John Guckenheimer, Ph.D. - US grants

This project supports resear5ch in dynamical systems with significant emphasis on computation. The specific projects discussed in the proposal are: 1. Hilbert's 16th Problem: Numerical and symbolic computations to obtain bounds on the number of limit cycles possessed by polynomial vector fields in the plane. 2. Dynamical studies of vector fields derived from equations of fluid boundary layers. 3. The implementation of perturbation methods for ordinary differential equations problems using MACSYMA. 4. The study of multiparameter systems of differential equations near points of multiple bifurcation. 5. Numerical studies of the complex Henon mapping, holomorphic diffeomorphism of complex two space. 6. Symbolic computation of polynomial knot invariants associated to periodic orbits of three dimensional flows. More broadly, the project will support the development of algorithms and efficient computing environments for the studies of dynamical systems.

John Guckenheimer will carry out research in a number of different areas. These include one dimensional mappings, quadratic vector fields in the plane and multiple bifurcations. He has already established himself as an international leader in all of these areas. The current proposal is a continuation of this very successful program. Over the last decade there has been a rapid growth in the mathematical theory of iterations of one dimensional mappings. This theory has been successfully applied in such diverse areas as the theory of chemical reactors, the dynamics of semiconductor devices and the transition to chaotic behaviour in fluid dynamics. Guckenheimer has a good record of interaction with scientists in other fields. His research into multiple bifurcations will be aimed at classifying types of degenerate bifurcations which occur near quadratic Hamiltonian systems. This will be used to find solutions to a restricted form of Hilbert's 16th problem.

This is a grant under the Scientific Computing Research Equipment for the Mathematical Sciences program of the Division of Mathematical Sciences of the National Sciences Foundation. This program supports the purchase of special purpose computing equipment dedicated to the conduct of research in the mathematical sciences. This equipment is required for several research projects and would be difficult to justify for one project alone. Support from the National Science Foundation is coupled with discounts and contributions from manufacturers and with substantial cost-sharing from the institutions submitting the proposal. This program is an example of academic, corporate, and government cooperation in the support of the basic research in the mathematical sciences. This equipment will be used to support three research projects in the Center for Applied Mathematics of Cornell University: dynamical systems theory, the investigators are Guckenheimer and Philip Holmes; statistics of image processing, the investigator is Richard Liu; three dimensional geometry, the investigator is Robert Connelly.

In addition to the mathematical aspects of the work, the project has the potential for wide impact on the scientific community through the development of a comprehensive computer environment for the exploration of dynamical systems. The intention is to provide the user of this environment with interactive, efficient access to all currently available algorithms for studying the dynamics of general systems together with the tools for effectively managing the data produced in these studies. A prototype for such an environment has been developed in the context of the first problem listed above, but further work is required to produce a package that meets the needs of a broad spectrum of applications and users. These further developments will occur in conjunction with studies of problems of significant theoretical interest.

The principal investigators undertake studies of a variety of problems in mechanics and biology, using techniques of nonlinear dynamical systems. The topics include the dynamics of dissipative partial differential equations and pattern formation in fluid systems, global dynamics of symmetric systems, knots and orbit genealogy in three-dimensional flows, dynamics of oscillators with slowly varying potentials, modelling of neurobiological oscillators, iterations of one-dimensional mappings, nonlinear stability and bifurcation in Hamiltonian and weakly damped systems. Arnold diffusion in four-dimensional symplectic maps, and dynamics of the growth of certain plants. These topics have applications in the design and development of "smart" materials, in aerospace (particularly satellites and spacecraft), in the control of turbulent boundary layers,and in biology.

This project will bring twelve undergraduate students to work at the Cornell Theory Center for a ten week period in the summer of 1992. Recruiting will be national in scope, and there will be special efforts to bring in women, minorities, and students from institutions without rich computing resources. Participants will work under the joint guidance of a faculty member from Cornell University, and a member of the scientific and technical staff of the Theory Center. Problems in the area of computational science research will be investigated.

A major challenge in neural network research is to build models that are simple enough for analysis but describe the biological behavior of the network and can be tested experimentally. Recent experimental and theoretical studies of small neural networks make the building of such models possible. In the proposed research we will build a detailed biological model of the pyloric Central Pattern Generator of the lobster. This model will be an extended version of the Hodgkin-Huxley model which includes the important currents and synaptic connections. Using methods from dynamical system theory and physical approximation we will obtain a simpler model which still describes the experimental results. A phase model will be developed from the full model. The effects of synaptic modification and changes of intrinsic properties of the neurons will be tested, and the effect of neurotransmitters will be investigated. The theoretical study will combine analytical and numerical methods from neural network and dynamical system theory, numerical simulations and will be compared to neurobiological experiments. The research will contribute to bridging the gap between regular neural network models and the biological reality.

This project will explore the dynamics of a small neural network. There are three major aspects to the project: 1. modelling the neural network as a system of nonlinear differential equations with explicit representation of ion channels for each neuron within the network 2. experiments on the pyloric circuit of the stomatogastric ganglion of spiny lobsters that will be used to provide input data for parameter values of the models and to test predictions of the models 3. mathematical research to develop algorithms that will lead to tools for the explicit computation of geometric features within the models A mathematical software package called dstool, developed at Cornell for the exploration of dynamical systems, will be used, and will incorporate algorithmic advances into this environment. The initial emphasis will be the pacemaker neuron within the circuit. Guckenheimer will investigate how its response depends upon multiparameter variations of its environment that come from application of channel blockers and neuromodulators and from changes in extracellular ionic concentrations. Data from previous studies of synaptic efficacy and their dependencies upon amines will be used in modelling the interactions of neurons within the network.

9322321 Guckenheimer The National Science Foundation's initiative in high performance computing has had a tremendous impact on scientific opportunities for the national research community. The Theory Center's Supercomputing Program for Undergraduate Research (SPUR) complements this initiative by extending the opportunities of computational research to undergraduate students. By providing students with an early research experience, this program is an effective mean of motivating students to pursue careers and graduate education in mathematics, science, and engineering. Through this program, undergraduate students will investigate problems in an area of computational science research under the joint guidance of a Cornell University faculty member and a member of the Cornell Theory Center's scientific and technical staff. While working on their research projects, the students will not only participate in scientific research but will also learn about computational methods and receive training in high performance computing techniques. The research projects will take advantage of the advanced computational tools of the Theory Center, including the use of scalable parallel processing and visualization tools. The research experience will be enhanced by the interaction among students from diverse institutions with multidisciplinary backgrounds and research projects. Building on longstanding relationships, the Center will aggressively recruit from historically black colleges and women's colleges and also target thoseinstitutions which lack a strong research orientation. Students at colleges and universities with otherwise limited computational research resources will thus have the opportunity to undertake significant research projects as part of their undergraduate education.

9424195 Guckenheimer Under this three-year program, undergraduate students will investigate problems in a variety of areas of computational science research under the joint guidance of Cornell University faculty member and members of the Corand technical staff. While working on their research projects, students will not only participate in scientific research but will also learn about computational methods and receive training in high performance computing techniques. The research projects will take advantage of the advanced computational resources of the Theory Center including scaleable parallel processing and visualization tools. The research experience will be enhanced by the interaction among students from diverse institutions with multi-disciplinary backgrounds and research projects. Building on long-standing relationships, the project will aggressively recruit from historically black colleges and women's colleges; it will also target those institutions which lack a strong research orientation. Students at colleges and universities with otherwise limited computational research resources will thus have the opportunity to undertake significant reseapch projects as part of their undergraduate education. ***

Guckenheimer 9418041 In this multidisciplinary project, the investigator and his colleagues at Cornell University together with Peck at Ithaca College focus upon studies of the stomatogastric ganglion. This small neural network with approximately 30 neurons is a central pattern generator that coordinates movement of the gastric mill and pylorus of crustacea. The investigators construct compartmental models of individual neurons based upon experimental measurements and amalgamate these into models for subnetworks. These systems provide a testbed for the development of algorithms to explore the bifurcations of dynamical systems. Particular attention is given to those aspects of the systems that involve multiple time scales. The goal of the project is to improve understanding of dynamical processes in biological systems, with an emphasis upon neural systems. The investigators work in a multidisciplinary setting, combining experiment and mathematical modeling, giving due attention to both mathematics and neuroscience, and ensuring that the two interact at all times in the work. The larger goals are pursued in the context of specific qystems, namely the study of a small neural circuit that controls rhythmic motions in the foregut of lobsters. This model system has been chosen for its modest size and accessibility for experimental manipulation. The project aims to understand how neural systems switch among motor patterns and how they regulate these patterns. ***

The Center for Applied Mathematics at Cornell University will purchase computer equipment which will be dedicated to the support of research in the mathematical sciences. The equipment will be used for several research projects in mathematics, including in particular: Stochastic spatial models in the biological sciences Computational analysis of dynamical systems Computational research in coding theory Computational Algebraic Geometry Computation of Eigenvalues and Applications

Guckenheimer 9705780 The investigator studies dynamical systems with multiple time scales as models of neural systems. Data and models from a small invertebrate neural network, the stomatogastric ganglion of lobsters, guide the work and provide case studies. Phenomena observed in the stomatogastric ganglion, such as bursting oscillations and spike frequency adaptation, can best be modeled as dynamical systems with multiple time scales. Previous mathematical analyses of qualitative properties of multiple time scale dynamical systems have dealt mainly with local phenomena that occur in low dimensions. The dynamics of neural systems raise questions that are not addressed by existing theories of multiple time scale systems. The aim here is to extend the theory by classifying qualitative features of the global dynamics and bifurcations for systems with two time scales. This is an ambitious endeavor to extend theories of nonlinear dynamical systems and singularly perturbed systems of ordinary differential equations that have been developed over the past thirty years. Using numerical investigations as a guide, the dictionary of patterns that occur in this setting is described and their analytical properties are characterized. This work draws heavily upon the theories of bifurcations of dynamical systems and models of hybrid dynamical systems that combine continuous and discrete components. As needed, numerical algorithms are developed that facilitate the simulation and analysis of multiple time scale systems. The initial emphasis of the mathematical work is upon systems that have two slow variables and two fast variables. Numerical investigations of conductance-based models for the stomatogastric ganglion also are performed. The results of these studies are compared with data and used to guide the refinement of the models. Nervous systems of animals regulate and control muscular activi ty such as locomotion. Well developed theories enable the construction of models for the electrical properties of nerve membranes in these processes, but there is little understanding of the dynamical principles used by organisms. This project investigates dynamical models of a small neural system consisting of fourteen neurons that control rhythmic motions of the foregut of lobsters. This system is used because it is small enough that unique properties of each neuron within the system have been identified, but large enough that the network architecture of the interactions among neurons is also important. The system displays a rich repertoire of rhythmic behavior. The focus of this project is on features of the behavior that involve different time scales. Mathematical theories have been successful at describing universal properties of the dynamics observed in an astounding array of physical and natural systems, but these theories need to be extended to systems with more than one time scale. The goal is to construct classifications of dynamical patterns that are the components from which the complex behaviors of neural systems are formed. Models of these systems are also complex. Computational investigations are required to predict their dynamical behavior. This project seeks to implement algorithms that improve our ability to extract useful information from the models and guide the improvement of their fidelity. Both the theoretical analysis and the numerical methods that are developed encompass all models of dynamical systems with multiple time scales and can be used far beyond the context of the neural system that is the focus of this study.

Research Computing in the Cornell Mathematics Department


The Department of Mathematics at Cornell University will purchase a powerful compute server, storage, backup, workstations with shared facilities and color print services, which will be dedicated to the support of research in the mathematical sciences. The equipment will be used for several research projects, including in particular: the study of various aspects of dynamical systems, computational symbolic algorithms in algebraic geometry and commutative algebra, stochastic spatial models in probability, problems of enumeration in polyhedral combinatorics and hyperplane arrangements, and problems concerning the stability of configurations of points in space under natural distance constraints.

Abstract

The purpose of the VIGRE project at Cornell University is to foster working relationships among undergraduates, graduates, postdoctoral faculty and permanent faculty; to promote the transfer of skills and knowledge between mathematicians inside the Mathematics Department and scientists, engineers and mathematicians outside the Mathematics Department; and to enhance and expand communications and contacts among teachers of mathematics at all levels.

This project consists of four main components:

1. Postdoctoral positions with a one course per semester teaching load will bring top young mathematicians to Cornell to participate in our mathematical research and teaching community. Each postdoctoral fellow will be assigned a faculty mentor and will be exposed to a wide variety of the possible applications of mathematics at Cornell, but they will be completely free to choose the topics of their research.

2. Beginning in the fall of 2000, several new VIGRE trainees each year will receive two years of fellowship support over the first three years of their graduate career. The program will engage the students in mathematical research as early as possible and will allow them to learn about the applications of mathematics in biology, physical sciences, or engineering. Some one-semester trainee awards will be available to other students in the middle to late stages of their academic career. These awards will allow them to pursue applied research or simply to have a free semester to get their thesis research in order.

3. Undergraduates will be supported by a variety of mechanisms. VIGRE funds will make it possible to support Cornell undergraduate math majors during the summer before their senior years in order to facilitate their pursuit of honors programs and writing senior theses in mathematics. A second new opportunity for the undergraduate math major is to do work in collaboration with scientists at a variety of labs on campus. A third benefit of VIGRE funds is that it will allow more Cornell students to participate in the successful REU Program and conversely, it will bring the benefits of the summer REU Program to Cornell math majors during the school year.

4. We will support outreach activities in the local schools in order to expose talented students to the excitement of mathematics and the career opportunities that further study would open for them.
This program is being jointly funded by the Division of Mathematical Sciences and the Office of Multidisciplinary Programs from the Directorate of Physical and Mathematical Sciences.

Guckenheimer
0101208
The investigator studies slow-fast decompositions and
bifurcations of trajectories in dynamical systems with multiple
time scales. This extends the theory of bifurcation in generic
families of dynamical systems to those with two time scales.
Emphasis is placed upon relaxation oscillations, periodic orbits
that have both slow and fast segments. The initial stages of the
work seek a classification of degenerate decompositions appearing
in periodic orbits of one parameter families with relaxation
oscillations. Geometric methods are used to determine
bifurcations associated with each degenerate decomposition.
Numerical investigation of examples is used to motivate the work
and to ensure that the results are directly applicable to
biological models. Together with collaborators Kathleen Hoffman
and Warren Weckesser, the investigator is reexamining a classical
example, the forced van der Pol system that gave birth to the
discovery of chaos for dissipative dynamical systems, and expects
to give a full description of the bifurcations that occur within
this system. He also develops algorithms for the computation of
structures that are difficult to compute with existing methods.
Rhythmic phenomena are ubiquitous in biological systems.
Examples include the heartbeat, the cell cycle, circadian
rhythms, legged locomotion, and electrical signals in the nervous
system. Most of these involve multiple time scales. The
investigator pursues new mathematical theory and computational
methods that apply to dynamical systems with multiple time
scales. Emphasis is given to models of neural systems, an area
in which the presence and importance of complex dynamics are
manifest. The investigations draw upon decades of research in
characterizing generic phenomena observed in dynamical systems
with a single time scale. The resulting body of mathematics,
sometimes called chaos theory, needs extension and modification
to fully explain the behavior of systems with multiple time
scales. Those extensions are the goal of this project. On a
longer time frame, the project lays foundations for coming
generations of biological models for cellular processes such as
gene expression and signal transduction. New biotechnology leads
to ever more complicated reaction networks that are simulated as
dynamical systems. This project produces results that aid the
implementation and interpretation of such simulations of complex,
multiple time scale dynamical systems.

The Cornell University IGERT Program in Nonlinear Systems supports graduate education and research in the area of complex nonlinear systems. The research component of the program will be organized around interdisciplinary groups (IRTG) comprising faculty with expertise in theoretical, computational and empirical science, who will jointly mentor graduate student fellow projects. The research areas of the initial IRTG, including areas of applications, are (i) networks (social networks, gene networks, internet, electric power grid); (ii) gene regulation (cell signaling and gene expression networks); (iii) moving machines and organisms (manual dexterity and control of locomotion); and (iv) biological pattern formation (cardiac electrophysiology).

Nonlinear science has been a role model for interdisciplinary research. Principles arising from dynamical systems theory have revealed common features in seemingly unrelated phenomena across the breadth of science and engineering. The intellectual merit of this project lies in the extension of successful strategies employed in nonlinear dynamics to confront increasingly complex systems. A primary goal of the research is to understand how systems, especially those arising in the life sciences, can be more than the sum of their parts. For example, legged locomotion and manual dexterity will be studied through a combination of mechanical devices, observation of human and animal behavior and computer models. The broader impacts of this research will be in improving the performance of robots and the treatment of physical injuries. Another theme that will be explored is how network architecture influences dynamics of a system. The concept of small world networks, developed by the founder of this IGERT Program, Steve Strogatz and his students, has already influenced research on biological, social and communication networks. Applied to the internet, the results of this research facilitate efficient web searches. In general, the program will have broad impact in developing methods to predict the dynamics of complex systems, taking full account of underlying network structures and making extensive use of experimental data.

The primary mechanism of the IGERT program is the engagement of Ph.D. students in nonlinear systems research early in their studies. The program involves students in the conceptual phases of research, and it encourages faculty to develop long term collaborations, stimulated by their joint mentorship of students in the IRTG. The most direct impact of the program is in training a new generation of scientists with broad interests and expertise. In the words of a former IGERT fellow, "graduate students who go through the IGERT program learn to speak the language of two or more fields with considerable fluency, and all students are introduced to a common mathematical foundation so that even those who do not share the language of a specific field can interact meaningfully."

IGERT is an NSF-wide program intended to meet the challenges of educating U.S. Ph.D. scientists and engineers with the interdisciplinary background, deep knowledge in a chosen discipline, and the technical, professional, and personal skills needed for the career demands of the future. The program is intended to catalyze a cultural change in graduate education by establishing innovative new models for graduate education and training in a fertile environment for collaborative research that transcends traditional disciplinary boundaries. In this sixth year of the program, awards are being made to institutions for programs that collectively span the areas of science and engineering supported by NSF.

ABSTRACT

Algorithms for Large-Scale Simulation
of Turbulent Combustion

NSF-ITR Grant
PI's: Stephen B. Pope, Cornell University
Peyman Givi, University of Pittsburgh


The focus of this collaborative ITR project is the development and use of innovative computational algorithms for the simulation of turbulent combustion. This is a topic of extreme intellectual challenge as it combines highly complex and non-linear combustion chemistry with the multi-scale and stochastic aspects of turbulence. Addressing this challenge, the four components of the project are (1) Dimension Reduction Algorithms suitable for combustion chemistry (2) Storage-Retrieval Algorithms including the use of widely-distributed databases (3) Algorithm Implementation for efficient performance on large-scale parallel systems, and (4) performance of Turbulent Combustion Simulations. In combustion (and other applications) the computational cost can be dramatically decreased if the dimensionality of the problem can be reduced. Two new approaches to dimension reduction are being explored and developed. These are based on pre-image curves and iterated Taylor series. Storage-retrieval algorithms have proved extremely effective in turbulent combustion calculations, and there are many other applications ripe for their use. The basis of these algorithms is to re-use data that are costly to compute directly (e.g., the solutions to the stiff ODE's governing chemical reactions). Data generated early in a simulation are efficiently re-used later in the simulation. This idea is extended to widely distributed computing and databases, so that data generated worldwide in all previous simulations can be used. To achieve accurate and efficient simulations of turbulent combustion, several advanced methodologies are combined: the flow is treated by large-eddy simulation (LES) so that the large-scale, unsteady, 3D motions are explicitly represented; the statistical distribution of the subgrid scale compositions is fully represented by its joint probability density function (PDF) whose evolution equation is solved by a Lagrangian particle method; and realistic combustion chemistry is incorporated using the combination of dimension reduction and storage-retrieval. The objective of this aspect of the work is to develop a comprehensive implementation of these methodologies that performs efficiently on large-scale parallel systems. Finally, as part of an ongoing international collaborative workshop, simulations are performed for several "target flames" for which there exist high-quality experimental data. In addition to testing and demonstrating the methodology developed, these simulations serve to investigate the performance of the physical sub-models, and to shed light on the physics and chemistry of the processes involved.

Now, and for many decades to come, turbulent combustion is a topic of tremendous significance to society and to several major industries. Energy usage (in power production, transportation, process industry and elsewhere) occurs predominantly through the combustion of fuels in turbulent flows. While there is, appropriately, great current interest in fuel cells and the possible re-emergence of nuclear power, the reality is that combustion technologies will remain dominant for many decades. There are compelling reasons to seek improvements in combustion devices, environmental and economic, and the industry is looking increasingly to computer simulations as a means of achieving improved designs. Higher combustion efficiencies lead directly to reduced CO2 emissions (for given output); at the same time, lower emissions of pollutants such as NO and particulates are continually being sought. It is inevitable that computer simulation, already an integral part of the design process, will grow in importance, as computers continually increase in power and the fidelity of the simulations improves. In this project, computer algorithms are being developed to increase substantially our abilities to simulate combustion processes and hence to impact the design of improved combustion devices. While the focus of the project is on turbulent combustion simulations, the algorithms developed (especially for dimension reduction and storage-retrieval) have broad applicability in computational science and engineering in general.

[unreadable] DESCRIPTION (provided by applicant): The goal of this project is to discover principles underlying the coordination of limbs and opposing muscles during walking by combining biological experiments with new dynamical models of the spinal neural networks that generate hindlimb movements in the mouse. Since walking is impaired in the large group of spinal cord injured patients, it is of importance to know the normal function of locomotor networks. The isolated mouse spinal cord gives good access for electrophysiological investigation of the neural circuits that control walking. New genetic techniques for specifically labeling or ablating specific classes of spinal neurons are likely to produce novel information about the physiology and anatomy of these networks. Because the networks are complex, computer models are needed in order to understand how the network works. Research on improved methods for fitting models of rhythmic processes to dynamical data will lead to new algorithms for parametrizing these models. The computer models to be developed in this project will be coupled cell systems of differential equations for membrane currents, whose structure incorporates what is known about the spinal cord. This project establishes a new collaboration to develop realistic models that build upon Kiehn's long experience and expertise with this system. Experiments will be conducted both at Cornell University (Harris-Warrick) and at the Karolinska Institute, Sweden (Kiehn) to measure the physiological properties of neurons and their synapses. In addition, connectivity of the network will be studied in the laboratory. [unreadable] [unreadable] The specific aims of the project are to: [unreadable] [unreadable] 1. generate current-based Hodgkin-Huxley-type models of the interneurons that coordinate the oscillator networks in the spinal cord; [unreadable] [unreadable] 2. characterize intrasegmental coordination of spinal pattern generation of hind leg movements; [unreadable] [unreadable] 3. analyze left-right coordination in mutant "hopping" mice, and develop algorithms for the parameterization and analysis of network models of central pattern generation. [unreadable] [unreadable]

The Cornell Mathematics Department will purchase and install a
multiple processor computer to enable computationally intensive
research within the department. The computer will be available to all
faculty and students in the Mathematics Department engaged in
computational research requiring more than a single processor.

Specific problems described in the proposal will be pursued by
R. Keith Dennis, Gerhard Michler, Mike Stillman, Dan Barbasch,
Alexander Valdimirsky, Rick Durrett and John Guckenheimer. Dennis and
Michler will investigate the structure of finite simple groups. Their
will work will help clarify the classification of finite simple
groups, one of the most complex proofs in mathematics. Durrett will
simulate probabilistic models for genomic and evolutionary
processes. This research will test approximations that significantly
reduce the computational time required to analyze large data
sets. Stillman will investigate algebraic problems arising from the
use of Bayesian networks in the statistical analysis of genomic data.
Barbasch will study irreducible representations of Lie groups, a
subject that finds important applications in quantum mechanics.
Vladimirsky will investigate parallelization of efficient algorithms
for solving of optimal control problems, for example problems to find
the minimum time or cost required for a process. Guckenheimer will
investigate dynamical systems that arise in neuroscience, studies of
locomotion and chemical combustion.

Insect flight is a complex system comprised of multiple interacting elements. Despite recent progress in understanding the underlying unsteady aerodynamics, which are notably different from conventional aerodynamics, relatively little is known about how insects execute flight maneuvers and far less is known about the main mechanisms that determine the diverse behaviors seen in natural flight. We propose an integrated approach to the study of insect flight that incorporates high resolution kinematics of freely flying insects, aerodynamic analysis of wing and body motions, and dynamical systems analysis of these data that will lead to reduced order models. Specifically, we propose three high risk pilot studies: 1) develop automated tracking techniques for 3D flight data that will be essential for aerodynamic and statistical analysis of cross-species comparison of free flight behaviors, 2) provide for the first time a direct connection between the internal muscles to the external observables in free flight through aerodynamics analysis and direct measurements, and 3) perform novel mathematical analysis of stability of free flight and build tools for constructing data-driven reduced order models.

We expect that this unprecedented integration of large experimental data sets and numerical analyses will transform research on insect flight. Our work will provide new tools to investigate fundamental issues relating to efficiency, stability, and control in animal locomotion. The techniques, algorithms and routines we develop will be distributed and made available to the research community through our web sites. The PI's will continue with their outreach activities which include: giving public lectures, providing K-12 teaching support, and organizing workshops.

This award will fund travel and local expenses for 25 invited participants for a workshop on ?Complex Systems,? to be held at the National Science Foundation in Arlington, VA, September 23-24, 2008. The workshop is being sponsored by the NSF Directorates for Mathematical and Physical Sciences (MPS) and Engineering (ENG).
The workshop will assess opportunities for research on complex systems in mathematics, the physical sciences, and engineering. Its primary emphasis will be on cross-cutting principles that will establish conceptual approaches applicable to diverse problems.
The workshop deliberations and conclusions will be summarized in a report to the NSF; the anticipated delivery date is October 31, 2008.

This project studies the mathematical theory of dynamical systems with multiple time scales and develops new computational methods for bringing this theory to bear upon models of biological phenomena. The research employs geometric approaches to study these problems. In particular, it investigates invariant manifolds that play a key role in organizing complex oscillations. New computational methods for computing these manifolds are one focus of the research. Indeed, there is a general lack of methods for computer investigation of manifolds. Creation of a comprehensive ``smooth computational geometry'' is a long term goal of the research. The project also seeks to develop methods for fitting models to data. It is rare that all of the parameters of a complex dynamical model can be measured or that systematic methods are used to estimate these parameters from empirical time series data. With multiple time scale systems, this is a particularly difficult optimization problem because abrupt changes in the dynamics are not readily fit by the quadratic models upon which smooth optimization algorithms are based. This project seeks to identify where these abrupt changes occur. The methods also enable accurate sensitivity analysis that describes the rates of change of model trajectories as parameters are varied. They are designed to contribute to the toolkit of methods available for designing engineered systems with periodic operating states rather than ones which are steady.

Dynamical systems theory is astonishingly successful in relating widely disparate phenomena observed in population dynamics, chemical reactions, lasers and much more. This project follows this tradition, seeking to explain universal dynamical behaviors observed in rhythmic processes, many of which display complex oscillations. Respiration, the heartbeat, circadian rhythms, menstrual cycles and animal locomotion are a few examples of biological rhythms to which the methods apply. All the primary modes of locomotion of higher animals: walking, running, slithering, swimming and flying result from cyclic motions of the body. Bursting oscillations that are ubiquitous in the nervous system exemplify temporal complexity: epochs of active firing of neurons alternate with quiescent periods. In mixed mode oscillations of non-equilibrium chemical reactors, epochs of large and small amplitude oscillations alternate. Multiple time scales are inherent in these complex oscillations. Thus, this project develops new methods for the analysis of dynamical systems with multiple time scales and the results yield a deeper mathematical understanding of how rapid changes in a system can result from variations of slow components. Geometric models reduce the mechanisms for such changes to their simplest forms and provide mathematical explanations for enigmatic results obtained from numerical simulations of systems with multiple time scales.