NORTH CAROLINA STATE UNIVERSITY
Department of Mathematics
Fall Semester, 2002
MA 532 --- Ordinary Differential Equations I
Text: Differential Equations and Dynamical Systems, 3rd edition, by Lawrence Perko
Topics: The general topics of the course in approximate order are as follows:
1. Statement of Existence-uniqueness theorem.
2. Solution of x' = A x, including e^{At} and phase portraits.
3. General theory of x' = A(t) x + f(t). fundamental matrix solutions and variation of parameters.
4. Equilibria of x' = f(x), linearization, stability, Hartman's theorem and stable manifold theorem
5. Limit sets
6. One and two dimensional autonomous systems, phase portraits
7. Liapunov theory, LaSalle invariant principle.
There will be two tests and one final exam. The grade is determined by:
1. Homework 30%
2. Tests
40%
3. Final
30%
Date
Sections
Topics
1
2
19-Aug
21-Aug
1.1-1.2
1.3-1.4
Uncoupled system, diagonalization
Exponential of Operators, Fundamental theorem for linear systems
3
23-Aug
1.5
Linear systems in
4
5
6
26-Aug
28-Aug
30-Aug
1.6
1.7
Complex eigenvalues
Multiple eigenvalues, generalized eigenvectors
Multiple eigenvalues, generalized eigenvectors
7
8
2-Sep
4-Sep
6-Sep
Holiday
1.8
1.9
9
10
11
9-Sep
11-Sep
13-Sep
12
13
14
16-Sep
18-Sep
20-Sep
Review
Test 1
2.1, 2.2
15
16
23-Sep
25-Sep
2.3
2.4
R2
Jordan forms and practical method of finding Jordan forms
Stability theory
1.10
Nonhomogeneous linear systems
Supplement Non-autonomous systems
Non-autonomous systems
Fundamental existence-uniqueness theorem
Dependence on initial conditions and parameters,
Maximal interval of existence
17
18
19
20
Date
27-Sep
30-Sep
2-Oct
4-Oct
Sections
2.5, 2.6
2.7
2.8
2.9
21
7-Oct
2.10
saddles, nodes, foci and centers
22
9-Oct
2.11
23
11-Oct
2.12
Nonhyperbolic critical points in
Center manifold
24
25
14-Oct
16-Oct
18-Oct
Fall Break
2.13
2.14
26
27
28
21-Oct
23-Oct
25-Oct
Review
Test 2
29
30
31
28-Oct
30-Oct
1-Nov
3.1
3.2
32
33
34
4-Nov
6-Nov
8-Nov
3.3
3.4
3.5
35
11-Nov
3.6
36
13-Nov
3.7
37
15-Nov
3.8
Poincare Bendixson theory in
Lienard systems
38
39
40
18-Nov
20-Nov
22-Nov
3.9
3.10
3.11
Bendixson's criteria
Poincare sphere and behavior at infinity
Global phase portraits and separatrix configurations
41
42
25-Nov
27-Nov
29-Nov
3.12
Index theory
Thanksgiving starts 1:15pm
43
44
45
2-Dec
4-Dec
6-Dec
Topics
Flow defined by diff equations, Linearization
The stable manifold theorem
The Hartman-Grobman Theorem
Stability and Liapunov functions
R2
Normal form theory
Gradient and Hamiltonian systems
Dynamical systems and global existence theorems
Limit sets and attractors
Periodic orbits, limit cycles and separatrix cycles
Poincare map
Stable and unstable manifold for periodic orbits
Hamiltonian systems with two degrees of freedom
R2
Thanksgiving
Supplement
Review
Liapunov theory, LaSalle invariant principle