Tasso J. Kaper - US grants

Recent advances in geometric singular perturbation theory have established a powerful set of tools for analyzing resonance phenomena and homoclinic orbits in systems of ordinary differential equations with two time or length scales. Among these new tools is the Exponentially Small Exchange Lemma (called EXSEL), which was developed by the author with C. Jones and N. Kopell, and which has been used successfully to prove the existence of multiple fast-pulse homoclinic orbits in perturbed multi-degree-of-freedom Hamiltonian systems, as well as in traveling wave problems for coupled reaction diffusion equations. It is proposed here to analyze four fundamental aspects of resonance phenomena in singularly-perturbed systems: (1) the existence of resonant sub- and super-harmonic orbits, (2) the geometry of islands of stability, which are obstructions to mixing in adiabatic chaos, (3) the resonant response of homoclinic structures in the forced, damped sine-Gordon equation which are conjectured to be sources of chaos, (4) the existence of periodic orbits in problems of passage throughresonance. Physical problems with multiple time scales arise in many branches of science and technology, including fluid mechanics, accelerator dynamics, plasma physics, neurophysiology, and biology. These problems are modeled mathematically as singularly-perturbed or adiabatic systems. Despite the long history of progress and the considerable continuing interest in this area, a wealth of open problems exists. This proposal concerns a group of four fundamental mathematical questions concerning resonance phenomena and homoclinic behavior that are of direct significance in the applications. Several new mathematical techniques, some developed by the author, have recently become available which offer substantial promise of solving these problems. While the main goal of the work proposed here is to use and extend the scope of these new tools, it is also endeavored to answer the above questions with an eye toward the technological applications. Toward this end, the proposer points to his successful development of enhanced fluid-mixing technology as an application of his earlier work on the geometry of homoclinic tangles in singularly-perturbed systems.

NSF Award Abstract - DMS-0072596
Mathematical Sciences: Applied Dynamical Systems and Singular Perturbation Theory for Patterns, Bubbles, and Chemical Reactions

Abstract

0072596 Kaper


This research project encompasses problems in chemical pattern formation, chemical kinetics, and nonlinear dynamics of gas bubbles. Self-replicating pulses have recently been discovered as new chemical patterns, and a central role in self-replication is played by strong, nonlinear pulse interactions. Pulses, such as bumps, annular rings, and circular spots in one and two dimensions, are localized large-amplitude perturbations of globally stable homogeneous states in the governing coupled reaction-diffusion equations. Specific aims include locating the hierarchies of saddle-node (disappearance) bifurcations that govern splitting, for example, when a ring solution splits into two rings, two rings into four, etc., and determining the underlying splitting mechanisms in two dimensions. Another aspect of this research focuses on stability of these patterns. Control over the stabilization of pulses in physically important systems of coupled reaction-diffusion equations is achieved by varying the strength of the coupling of the slow inhibitor field to the faster activator field, and by exploiting a recently discovered zero-pole cancellation in the nonlocal eigenvalue problems. In chemical reaction theory, this project focuses on large-scale systems involving many species and reactions and on the development of reduction methods that decrease the number of effective species and reactions that need to be modeled. The project investigates iterative numerical methods to find low dimensional manifolds in systems of reaction-diffusion equations using geometric singular perturbation theory. Finally, the project develops and analyzes a fully nonlinear model of the interactions of gas bubbles in liquids.

The fields of chemistry and fluid mechanics have long had a strong influence on the development of mathematics; and in turn, mathematics has led to many useful developments in both chemistry and fluid mechanics. This research project uses mathematical theory, specifically applied nonlinear dynamical systems theory, to gain new insights and make quantitative predictions for fundamental problems in pattern formation and large-scale reaction systems in chemistry and for nonlinear interactions between gas bubbles in fluid mechanics. A nonlinear control mechanism for stabilizing patterns in which the concentrations of the
reacting compounds are maintained at desirable levels in localized regions is under development. In addition, the project designs, implements, and tests reduction methods, known to be essential for modeling the large-scale systems of chemical reactions that arise in combustion, reacting flows, and other technologically important problems. Finally, the project carries out fundamental theoretical research on the nonlinear interaction of gas bubbles in liquids. Over the long term, this work will lead to deeper understanding of the complex problems of bubble clouds that generate noise behind submarines and damage turbine blades.

The investigator and her colleagues collaborate in a group
project at the Center for BioDynamics (CBD) to provide
interdisciplinary education and training for graduate students
and postdoctoral-level investigators in the context of a vigorous
interdisciplinary research program that focuses on areas of
mutual interest in mathematics (especially dynamical systems),
biology, and engineering. Disciplines include mathematics,
biomedical engineering, aerospace/mechanical engineering,
biology, psychology, and physics. Training extends beyond the
usual classroom activities by engaging participants in a variety
of research projects as well. One of the major topics is dynamics
of the nervous system. The projects, which involve experiments,
modeling, and analysis, all deal with the variety of rhythms in
the nervous system and the potential functions of these rhythms
in key cognitive states and processes such as attention,
awareness, learning, and recall. A second major topic is
dynamics of gene expression. Progress in genomic research is
leading to maps of the building blocks of biology and fueling the
study of gene regulation, where proteins often regulate their own
production or that of other proteins in a complex web of
interactions. CBD projects focus on using techniques from
nonlinear dynamics, statistical physics, control theory, and
molecular biology to model, design, and construct synthetic gene
regulatory networks, and to probe naturally occurring gene
regulatory networks. The third major topic is the dynamics of
patterns and waves. Training activities include two weekly
working seminars, extra journal clubs and reading groups,
seminars to educate the CBD members in the research going on
within the Center, and a CBD-initiated team-taught course.
The Center for BioDynamics (CBD) helps to advance
understanding of difficult interdisciplinary problems at the
intersection of mathematics, biology, and engineering, and it
trains mathematicians, scientists, and engineers for the 21st
century workforce. It does this by combining traditional
classroom education with significant engagement of students and
postdocs in interdisciplinary teams working on current problems.
The disciplines involved are mathematics, biomedical engineering,
aerospace/mechanical engineering, biology, psychology, and
physics. One of the major topics is dynamics of the nervous
system. The projects in this topic seek to shed light on the
origin of the electrical activity in the brain, and how the brain
uses this activity to process sensory information, to think, and
to regulate movement. A second major topic is dynamics of gene
expression. The web of interactions among the proteins that are
produced by genes is complex; the projects associated with this
topic involve the design and construction of artificial gene
regulatory networks, and techniques to better understand
naturally occurring gene regulatory networks. The third major
topic is the dynamics of patterns and waves, occuring in a
variety of applications. Training activities include two weekly
working seminars, regular sessions to read scientific journals,
seminars to educate the CBD members in the research going on
within the Center, and a CBD-initiated team-taught course. The
project is supported by the Computational Mathematics, Applied
Mathematics, Computational Neuroscience, and Biological Databases
and Informatics programs and by the MPS Office of
Multidisciplinary Activities.

Kaper
0306523
This project concerns systems with multiple length and time
scales, with the goals of analyzing recent experiments, of
improving computational methods, and of establishing new
mathematical theory for such systems. First, in the area of
chemical patterns with multiple length scales, the
recently-discovered phenomena of self-replicating spots and
pulses has posed new challenges for modeling and for stability
analysis of solutions of partial differential equations. The
investigator and collaborators build on their analysis of the
dynamics, time scales, and mechanisms responsible for
self-replication to study the underlying bifurcation hierarchies
that organize the self-replication regime, to further examine the
zero-pole cancellation phenomenon in the nonlocal eigenvalue
problem stability analysis, to extend the renormalization group
technique to establish the fully-nonlinear stability of pulses,
and to develop extensions to systems with more than two length
scales. Second, in the increasingly-important area of reduction
methods for large systems of chemical reactions with multiple
time scales, the validity and accuracy of certain methods are
analyzed, with special focus on the computational singular
perturbation method of Lam and Goussis. Third, the investigator
analyzes the Oya-Vallochi model of subsurface bioremediation.
Bioremediation is a process in which microorganisms, in the
presence of electron acceptors, degrade environmentally-harmful
organic compounds. The investigator studies traveling waves of
biomass activity and advection versus dispersion. Fourth, he
conducts fundamental studies of nonspherical deformations of gas
bubbles in Newtonian fluids. Finally, a challenging open problem
concerning the existence of self-similar, blow-up solutions of
the nonlinear Schroedinger equation in spatial dimensions between
two and four is attempted.
This project concerns mathematics for problems of
significant current interest in biology, chemistry, engineering,
and physics, which exhibit both fast and slow dynamical
processes. First, with collaborators and a doctoral student, the
investigator analyzes computational methods used to simulate
large, complex systems of reactions in biochemistry, combustion,
and air pollution engineering. These processes, such as the
production of certain proteins, the burning of natural gas, and
the formation of nitrous oxides in the atmosphere, typically
involve a few hundred species, each of which participates in
several reactions, with the reaction times ranging from
nanoseconds to milliseconds, even to minutes. Methods that reduce
the system complexity, while retaining a desired accuracy, are
critical for modeling these processes. The investigator aims to
show that there is a highly accurate method that can be used to
improve the accuracy of other widely-used methods, which are
embedded in major computer codes. Second, the investigator and a
doctoral student study mathematical models of bioremediation, in
which microorganisms are used to degrade environmentally-harmful
organic compounds. Mathematics provides an advantageous approach
to determine important quantities, such as the wave speed with
which the biologically-active zone propagates through a wet soil
column and how this speed depends on the many physical
parameters. Third, fundamental research is conducted on the
dynamics of gas bubbles in water. Deformations of spherical
bubbles lead to oscillations on time scales much shorter than
that on which the spherical mode itself oscillates, and the main
goal is to model the nonlinear transfer of energy between the
spherical and nonspherical modes that can lead to bubble
cavitation and the attendant production of underwater sound by
turbine blades, for example. Finally, the investigator develops
further theory for self-replicating chemical patterns and for a
prototypical equation that governs nonlinear wave propagation.

This research project concerns a series of interdisciplinary and fundamental mathematical methods for reaction-diffusion problems with multiple length and time scales. The first project concerns the Oya-Vallochi model of bioremediation, a system of advection, reaction, diffusion equations for the concentrations of the substrate to be degraded, the nutrient, and the active bacteria. Numerical simulations and mathematical analysis will be carried out to determine parameter domains in which stable traveling waves exist, as well as which instabilities develop on the boundaries of these domains. The second project involves reduction methods for chemical reaction and diffusion problems arising in biochemistry, chemical engineering, combustion, and air pollution engineering. These reactions typically involve 30-100 species, 50-400 reactions, and time scales ranging from nanoseconds to seconds. Moreover, these reactions typically occur in reactive flows, over extended domains, so that one must model the chemistry in each part of the domain. Reduction methods are indispensable to the analysis of this large amount of data, because they identify key progress variables and low-dimensional manifolds on which the long-term dynamics play out. The third project concerns pattern formation in activator-inhibitor reaction-diffusion equations, specifically the analysis of competing instabilities of spot patterns, self-replicating spots, newly-discovered `separator' solutions that govern the scattering of pulses and spots. Also, nonlinear stability of strongly interacting pulses will be established using a novel renormalization group approach the PI and collaborators have developed.





This applied mathematics research project concerns mathematical methods for key applications of multiple-scale reaction diffusion systems in engineering and science. The goals of these projects are to explain recent experiments, to improve computational methods, and to develop new mathematical theory that will be essential for the next-generation of engineering techniques. The first project concerns bioremediation, a process in which microorganisms in soil are induced to degrade environmentally-harmful organic compounds. Operating conditions in which remediation takes place in a regular and optimal fashion will be identified. The second project concerns reduction methods that are essential for understanding complex chemical reactions in biochemistry, chemical engineering, combustion, and air pollution engineering. The mathematical theory the PI has developed to date has helped to improve reduction methods currently used by engineers, and the proposed new theory will help the development of the next generation of faster and more accurate reduction methods. The third project centers on the dynamics and stability of chemical patterns involving spots and interacting coherent structures. Mathematical methods from dynamical systems theory, singular perturbations, and differential equations will be used, and new mathematical techniques will also be developed. The projects also involve PhD students and postdoctoral fellows, a significant percentage of whom are women, as well as collaborations at Argonne National Laboratory and foreign universities.

This project will advance the creation and support of a community of scholars, from undergraduate to faculty, working at the interfaces among dynamical systems and biological applications. The three main areas of focus are: 1. Analysis of systems with multiple length and time scales, including applications to pattern formation; 2. Mathematical neuroscience, including analytical methods for working with small networks and reduction of dimension techniques; 3. Gene regulatory networks, including the development of RNA switches, transcriptional bursting and programmable cells. These areas have major applications to issues concerning health and medicine. The project will build on the previous research and training experience of the Center for BioDynamics, co-directed by the Principal Investigator and one of the other senior faculty members. Trainees will be pre- and post-doctoral students who will take part in a wide variety of formal and informal activities, including special seminars, working groups, mini-symposia, laboratory work, journal clubs and social events, which will enable them to acquire the multiple scientific cultures needed to work in a trans-disciplinary manner. The pre-doctoral students will be from the departments of Mathematics or Biomedical Engineering; the postdoctoral associates will be drawn from a wide range of backgrounds, with a focus on applied math. In addition to their research activities, trainees will obtain experience teaching at different levels. Math department faculty and trainees will be involved in the construction of new interdisciplinary curricula for undergraduates in other departments, including Biology; the faculty will mentor the trainees in teaching the new curricula.

This research project encompasses a series of critical mathematical and scientific questions for multiscale problems arising in the fields of pattern formation, chemistry and combustion, neuroscience, electrical engineering, and multi-particle systems. The first project is on pattern formation and analyzes the dynamics and stability of fronts, pulses, and spots in paradigm reaction-diffusion systems. The second project studies model reduction methods used in complex multiscale chemical reactions, biochemical networks, and combustion by incorporating also the effects of diffusion. The third project involves a completely new class of solutions, known as torus canards, found in models from neuroscience. These solutions help understand the transitions between periodic spiking and bursting. The fourth project will focus on ways to model and analyze the impacts of cut-offs on the dynamics of fronts. The project involves graduate students and postdoctoral fellows in the research, as well as collaborations with scientists at national laboratories.

This research project addresses a series of questions concerning multiscale problems in pattern formation, chemistry and combustion, neuroscience, electrical engineering, and multi-particle systems. In the pattern formation project, paradigm reaction-diffusion systems will be studied. The goals are to develop new analytical techniques and mathematical theory for determining the boundaries of the stable pattern-forming regimes, analyzing the stability of semi-strong pulse interactions, modeling the scattering of pulses in 1-D systems and spots in 2-D systems, extending renormalization group methods for stability of modulating pulses, and predicting the dynamic bifurcations of pulses and fronts. The second project centers on accurate model reduction methods for large-scale combustion, chemical, and biochemical systems exhibiting multiple time scales. The goals are to analyze, develop, and improve cutting-edge model reduction methods for finding the low-dimensional manifolds that govern the effective system dynamics in the presence of diffusion. In the third project, the new phenomena of torus canards and canards in partial differential equations will be investigated. A theory of generic torus canards will be developed for fast-slow systems with multi-dimensional fast and slow variables. Known to exist in many neuroscience models, such as the Hindmarsh-Rose equations, the Morris-Lecar-Terman model, the Wilson-Cowan-Izhikevich system, and the forced van der Pol equation, torus canards are critical in the transition regimes between tonic spiking and bursting. A detailed study will also be carried out of the new bursting rhythms known as amplitude-modulated bursting rhythms. The fourth project will study the impacts of cut-offs on the reaction terms, introduced to accurately model regions of low particle densities, on the speeds, shapes, and stability of propagating fronts. A series of important problems related to fourth-order models, two-dimensional space dynamics, front initiation, and front pre-cursors will be studied.