Master in Applied Mathematics
with
concentration in Dynamical Systems
MS code: 776316
Program:
Students enrolled in the Dynamical Systems
program are required to complete a total of 30 credits
broken down as follows: A) 15 units of core
courses + B) 12 units of electives + C)
3 units of Thesis/Project.
A.
Core courses -- 15 units:
(3 units each)
- MATH-531
Partial Differential Equations
Prerequisites: Mathematics 252 and
337.
Boundary value problems for heat and wave equations:
eigenfunction expansions, Sturm-Liouville theory and
Fourier series. D'Alembert's solution to wave equation;
characteristics. Laplace's equation, maximum principles,
Bessel functions.
- MATH-537
Ordinary Differential Equations
Prerequisites: Mathematics 337.
Theory of ordinary differential equations: existence
and uniqueness, dependence on initial conditions and
parameters, linear systems, stability and asymptotic
behavior, plane autonomous systems, series solutions
at regular singular points.
- MATH-538
Dynamical Systems & Chaos I
Prerequisites: Mathematics 252 and 254.
Phase space analysis, equilibria and
stability in one and two dimensions, limit cycles, Floquet
theory, Poincaré maps, one and two dimensional
maps, chaos, period doubling, chaotic attractors. Applications
in biology, chemistry, physics, engineering and other
sciences.
- MATH-636
Mathematical Modeling
Prerequisites: Mathematics 254 and 337
or Mathematics 342A and 342B or Engineering 280.
Advanced models from the physical, natural and social
sciences. Population dynamics, mechanical vibrations,
planetary motion, wave propagation, traffic flow. Phase
plane analysis, Inverse problems, fitting a model to
experimental data. (Formerly numbered Mathematics 536.)
- MATH-638
Dynamical Systems & Chaos II
Prerequisites: Mathematics 538 or 337;
or consent of instructor.
Nonlinear systems of differential equations,
Potential fields, periodic solutions, Lyapunov functions.
Chaos in differential equations, Lyapunov exponents,
chaotic attractors, Poincaré maps. Lorenz and
Rossler attractors, forced oscillators, Chua's circuit.
Stable manifolds, bifurcations. Applications in science
and engineering.
B. Electives
-- 12 units:
The 12 units of electives are
to be chosen with the approval of the graduate adviser.
Depending on demand and resources, Special topics
in Dynamical Systems might be offered periodically
(one per year).
Sample
of possible Special topics in Dynamical Systems:
(3 units each)
(offering of these courses depending on demand
and resources)
-
M-639 Nonlinear Waves:
Linear waves, dissipation, dispersion. Conservation
laws. Water waves. KdV equation, solitary waves,
cnoidal waves. Scattering and inverse scattering.
Perturbation theory. Nonlinear Schroedinger equation,
dark and bright solitons, vortex solutions.
Variational techniques, modulational instability, stability.
Prerequisites: Mathematics 531 or 537; or consent by instructor.
-
M-635 Pattern Formation:
Linear stability, marginal stability curves, classification.
One dimensional patterns, bifurcations. Two dimensional
patterns, square and hexagonal patterns, spirals, defects.
Diffusion driven instability, Turing patterns. Spatio-temporal
chaos. Applications in biology, chemistry, and physics.
Prerequisites: Mathematics 337 or 531 and Mathematics 254 or 342A, 342B; or consent
by instructor.
-
Applied Bifurcation of Dynamical
Systems: Bifurcations and
structural stability of dynamical systems. One-parameter
bifurcations of equilibria/fixed points in continuous-time/discrete-time
systems. Bifurcations of periodic orbits. Homoclinic
and heteroclinic orbits. Two-parameter bifurcations.
Applications in science and engineering.
-
Numerical Experiments and Methods
in Dynamical Systems: Practical
methods for analysis and exploration of dynamical
systems. Chaos, universality, self-similarity.
Chaotic ODEs, unimodal maps, stable and unstable
manifolds. Lyapunov exponents, attractors fractal
dimension, renormalization operators. Quasi-periodicity,
phase locking, KAM theory.
-
Nonlinear Time Series:
Linear models, stationarity, correlation. Phase
space methods, time-delay reconstruction. Determinism
and predictability. Instability and Lyapunov exponents,
sensitive dependence, exponential divergence.
Self-similarity, correlation dimension, Lyapunov
dimension. Whitney's embedding theorem, Taken's
embedding theorem. Chaotic data, nonlinear noise
reduction.
-
Fractal Geometry: Metric
spaces and space of fractals, transformations
on metric spaces, contraction mappings, iterated
function systems, computer algorithms for constructing
fractals, chaotic dynamics on fractals, theoretical
and experimental determination of fractal dimension,
fractal interpolation, Julia and Mandelbrot sets.
-
Mathematical Biology / Neural
Modeling: Cellular physiology
and cell structure, the cell membrane, Fick's
law, membrane potential, Nernst potential, the
Goldman-Hodgkin-Katz equation, modeling cell membrane,
membrane ion channels, channel gating, excitability,
the Hodgkin-Huxley model, FitzHugh-Nagumo model,
Morris-Lecar model, phase-space behavior, calcium
dynamics, bursting behavior.
Other
recommended electives for the remaining units:
(3 units each)
(other courses, even
in other departments, may be approved by adviser)
- MATH-541
Introduction to Numerical Analysis and Computing
- MATH-542
Introduction to Numerical Solutions of Differential
Equations
- MATH-637 Theory of Ordinary Differential Equations
- MATH-668 Applied Fourier Analysis
- MATH-797 Research
- CS-553: Neural Networks
- PHYS-580 Computational Physics
- PHYS-585 Computer Simulations in the Physical Sciences
- PHYS-608 Classical Mechanics
C.
Thesis or Project -- 3 units:
- MATH-799A Thesis or Project
Prerequisites: An officially appointed
thesis committee and advancement to candidacy.
Preparation of a project or thesis in the field of Dynamical
Systems for the master's degree.
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times since February 2002.
Last update: 05 Mar 2003.
Maintained by
Ricardo Carretero
Ricardo Carretero Gonzalez
Antonio Palacios Peter Blomgren Joe Mahaffy Diana Verzi San Diego
San Diego State University SDSU California West coast MS master
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