Suggested Syllabus Math 290 A-B-C
A:
Dynamical systems.
Introduction
to dynamical systems. Definition and examples. Gradient flows and Hamiltonian
systems. Existence and uniqueness of solution. Equilibria and periodic
solutions. Introduction to
bifurcation: Types of bifurcations and Exchange of Stability. Bifurcation of
stationary solutions. Bifurcation of periodic solutions.
B:
Perturbation and Asymptotic Methods.
Introduction
to asymptotic approximations,
examples of regular and singular perturbations. Initial value problems
for ODEs. Methods of multiple scales and averaging. Boundary value problems for ODEs. The WKB method, turning points and matched asymptotic
expansions. High frequency wave
propagation. - Homogenization and effective medium theory.
C:
Calculus of variations.
Basic
problems in the calculus of variations and direct methods. Euler Lagrange equations. Theorems of
DuBois-Reymond and Haar. Erdman's corner condition and Euler boundary
conditions. The second variation
and the Legendre condition. Weak and strong minima. The Hamiltonian and Hamilton-Jacobi equations. Variational problems with subsidiary
conditions. Applications.
Part
A:
Books:
[L] G.
Looss & D. D. Joseph: Elementary Stability and Bifurcation Theory, Springer
1980.
[A] A.
Amann: Ordinary Differential Equations, de Gruyter 1983.
Schedule:
Introduction
to dynamical systems. Examples. (3 weeks)
[A] Chapter 1,3 and Notes. Existence and uniqueness of solutions. (1
week) [A] Chapter 2.
Introduction
to bifurcation: Types of bifurcations and exchange of stability (2 weeks) [L]
Chapter2 I-III. Bifurcation of
stationary solutions. Bifurcation of periodic solutions. (3 weeks) [L] Chapter2
IV-VI.
Part
B:
Books:
[B-O] C.
M. Bender & S. A. Orszag: Advanced Mathematical Methods for Scientists and
Engineers, McGraw-Hill 1991.
[K-C] J.
Kevorkian & J. D. Cole: Perturbation Methods in Applied Mathematics,
Springer-Verlag, 1981.
[H] M.
H. Holmes: Introduction to Perturbation Methods, Springer 1991.
Schedule:
Introduction
to asymptotic approximations.
Method of stationary phase. Regular and singular perturbations. (2
weeks) [H] Chapter 1 and [B-0] Chapter 1.
Method
of multiple scales. Application to
nonlinear oscillator. Method of
averaging. Eigenvalue problems and turning points. (4 weeks) [K-C] Chapter 3
Wave
propagation the WKB method. (2 weeks) Notes.
Homogenization. (1 week) [H] Chapter 5.
Part
C:
Books:
[C-H]
Courant and Hilbert: Methods of Mathematical Physics, Vol. I.
[W] F.
Wan: Introduction to the calculus if variations and its applications. Chapman
& Hall, 1995.
Schedule:
Basic
problems in calculus of variations. Direct methods. (2 weeks) [W] chapter 1 and [C-H] chapter 4 section 1, 2.
Euler-Lagrange
equations. Theorems of DuBois-Reymond and Haar. Erdmann's corner condition and Euler boundary conditions. (2
weeks) [W] chapter 2, 3 and [C-H] chapter 4 section 3, 5.
The
second variation and the Legendre condition. Weak minimum and strong minimum. (3 weeks) [W] chapter 4, 5.
The
Hamiltonian and Hamilton-Jacobi equations. (1 week) [W] chapter 6 and [C-H] chapter 4 section 10.
Variational
problems with subsidiary conditions. (1 week) [W] chapter 10 and [C-H] chapter 4 section 7.