ChE 448
PROJECTS
SPRING 2013
FIRST VERSION, Tuesday, April 9, 2013
Here is a number of projects to choose from.
First take a look at all of them.
1) You are not limited to these - you can suggest your own, as some of you have already done.
2) You can choose to do a project together with somebody else if you wish. I strongly recommend
this, it would make it more fun.
3) Some projects are more difficult than others - e.g., some involving PDEs I would suggest for
graduate students.
4) Two people could do the same project independently if they want.
5) DURING this and next week I expect you to come after class or during office hours, and tell me
your first and second preference. We will discuss it, and choose something together.
6) IMPORTANT. Either I, or Matt Williams, or possibly one of my students and/or postdocs will
work closely with you on your project - you will not be alone.
Coexistence of Microbial Populations Under Cycling
Two bacterial populations competing for the same nutrient in a chemostat (a "biochemical CSTR") cannot
in general manage to coexist; one of the two populations "wins" and the other is washed out from the
chemostat. But the coexistence of two populations is often desirable because of some byproduct of their
common growth in the same environment. It was found that operating under periodically varying
conditions (so that one population "wins" half the time and the other population "wins" the second half of
the operating period) could lead to coexistence and stable periodic oscillations (see G. Stephanopoulos, A.
G. Fredrickson and R. Aris, AIChE J. 25 p.863 (1979)).
Work on this project would involve setting up the equations for "pure and simple competition" as it is
called, in a chemostat (a set of a few ODEs) and find using simulations and AUTO/MACONT the types
of solutions and average biomass production rates when certain parameters (like the feed concentration of
nutrients or the dilution rate -flow rate-) are varied periodically.
Bifurcations in Data Driven Reduced Order Models
In addition to systems of ODEs, we are often interested in the solution branches and bifurcations of PDEs,
but an accurately discretized version of a PDE is often too large to be handled by “off the shelf” tools
such as AUTO or MATCONT. To make the study of PDEs more tractable, data driven model reduction
tools such as the Proper Orthogonal Decomposition are used to generate a low-order (but accurate!)
system of ODEs from a given PDE model and a set of precomputed orbits (see E. Shlizerman, Int. J. Opt.
2012 p.831604 (2012)).
Work on this project would involve taking a PDE such as the FitzHugh-Nagumo equation and finding
orbits that appear to be stationary or periodic orbits. This data combined with the Proper Orthogonal
Decomposition should be used to create a reduced order model whose solution branches and bifurcations
can be explored in AUTO or MATCONT. These results can then be compared to approximating the PDE
on a coarse grid.
Nonlinear Model Reduction
In recent years, model reduction techniques have moved beyond linear techniques such as the Proper
Orthogonal Decomposition due to the advent of nonlinear manifold learning techniques such as ISOMAP
and diffusion maps (see Bollt, Int J. Bifur. Chaos 17 p. 1199-1219 (2007)).
Work on this project would involve learning some nonlinear methods for model reduction and applying
them to data taken from several system including the a Duffing oscillator in a paraboloid, the Lorenz
equations, and Chua’s circuit equations. We will then test the efficiency and accuracy of the nonlinear
methods by comparing them to the more traditional linear methods for model reduction.
Computing Periodic Orbits with the Poincare Lindstedt Method
In Chapter 7, Strogatz gives a nice introduction to “two-timing” (a.k.a., the method of multiple scales).
The Poincare-Lindstedt method (problem 7.6.19) is a nifty analytical technique for approximating
periodic orbits in perturbed differential equations based on the method of multiple scales. However, it
can also be used as an extremely accurate numerical method for computing periodic orbits even when the
perturbation to the differential equation is not small.
This project would involve taking a relatively small differential equation such as Hill’s problem for the
orbit of the moon around the Earth or the Lorenz equation, and computing the periodic orbits to extreme
levels of accuracy. The performance of the Poincare-Lindstedt method could then be compared to
collocation methods like those found in AUTO or MATCONT or traditional shooting methods.
Matrix Free Numerical Continuation
In a number of situations, we either do not have access to the governing equations in explicit form or must
build our numerical code around some pre-existing black-box simulator. In this situation, the prepackaged continuation packages, which assume access to some “right hand side” function, cannot be
used. However, numerical continuation can still be performed as long as only matrix-free methods are
used.
In this project, we will learn about specific matrix-free methods for accomplishing numerical continuation
tasks (i.e., approximate pseudo-arclength continuation, locating special points, branch-switching, etc.) and
apply them to black-box systems. These techniques may also be useful for graduate students whose
research involves (or will involve) large systems.
On the coupling of immune responses via interleukins
Modeling of "pieces" of the immune response (antigens that are attacked by T -cells, that are stimulated
and differentiate based on the antigens) can be modeled with differential equations very reminiscent of
the ones used for modeling chemical reactions. The immune system has various types (a "repertoire") of
antibodies, and T-cells that produce them, each of which is specific towards a particular class of antigens.
Usually, when an antigen comes in the system, an antibody specific to it will be produced through the
stimulation of the T -cells specific to that antigen.
While, therefore, the response to unrelated antibodies involves different families of T cells, some
chemical molecules (lymphokines, like IL-2), chemical messengers involved in the activation cycle, are
common for all different responses. A response can therefore be affected by another, unrelated one, just
because the unrelated one produces a lot of "chemical messengers" signalling activation.
This project involves some encyclopaedic reading about the immune system and its modeling, and a
bifurcation analysis of (a) a single immune response and (b) two unrelated immune responses coupled
through lymphokines.
All that jazz in the CSTR – four possible projects
I would like somebody who was intrigued enough by the CSTR to want to work through all the
"intricacies" of the various CSTR diagrams (from the Uppal Ray and Poore papers I gave you, and
from a couple of other CSTR papers), learn "everything there is to know" about them, and reproduce
them.
There are several projects here: the “main” CSTR, the CSTR with two consecutive reactions, which
can have chaotic behavior, the periodically forced CSTR and two coupled CSTRs
"More on the dynamics of a stirred tank with consecutive reactions" ( D.V. Jorgensen, W.W. Farr and R.
Aris ). Chem. Eng. Sci. 39, 1741–1752 (1984).
"The stirred tank forced" ( I.G. Kevrekidis, L.D. Schmidt and R. Aris). Chem. Eng. Sci. 41,1549–1560
(1986).
Couple, Double, Toil and Trouble: A computer-assisted study of two coupled CSTRs, Chem. Eng. Sci
48(11), pp.2129-2149 (1993) M. A. Taylor and I. G. Kevrekidis
Four projects for people interested in dynamics on lattices, waves on lattices, and Hamiltonian
systems (possibly Physics majors or EE students)
1) Examination of traveling Waves in the discrete sine-Gordon equation/coupled torsion pendula:
see e.g. the pioneering work of Physica D 14, 88 (1984) and the recent developments of Physica
D 237, 551 (2008).
2) Examination of traveling fronts and propagation failure in neural network / cardiac
tissue reaction-diffusion models, see e.g., Physica D 136, 1 (2000), SIAM J. Appl. Math
61, 317 (2000), or Physica D 155, 83 (2001).
3) Study of solitary traveling waves/discrete breathers in granular crystals within material science,
see e.g. Phys. Rev. E 56, 6104 (1997), and more recent modeling developments such as Phys.
Rev.E. 79026209 (2009), as well as Phys. Rev. E 82, 056604 (2010).
4) Study of finite-temperature dynamics of matter-wave dark solitons in Bose-Einstein
condensates within atomic physics: the case of linear and periodic potentials,
connections with anti-damped Josephson junctions and with Ch. 8 of Strogatz' book;
see: Phys. Rev. A 86, 033616 (2012).
Characterizing the collective activity of large neural networks. (two or more projects)
While a lot is known about the modeling of the dynamics of individual neurons, when large numbers of
them are connected it becomes both interesting and important to describe their collective dynamic
behavior – can we say something about the statistics of their dynamic activity? We will explore
computationally the coarse-graining of the dynamics of large connected networks of neurons (actually,
neuron models), searching for synchronization and for transitions in their collective behavior. See, for
example, E. Brown, J. Moehlis, and P. Holmes. On the phase reduction and response dynamics of neural
oscillator populations. Neural Computation 16:673-715, 2004, and also "On the application of "equation-
free" modelling to neural systems." C. R. Laing. J. Comput. Neurosci. (Vol. 20, 2006) and "Coars
egrained dynamics of an activity bump in a neural field model." C. R. Laing, T. A. Frewen and
Y. G. Kevrekidis, Nonlinearity 20 pp.2127-2146 (2007) which can be found at
http://www.massey.ac.nz/~crlaing/stickslip.pdf .
Another interesting “twist” in this direction is the study of the dynamics of heterogeneous populations of
neurons – instead of assuming all neurons of a population group to be identical, we build in heterogeneity
by allowing, for example, the kinetic parameters in the models to have a distribution in their values. Here
one can use techniques from the field of uncertainty quantification in order to describe the collective
behavior of such populations. An example of such work (involving a different type of cells) can be found
in: K. A. Bold, Y. Zou, I. G. Kevrekidis and M. A. Henson, An equation-free approach to analyzing
heterogeneous cell population dynamics, J. Math. Biol. 55(3) pp.331-352 (2007).
Mixed-Mode Oscillations
Koper and Gaspard proposed a three equation model for the complicated (periodic mixed-mode, as
they are called, and chaotic) oscillations in the electrochemical reduction of indium(III) at mercury in
the presence of thiocyanate (J. Phys. Chem. 95 pp.4945-4947, 1991).
They presented some simulations of the system - I'd like somebody to perform a more detailed
stability and bifurcation analysis of the model using AUTO/MATCONT; there is a possibility to
expand the model to a hierarchy of models with 4, 5, ...... equations. It would be nice to try and see
how much of the behavior remains there independent of the number of equations !
Noninvertibility and Adaptive Control (this could be more than one projects).
When we integrate ODEs, the trajectories in time can be integrated also backward uniquely; in the
same way that trajectories do not cross forward in time they also do not cross backward in time, or, in
other words, a state can only be "coming from" a single previous state.
It turns out that a large class of discrete time systems (maps) are noninvertible: one has a choice of
possible "pasts", even though one has a unique "future". This happens in numerical integrators, it
happens in adaptive control problems, it happens when one uses neural networks for model
identification purposes.
These "noninvertible" systems can have new types of bifurcations, different from the ones usually
encountered in "normal" systems. The project involves study of physical models (background and
assumptions, simulations, bifurcation analysis) of various types of systems -especially some
problems that come from control that are noninvertible, and what special features the dynamics of
these systems have.
Synchronization and adaptation
Strogatz' book has a lovely little chapter on synchronization: take two systems that are similar and
have therefore comparable dynamics and couple them - will they synchronize ? There are many
people studying this phenomenon today- we have a question and a "twist" that we will address in this
project: (a) how do we quantify the degree of "nonsynchronization" for nearby systems? and (b) how
do we use the degree of nonsynchronization between a system and its model to adjust the model
parameters and get the model to better fit the system ? Again here there is simulation and numerical
bifurcation analysis.
Controlling Chaos
Some simple feedback control methods have been proposed for "taming chaos": that is, stabilizing
unstable steady states or periodic solutions that "underlie" chaotic attractors. This project will involve
studying some of the literature, examining critically the types of algorithms suggested, and their
application in the case of both Ordinary and Partial differential equations. It is also possible to study how
to use such methodologies to find not only (unstable) steady states, but to find bifurcation points
experimentally.
Obtaining models from time-series processing
In recent years several methods have been proposed for the identification of nonlinear systems based
on experimental time series. These methods include (and are not confined to) local linear models,
polynomial models, artificial neural networks etc. This project will involve studying some of the
relevant time-series processing literature, and implementation and comparison of a couple of the
various methodologies on computational time series. We will then test whether the models identified
based on time series only have the same bifurcations as the real systems.
READING and NUMERICAL BIFURCATION PROJECTS
(a) What are the details of the scientific computing algorithms underlying AUTO/MATCONT and
its calculations? (boundary value problems, numerical bifurcation calculations)
(b) Proof of Sarkovski's theorem for period doublings
(c) Computation of invariant circles of maps.
(d) Homoclinic and heteroclinic orbits, and their relation to travelling waves and to
one-dimensional "turbulence" in a reaction-diffusion system.
(e) Analytical computations of Hopf bifurcations and its super/subcriticality
(f) Analytical computations of simple normal forms
ADDITIONAL POSSIBILITIES
(g) periodically forced chemical reactions: quasiperiodicity and entrainment
(h)
……