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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.



Maintained by Ricardo Carretero
Ricardo Carretero Gonzalez Antonio Palacios Peter Blomgren Joe Mahaffy Diana Verzi Chris Curtis San Diego San Diego State University SDSU California West coast MS master masters PhD doctorate doctoral graduate undergraduate concentration emphasis applied mathematics chaos chaotic fractal fractals dynamics dynamical systems nonlinear nonlinear dynamical systems nonlinear dynamics NLDS model modeling modelling publication publications research preprints analysis adaptivity aggregation bifurcation bifurcations bioloby blowup blow up blow-up bose bose-einstein breather breathers CML CMLs condensates coupled map lattices delay differential determinism deterministic differential einstein embedding equation equations fluidization fluidized GPE heteroclinic homoclinic ILM ILMs image restoration intrinsic localized modes lattices manifold map maps math mathematical bioloby metastability moving mesh NLS nonlinear waves numerics numerical ODE ODEs orbit orbits pattern patterns PDE PDEs POD prediction proper orthogonal decomposition reconstruction soliton solitons spatio temporal stable stochastic studies study systems tangle temporal time series unstable