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

Website:

http://web.njit.edu/~matveev/Courses/M676_F11/

Course Outline

Week

Lecture

Sections

Topic

H/W

1

1 (9/1)

1.1-1.7

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

p. 23

2

2 (9/6)

2.1-2.3

Review: Linear Systems and Diagonalization

p. 67

3 (9/8)

2.4-2.6

Review: Fundamental Solution Theorem for Linear Systems

p. 67

3

4 (9/13)

2.7-2.8

Linear Systems: Stability and Non-autonomous Systems

p. 67

5 (9/15)

3.1-3.3

Existence and Uniqueness Theorem

p. 101

4

6 (9/20)

3.4-3.5

Dependence on Parameters; Maximal Interval of Existence

p. 101

7 (9/22)

4.1-4.4

Flows, Global Existence, Linearization

p. 159

5

8 (9/27)

4.5-4.6

Stability; Lyapunov Functions and Hamiltonian Systems

p. 159

9 (9/29)

4.7-4.8

Topological Equivalence; Hartman-Grobman Theorem

p. 159

6

10 (10/4)

4.9-4.10

Limit Sets, Attractors & Basins

p. 159

11 (10/6)

4.11-4.12

Stability of Periodic Orbits; Poincare Maps

p. 159

7

12 (10/11)

5.1-5.3

Stable and Unstable Manifolds; Heteroclonoc Orbits

p. 192

13 (10/13)

5.4

Local Stable Manifold Theorem

p. 192

8

14 (10/18)

Review for the Midterm Exam

15 (10/20)

MIDTERM EXAM

9

16 (10/25)

5.5-5.6

Global Stable Manifolds and Center Manifolds

p. 192

17 (10/27)

6.1-6.4

Nonhyperbolic Equilibria & Nodes; Centers; Symmetries & Reversors

p. 238

10

18 (11/1)

6.5-6.6

Index Theory; Poincare-Bendixson theorem

p. 238

19 (11/3)

6.7-6.8

Lienard Systems; Behavior at Infinity

p. 238

11

20 (11/8)

7.1-7.3

Chaos: Lyapunov Exponents, Strange Attractors; Hausdorff Dimension

p. 265

21 (11/10)

8.1-8.2

Bifurcations of Equilibria

p. 325

12

22 (11/15)

8.3-8.4

Unfolding Vector Fields; Saddle-Node Bifurcation in 1D

p. 325

23 (11/17)

8.5

Normal Forms

p. 325

13

24 (11/22)

8.6-8.7

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

p. 325

14

25 (11/29)

8.8-8.9

Andronov-Hopf Bifurcation; the Cusp Bifurcation

p. 325

26 (12/1)

8.10-8.11

Takens-Bogdanov Bifurcation; Homoclinic Bifurcations

p. 325

15

27 (12/6)

8.12

Melnikov’s Method

p. 325

28 (12/8)

Review for Final Exam

IMPORTANT DATES

FIRST DAY OF SEMESTER

September 1

Midterm Exam

October 20, 2011

LAST DAY TO WITHDRAW

November 3, 2011

LAST DAY OF CLASSES

December 12, 2011

FINAL EXAM PERIOD

December 14-20, 2011

Assignment Weighting

Tentative Grading Scale

Hand-in Hw

 30

A

87 -- 100

Common Exam I

30

B+

81 – 86

Final Exam

40

B

74 – 80

C+

68 – 73

C

60 – 67

F

Below 60