Research Interests
My work focuses both on the mathematical theory for singularly
perturbed dynamical systems and on applications to problems
arising in pattern formation, fluid mechanics, systems of
mechanical oscillators, and neurophysiology.
Pattern formation in chemistry
(Joint work with Arjen Doelman, Wiktor Eckhaus, Rob Gardner, Paul Zegeling)
Over the past 25 years, the field of pattern formation
has undergone a revolution fueled by experiments in chemistry
and fluid mechanics. Perhaps one of the most intriguing
outstanding questions is to understand the phenomena of
self-replicating spots and pulses, which were discovered
in 1993 on the Gray-Scott model by J. Pearson and H. Swinney.
Spots and pulses are regions of high activator and low inhibitor
concentration. In these autocatalytic reactions, the spots and
pulses can shrink, grow, oscillate in size, and even self-
replicate, depending on the rate at which the inhibitor is
supplied to the reactor, on the rate at which the activator
decays, and on the relative diffusivities. Visually, self-
replication looks much like cell reproduction: a spot enlarges,
its length growing perpendicular to the direction of its motion,
and then pinches into two. Mathematically, the Gray-Scott model
consists of a pair of coupled reaction-diffusion equations.
My collaborators and I establish the existence and stability
of pulses. This analysis identifies the regimes in which stationary
and slowly-modulating pulses are observed, and it establishes
the boundaries (both external and internal) of the self-replication
regime. In addition, we identify the mechanism in 1-D by which a
one-pulse solution splits into a two-pulse solution, which is the
fundamental event during self-replication, and show that there is
a large class of coupled reaction-diffusion equations that exhibit
the same type of phenomena as the Gray-Scott model does. Finally,
with a Ph.D. advisee of mine, D. Morgan, we prove the existence
and stability of spatially-periodic patterns that are the attractors
in the self-replication regime.
Pattern formation in bacterial colonies
Jointly with our Ph.D. thesis advisee G. Medvedev, N. Kopell and I
have investigated the periodic-terracing of Proteus mirabilis bacteria
observed in experiments by J. Shapiro of the University of Chicago.
This fascinating phenomena involves the dynamics of single-cell and
multi-cellular forms of the bacteria, and the collective evolution
of the colony as it alternately expands and consolidates. We identify
waves of diffusivity as being essential switching mechanisms from
the consolidation phase to the expansion/swarming phase.
Bubble dynamics
Applications of existing dynamical systems techniques and the development
of new ones for problems in fluid mechanics has been another ongoing theme.
Jointly with A. Harkin and A. Nadim, we study gas bubbles in liquids,
focusing in particular on the acoustic cavitation of bubbles and
on the nonlinear interactions of bubbles in small clusters. Cavitation
occurs when bubbles expand and pop in response to changes in the
ambient acoustic pressure. We have taken one of the first steps
beyond the 50-year-old quasistatic criterion for cavitation and
developed a criterion for the onset of dynamic cavitation, which
occurs when the external pressure changes on the same time scale
as that of the natural breathing oscillations, for slightly
subcritical bubbles. Presently, we are
investigating the coupled oscillations of two bubbles and the nonlinear
interactions between shape and breathing modes. In both cases, there
are resonant and nonresonant mechanisms for energy transfer, and we
endeavor to find thresholds for them.
Mathematical theory for singularly perturbed systems
Singularly perturbed systems have a rich geometry, and the
overarching goal of my work has been to uncover and exploit it.
Work of my first Ph.D. student, C. Soto-Trevino (1997), develops
a geometric method to prove the existence of periodic orbits in
these systems. I have also helped to develop, with C. Jones and N. Kopell,
the exchange lemma, which is a major theoretical tool for tracking
invariant manifolds in singularly perturbed systems. In addition,
I have analyzed applications to resonant dynamics in various N-degree-
of-freedom Hamiltonian systems (with G. Kovacic), to rapid oscillations in
coupled mechanical oscillators (with S. Weibel and J. Baillieul), and
to 2-point boundary value problems (with M. Hayes and N. Kopell).
Adiabatic Hamiltonian systems
Separatrix-swept regimes are the regimes complementary to those in which
the famous theory of adiabatic invariants (discovered by Einstein and Alfven)
applies. I discovered numerically (and analyzed mathematically) the rich
structure of the homoclinic tangles responsible for chaos in these systems.
I have also focused on a number of applications in fluid mechanics
of the mathematical
theory for these systems: to develop a mixing theory
for low Reynolds number flows
and to develop a scattering theory for oceanic internal waves.
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