My research blends
mathematics with
other disciplines. I study
fundamental mathematical and algorithmic questions that arise
from the investigations in these other disciplines, and I
engage
in research that attempts to answer questions within disciplines
outside of mathematics, especially in the life sciences. The core of my
interests is dynamical systems theory, the study of long time
properties of solutions to ordinary differential equations and
generalizations of ordinary differential equations. Bifurcations, the
qualitative changes in system behavior that occur when varying
parameters, have
been a continuing theme throughout my research. I have investigated
bifurcation theory, algorithmic
methods for computng bifurcations and the characterization of
bifurcations in applied problems. Another theme in my work has been the effects of multiple time scales in dynamical
systems. Systems with multiple time scale systems are often referred to
as singularly perturbed systems or slow-fast systems.

- Neuroscience and Locomotion

I study dynamical models of a small neural system, the stomatogastric
ganglion of crustaceans, in collaboration with the laboratory of Ronald
Harris-Warrick. The STG is a *central pattern
generator, *a
group of neurons that control movement. Through this research, we hope
to learn more about neuromodulation, the ways in which the rhythmic
output
of the STG is modified by chemical and electrical inputs. The models we
investigate are systems of differential equations that describe the
currents
contributing to the membrane potential of each neuron in the network.
The
interaction of these currents is complex, so simulation and analysis of
these models is used to predict
what the effects of varying the conductances or other properties of the
currents will be on the oscillations of an individual cell or the
entire
network.

We have also work on modeling neural networks in mouse spinal cord that control locomotion. Our focus has been on the role of intercomissural neurons, neurons whose axons cross the spinal cord. These neurons play a fundamental role in coordinating motion of the right and left limbs of the animal. Rhythmic behavior ("ficitive locomotion") is elicited and recorded in a spinal cord preparation by bath application of neuromodulators. The laboratory uses genetic techniques to identify neurons in the spinal network, investigate their electrophysiological properties and determine the effects of their modification on the network rhythms. Experimental data provides the foundation for network models that we use to gain insight into how the rhythmic activity of this network arises from the intrinsic properties of its individual neurons and their synaptic interactions with each other.

I particpated in an NSF sponsored FIBR
project led by Robert Full to
study animal locomotion. The other senior investigators were Philip
Holmes, Dan Koditeschek and John Miller. The project investigated
dynamical principles used by cockroaches to control locomotion. A
set of hypotheses
about locomotion were proposed that we studied using animals, robots
and computer
models. Our starting point was the observation many animals have
evolved
behaviors that resemble a biped whose legs act like pogo sticks, or
spring loaded inverted pendulua. Insects with six legs frequently run
with a gait in which tripods of legs move together. To the
extent
that these tripods are coordinated to act like inverted pendula, there
is a dimension reduction that makes the system easier to control.
We hypothesized that neural control acts on this mechanical
plant
to modify its intrinsic behavior as contrasted with directing all
aspects of the motion of legs as is done with many legged robots. Shai
Revzen, a former student in the Full lab, and I analyzed experimental
data to assess whether cockroaches stabilize their movement around a
periodic orbit as predicted by the theory.

More recently I have joined Jane Wang and Itai Cohen in
investigations of insect flight, primarily with fruit flies of the
genus Drosophila. HIgh spped
films are made in Cohen's laboratory of freely flying insects. In some
experiments, the insect is stimulated, for example by gluing a small
iron thread to the animal and giving a brief magnetic pulse that exerts
a torque via the iron thread. A goal of the research is to infer the
mechanical forces that the insect uses in executing the maneuver that
follows a stimulus. Typically, a fly reorients its direction to the
original flight path within about one tenth of a second. Our analysis
draws upon two advances that were developed by the group before I began
working with them. First, Jane developed a quasisteady model for the
forces acting on a plate moving through a fluid. These models
approximate the fluid forces acting on the plate as a function of the
veolcity and angular velocity of the plate. Second, automated tracking
algorithms were created to reconstruct the motion of a fly as a
rigid body with two rigid wings attached to the body via universal
joints. We model the fly as such a finite degree of freedom mechanical
device. It executes a maneuver by exerting torques on its wings with
its muscles. A key part of the analysis is to infer these torques by
applying the quasisteady model to the measured flight paths of flies,
using an inverse dynamics calculation to determine the combined fluid
forces and muscle torques that produced the observed motion. We find
that small asymmetries of the order of ten degrees in variables such as
wing pitch or stroke amplitude are sufficient to produce the observed
maneuvers.

__Periodic Orbits and Their Bifurcations__

Periodic orbits are fundamental objects within dynamical systems. They represent cyclic processes in dynamical models of biological systems. Attracting periodic orbits can be found by simulation; i.e., solving initial value problems for long times. However, simulation alone is insufficient to obtain all of the information that we would like to have about periodic orbits. There are many circumstances in which it is desirable to have methods that compute periodic orbits directly. The algorithms used by such methods are examples of boundary value solvers. In addition to solving for individual periodic orbits, the methods enable one to determine local stability information, compute sensitivity information that describes how the orbit changes with perturbations of parameters, identify parameter values at which bifurcations occur and track families of periodic orbits as parameters vary. Our contributions in this area have centered on the use of a technique called automatic differentiation in the solution of the boundary value problems. Automatic differentiation enables the computation of derivatives of functions without the truncation errors that arise from the use of finite differences. This facilitates the implementation of methods of very high order that are conceptually simple and utilize coarse meshes in discretizing the periodic orbits. Another aspect of our research on periodic orbits has been to generate rigorous proofs of the qualitative properties of numerically computed dynamical systems.

__Dynamical
Systems with Multiple Time Scales__

Multiple time scales present challenges in the simulation of
dynamical
systems. Resolving the dynamics of fast time scales while computing
system
behavior for long times on slower time scales requires special i*mplicit*
algorithms
that make assumptions about the character of the fast time dynamics.
There are also qualitative features of the dynamics in systems with
multiple
time scales that do not appear in systems with single time scales. My
research
is directed at extending the qualitative theory of dynamical systems to
apply to systems with multiple time scales. Much of this work
began as a collaborative effort with Kathleen Hoffman and Warren
Weckesser.
We studied the classical forced van der Pol system intensively.
The pheomenon of chaos in dissipative dynamical systems was first
discovered and analyzed in this example by Cartwright and Littlewood
during the 1940's and 1950's. Our work and that of my former student
Radu Haiduc simplifies and extends the results of Cartwright and
Littlewood. Haiduc's thesis establishes parameter values for which the
forced van der Pol equation has a chaotic invariant set and is
structurally stable. In joint workwith Martin Wechselberger and
Lai-Sang Young, I explored the occurrence of chaotic attractors in
concrete vector fields. We developed a general theory relating
"Henon-like" attractors of two dimensional diffeomorphisms to three
dimensional flows with two slow and one fast variable, and then
illustrated the theory with a modification of the forced van
der
Pol system.

Mathieu Desroches, Bernd Krauskopf, Christian Kuehn, Hinke Osinga,
Martin Wechselberger and I have systematically brought multiple time
scale methods to bear on understanding mixed mode oscillations. There
is no precise definition of mixed mode oscillations, but they have been
identified in systems of chemical reactors, neural oscillations and in
other application domains. We give examples of slow-fast systems where
the interactions of slow and fast time scales leads to small
oscillations that are not apparent in the dynamics of either the slow
or the fast time scales individually. This work analyzes several
example more extensively than previous studies and includes a
comprehensive list of references about mixed mode
oscillations.

Invariant Manifolds

Computation of invariant manifolds of dynamical
systems with
dimension larger than one has proved to be a challenging problem. There
are two sources of the difficulty in computing these manifolds. First,
they have an intricate geometric structure with folds and spiraling
structures. Second there is "geometric stiffness" reflected in
trajectories that evolve with very different speeds. Thus, attempts to
compute the manifolds by solving initial value problems for a few
trajectories tend to produce poor results because the trajectories
don't spread uniformly, and where they do spread they diverge rapidly
enough to leave large gaps in the manifolds. Alex Vladimirsky
and
I have used "ordered upwinding" methods from the numerical
analysis of partial differential equations to develop and implement
efficient methods for computing these manifolds. The methods as we have
developed them are low order, but very fast.
Several research groups have introduced different approaches to the
computation of invariant two dimensional stable and unstable manifolds
of equilibrium points of vector fields. Krauskopf et al. give a survey
of these methods using computation of the unstable manifold of the
origin in the Lorenz systems as a benchmark problem.

Invariant manifolds also play a key role in systems with multiple
time scales. Christian Kuehn and I developed an algorithm for computing
slow manifolds of saddle-type. These manifolds are the subject of the
Exchange Lemma for slow-fast systems, and they are an important
component of classes of homoclinic orbits of systems with multiple time
scales. Homoclinic orbits arise as traveling wave profiles of partial
differential equations. Traveling waves of the FitzHugh-Nagumo model
for action potentials have been investigated intensively. Our algorithm
yields improved calculations of these periodic orbits. This work also
highlights how tangential intersections of invariant manifolds underlie
bifurcations that create both homoclinic orbits and mixed mode
oscillations.