Course Info:

• Course: Ordinary Differential Equations II, 1MA208
• Lecturer: Denis Gaidashev, Ång 14231, gaidash at math.uu.se
• First Day Handout

• Textbooks:

• Text I (HSD): M. W. Hirsch, S. Smale, R. L. Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos, Academic Press (Elsevier) 2004.

Things on the current agenda:

• TAKE HOME FINAL, DUE MARCH 12, 2018.
• Warwick Tucker's proof of existence of the Lorenz attractor
• Foldings of the Lorenz attractor. Look at the pictures.
• Iterations and the bifurcation diagram for the quadratic map
• THINGS YOU SHOULD KNOW AND REMEMBER BEFORE THE FINAL.
• WHAT YOU SHOULD EXPECT ON THE FINAL: Set 1 and SOLUTIONS; Set 2 and SOLUTIONS.
• Matrix exponentials, fundamentals matrices and generalized eigenvectors: handout I, Polytechnic Institute of New York, and handout II, by Xu-Yan Chen (GaTech)
• In "General Linear Methofs for Ordinary Differential Equations" by Zdzislaw Jackiewicz,
• page 16: Gronwall Lemma and Birkhoff-Rota Theorem on continous dependence.
• Green's function for the Sturm-Liouville problem, orthonormality of the eigenfunction set, compact operators and their spectral properties, lecture notes by John Erdos from King's College London
• Constructing a topological linearizing cordinate for a saddle using the Hartman Grobman theorem for the maps

• Course Objectives/Outcomes:

1) Understand the matrix methods for first order linear systems. Be able to solve the relevant problems.

2) Be able to state, prove and apply existence and uniqueness theorems.

3) Understand the non-linear systems and their stability properties; limit cycles and Poincare-Bendixson Theorem.

4) Understand the basics of the Sturm-Liouville theory. Be able to apply the theory in boundary very problems.

5) Understand and be able to approach first-order systems as continuous dynamical systems. Be able to describe the details of the dynamics of the Lorenz attractor and homoclinic phenomena.

A takehome final, 100% of the final grade.

Schedule

1. 2017-01-18
• Introduction, course information.
• Review of planar linear systems.

2. 2017-01-19
• Existence and uniqueness of solutions.
• Picard's method of successive approximations.

3. 2017-01-23
• Picard's method continued.
• Cauchy-Peano existence theorem.

4. 2017-01-25
• Continous dependence on initial conditions and parameters.
• Gronwall's Lemma.

Reading: 17.3 and "General linear methods..." on the Current Agenda, 6.4 in HSD

5. 2017-01-31
• Discrete dynamical systems. Quadratic family.
• Continous dynamics. ODE's as flows.

6. 2017-02-02
• Equilibria. Linearization of sinks.

7. 2017-02-08
• Stable/unstable manifold theorem.

8. 2017-02-10
• Stable/unstable manifold theorem.
• Stability.
Lyapunov function.

9. 2017-02-14
• Lyapunov function.
• Limit set.

10. 2017-02-17
• Limit set.
• Flow boxes.

Reading: HSD 10.2, 10.3, 10.4, 10.5, 10.6

11. 2017-02-22
• Flow boxes. Poincare map.

Reading: HSD 10.3, 10.4, 10.5, 10.6

12. 2017-02-24
• Flow boxes, Poincare-Bendixson Theorem

Reading: HSD 10.3, 10.4, 10.5, 10.6

13. 2017-02-28
• Poincare-Bendixson Theorem
• Selkov system

Reading: HSD 10.3, 10.4, 10.5, 10.6

14. 2017-03-02
• Selkov system
• Lorenz system.

15. 2017-03-06
• Lorenz system.