wang, Young.

Today's research in the theory and applications of dynamical systems all has its roots in the work of early innovators in differential equations and mathematical modeling such as Newton, the Bernoullis, Euler, Lagrange, Laplace, Legendre, Gauss, Cauchy, Abel, Fourier, Liouville,Weierstrass, Dirichlet, Hamilton, and Riemann. But we have come a long way since the middle of the nineteenth century in terms of our understanding and the variety of applications of both finite-dimensional dynamical systems (ordinary differential equations) and infinite-dimensional dynamical systems (partial differential equations).

A major revolution in dynamical systems research took place during the late nineteenth and early twentieth century characterized by innovations in the study of integrability such as those of Kovalevskaya, and culminating in the ground-breaking work of Poincare on nonintegrable Hamiltonian systems. Poincare brought a new infusion of topological methods to dynamical systems research that has illuminated and served as a source of inspiration for virtually all subsequent investigations. In the process he introduced a new perspective on nonlinearity and complex motion that predated chaos theory. This new topological trend continued and was greatly advanced by such notables as Birkhoff, Kolmogorov, Arnold and Moser.

Then in the 1960's the face of dynamical systems research was dramatically altered by Smale and others with the introduction of a variety of techniques from differential topology that provided amazing new insights into the nature of chaotic dynamics. At about the same time, a dramatic advance in research on infinite-dimensional Hamiltonian systems was occuring as a result of several extraordinary discoveries concerning integrability, solitons and the inverse scattering transform made by the likes of Gardner, Greene, Kruskal, Lax, and our own Robert Miura. These remarkable breakthroughs established the foundations of what has come to be known as the modern theory of dynamical systems, and catalyzed an explosion of applied and fundamental research in nonlinear dynamics.

Dynamical systems research in CAMS has a decidedly applied focus, and is extremely active in a wide and diverse range of areas including mathematical biology, fluid dynamics, wave propagation, computational topology, nonlinear optics, and quantum field theory and its applications to such things as quantum computing. There are a significant number of researchers who employ techniques from nonlinear dynamics in their work, and a smaller but sizeable core group whose interests are centered around dynamical systems and their applications. One of the most appealing aspects of research in dynamical systems is the wealth of opportunities it provides for interdisciplinary studies, and our dynamical systems group is among the most active in such efforts.

CAMS research in dynamical systems can be described briefly as follows: Nadine Aubry uses methods from dynamical systems to characterize fluid flows and how they can be controlled by a variety of mechanisms such as the placement of vortex configurations. Denis Blackmore applies nonlinear dynamics to study the motion of vortices and vortex filaments in fluids and particles in granular flows, the chaotic evolution of biological populations, the computational topological nature of certain geometric objects, and quantum computing. He also does fundamental research in bifurcation theory, chaos theory, and algebraic and differential integrability analysis of infinite-dimensional Hamiltonian dynamical systems. Amithaba Bose employs dynamical systems techniques in his studies of coupled neuronal oscillators; in particular, he uses geometric singular perturbation theory to affect reductions in dimension of high dimensional systems, so that they can be more readily analyzed using such techniques as Poincare maps. Recently, he has studied the global effects of localized neuronal activity with regard to phase relationships and multi-stability.

Christopher Elmer's research focus is on both finite- and infinite-dimensional time dependent differential equations that are discrete in space. He employs both analytical and computational methods to analyze the regular and singular behavior of these systems, and to interpret his findings with regard to several physical applications of these types of systems. Daniel Goldman uses a variety of techniques from dynamical systems theory to study dynamical systems arising from the modeling of fluid mechanical phenomena related to biological applications. Jorge Golowasch employs approaches from nonlinear dynamics to investigate the cellular mechanism of activity-dependent regulation of ionic currents, neuronal excitability, and neural network activity. Dynamical systems methods applied to nonlinear waves and optics is the focus of Roy Goodman's research. A key ingredient in his work is the development of methods for obtaining insights from finite-dimensional reductions of infinite-dimensional systems such as the nonlinear Schrodinger equation.

Shidong Jiang applies methods from nonlinear dynamics in his research on mathematical fluid dynamics, and wave propagation. Jay Kappraff has used dynamical systems techniques to uncover interesting relationships among regular geometric figures, matrix groups, chaotic regimes, and fractal geometry. Lou Kondic employs a variety of dynamical systems approaches in his research on interfacial fluid dynamics, and granular flows. Gregory Kriegsmann's research in applied mathematics has involved the application of bifurcation theory and differential equation techniques in several problems related to wave propagation and electromagnetics. Victor Matveev's work in computational neuroscience, stochastic process theory, and statistical mechanics has employed several methods from nonlinear dynamics. In his research on the kinetic theory of gases, mathematical biology, interfacial surface tension, and direction reversal in Brownian motion, Robert Miura has employed a variety of techniques from dynamical systems theory. For example, some of his recent work in mathematical biology has made use of the theory of Hopf bifurcations and saddle-node bifurcations.

Richard Moore studies nonlinear wave equations with both deterministic and stochastic perturbations with the aid of a variety of techniques from dynamical systems theory. Cyrill Muratov studies, among other things, traveling wave solutions and propagation phenomena in gradient reaction-diffusion systems using both variational and dynamical systems methods. He also studies several other types of infinite-dimensional dynamical systems arising from areas such as mathematical biology and fluid dynamics. Farzan Nadim makes liberal use of techniques from nonlinear dynamics in his research in computational and analytical neuroscience. Demetrius Papageorgiou employs ideas from infinite-dimensional dynamical systems theory, such as inertial manifolds and chaotic dynamics, in his research in fluid dynamics. Nonlinear dynamical techniques related to vortex dynamics play a key role in some of Michael Siegel's research in fluid dynamics. Louis Tao employs methods from dynamical systems theory in his work in neuroscience and mathematical biology. John Tavantzis has been investigating relationships between biologically generated time series and dynamical systems models for the associated phenomena, and also methods for capturing periodic orbits. Sheldon Wang has made several contributions to the literature in applications of dynamical systems, and is currently working on the development of methods for capturing periodic orbits of finite-dimensional dynamical systems. Yuan-Nan Young uses a variety of nonlinear dynamics approaches in his research in fluid dynamics and complex systems.