Math (and ECE) 777: Nonlinear Dynamics, Bifurcations and Chaos
Time: 12:05 MWF
At: B135 Van Vleck
Instructor: J. Robbin
Text: Dynamical systems: Stability, Symbolic Dynamics, and Chaos, by Clark Robinson (CRC Press).
Topics:
I hope to cover the first eight chapters of the text. The chapter headings
are
I. Introduction
(Henon map, Lorenz equations)
II. One Dimensional Dynamics by Iteration
(Cantor sets and symbolic dynamics)
III. Chaos and Its Measurement
(Sharkovskii's theorem, zeta function,
chaos, Liapunov exponents)
IV. Linear Systems
V. Analysis Near Fixed Points and Periodic Orbits
(Van der Pol equations, Hartman's theorem, Poicare-Bendixson
theorem)
VI. Bifurcation of Periodic Points
(saddle-node, period doubling, [Andronov-]
Hopf)
VII. Examples of Hyperbolic and Attractors
(shift spaces, horseshoes, Melnikov method, fractal
basin boundaries,
hyperbolic
toral automorphisms, solenoid attractor, DA attractor,
Plykin
attractor, Henon attractor, Lorenz attractor.)
VIII. Measurement of Chaos in Higher Dimensions
(topological entropy, Liapunov exponents,
Sinai-Ruelle-Bowen measure, fractal dimension)
Grading:
The text has a large number of exercises. The grade will be based on
homework assignments: each student will be required to do three of eight
problem sets chosen from the text.
Prerequisites:
The most important prerequisite is a familiarity with the lingo of
sets and maps. The text is elementary enough that it defines things like
manifolds and tangent bundles and Jordan normal form. A student who
has done well in a good undergraduate courses in linear algabra and and
vector calculus should be able to follow the course. If you are in doubt
as to whether you are ready for this course, talk to me or examine the
textbook.
Visit my homepage http://www.math.wisc.edu/~robbin for updates on the course content.
Statistics 840
Multivariate Function Estimation for Observational Data, with Emphasis on Splines
Fall 1999 T Th 4:00-5:15 CS 1207
Grace Wahba, Instructor http://www.stat.wisc.edu/~wahba
Course Description (also available via my home page above)
This course is about various aspects of multivariate function estimation and statistical model building given scattered, noisy, direct, and indirect data, mostly via the use of smoothing spline and reproducing kernel Hilbert space (rkhs) techniques. Use of public software will be included. Upon completing the course the student should be able to apply modern multivariate smoothing spline and related methods to medical, environmental, atmospheric and economic data sets. Open problems will also be discussed.
No prior knowledge of Hilbert space is required. Students who already have a background in rkhs (from Statistics 860) may take the course and do special reading assignments while the introductory material is covered.
1. Background, introduction to the theory of reproducing kernel Hilbert spaces. Polynomial and thin plate splines, splines on the circle and sphere, additive splines, ANOVA splines, regression splines, hybrid and partial splines, radial basis functions, sigmoidal basis functions, representers.
2. Adaptive estimation of multiple smoothing and tuning parameters and the bias-variance tradeoff. Generalized cross validation, unbiased risk and maximum likelihood estimates for Gaussian and non-Gaussian data. Degrees of freedom for signal and the bias-variance tradeoff. Tihonov regularization.
3. Model selection and model building methods suitable for spline and related models. Bayesian and bootstrap confidence intervals. Penalized GLIM models for risk factor modeling.
4. Numerical methods for medium sized to very large data sets. Monte Carlo trace estimation for the degrees of freedom for signal. Early termination of iterative methods as a form of regularization. Basis function selection methods.
5. Applications in biostatistics (risk factor modeling), environmental data analysis, economics, meteorology (ill-posed inverse problems, remote sensing, model tuning), computer sciences (supervised machine learning, data mining, support vector machines) will be discussed, according to the interests of the class.
Prerequisites: - Statistics Majors, mathematical maturity, evidenced
by, for example, readiness to take 709, and either multivariate analysis,
or, some exposure to Hilbert spaces, or cons. instr. Those unfamiliar
with Hilbert spaces will be asked to read the first 33 pages of Akhiezer
and Glazman, Theory of Linear Operators in Hilbert Spaces, vol. I at the
beginning of the course. Graduate students in biostatistics, CS, AOS and
other physical sciences, engineering, economics, math and business may
find some of the techniques
studied here useful and are welcome to sit in, or, take the course
for credit if they have exposure to linear algebra, sufficient math background
to read Akhiezer and Glazman, and are familiar with the basic properties
of the multivariate normal distribution, as found, e. g. in Anderson, Multivariate
Analysis, or Wilks, Mathematical Statistics. Otherwise, the development
will be self-contained. If in doubt, please contact the instructor by e-mail
(wahba@stat.wisc.edu) or come
to the first class. This will be a seminar-type course. There will be no
sit-down exams. Students taking the course for credit will be expected
to do one or two computer projects studying the behavior of some of the
methods discussed on simulated or experimental data, and one or two projects
in an area of application of their choice, with a possible project being
the presentation of a lecture in class on a recent paper or recent research.
Text: Spline Models for Observational Data, G. Wahba, SIAM (1990), and
recent papers, tba.
Note: Wahba(1990) is on reserve at Wendt, Akhiezer and Glazman is on reserve in Wendt and the Math Library.