Course Title:

Math 676 – Advanced Ordinary Differential Equations

Textbook:

James D. Meiss “Differential Dynamical Systems” ISBN 978-0-898716-35-1

Prerequisites:

Math 222, Math 337, and Math 545 or Math 645

 Tentative Course Outline

We will probably study some other topics and may not get to all of these

Homework assignments will be given in class. The pages below simply list the places in the text that have homework problems.

Week

Lecture

Sections

Topic

H/W

1

1

1.1-1.7

Review: 1D Flows; 2D Phase Space and Nullclines

p. 23

2

2

2.1-2.3

Review: Linear Systems and Diagonalization

p. 67

3

2.4-2.6

Review: Fundamental Solution Theorem for Linear Systems

p. 67

3

4

2.7-2.8

Linear Systems: Stability and Non-autonomous Systems

p. 67

5

3.1-3.3

Existence and Uniqueness Theorem

p. 101

4

6

3.4-3.5

Dependence on Parameters; Maximal Interval of Existence

p. 101

7

4.1-4.4

Flows, Global Existence, Linearization

p. 159

5

8

4.5-4.6

Stability; Lyapunov Functions and Hamiltonian Systems

p. 159

9

4.7-4.8

Topological Equivalence; Hartman-Grobman Theorem

p. 159

6

10

4.9-4.10

Limit Sets, Attractors & Basins

p. 159

11

4.11-4.12

Stability of Periodic Orbits; Poincare Maps

p. 159

7

12

5.1-5.3

Stable and Unstable Manifolds; Heteroclonoc Orbits

p. 192

13

5.4

Local Stable Manifold Theorem

p. 192

8

14

Local Stable Manifolds continued

p. 192

15

5.5-5.6 

Global Stable Manifolds and Center Manifolds

p. 192 

9

16

MIDTERM EXAM

17

6.1-6.4

Nonhyperbolic Equilibria & Nodes; Centers; Symmetries & Reversors

p. 238

10

18

6.5-6.6

Index Theory; Poincare-Bendixson theorem

p. 238

19

6.7-6.8

Lienard Systems; Behavior at Infinity

p. 238

11

20

7.1-7.3

Chaos: Lyapunov Exponents, Strange Attractors; Hausdorff Dimension

p. 265

21

8.1-8.2

Bifurcations of Equilibria

p. 325

12

22

8.3-8.4

Unfolding Vector Fields; Saddle-Node Bifurcation in 1D

p. 325

23

8.5

Normal Forms

p. 325

13

24

8.6-8.7

Saddle-Node Bifurcation in Rⁿ; Degenerate Saddle-Node Bifurcation

p. 325

14

25

8.8-8.9

Andronov-Hopf Bifurcation; the Cusp Bifurcation

p. 325

26

8.10-8.11

Takens-Bogdanov Bifurcation; Homoclinic Bifurcations

p. 325

15

27

8.12

Melnikov’s Method

p. 325

28

Review for Final Exam

Assignment Weighting

Tentative Grading Scale

Homework

 35%

A

87 -- 100

Midterm

25%

B+

81 – 86

Final Exam

40%

B

74 – 80

C+

68 – 73

C

60 – 67

F

Below 60