1MA208 Homepage
A collection of links to various ODE resources
Maple and MatLab worksheets
Timetable


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 20, 2017.
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 XuYan Chen (GaTech)
In "General Linear Methofs for Ordinary Differential Equations" by Zdzislaw Jackiewicz,
 page 16: Gronwall Lemma and BirkhoffRota Theorem on continous dependence.
Green's function for the SturmLiouville 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 nonlinear systems and their stability properties; limit cycles and PoincareBendixson Theorem.
4) Understand the basics of the SturmLiouville theory. Be able to apply the theory in boundary very problems.
5) Understand and be able to approach firstorder systems as continuous dynamical systems. Be able to describe the details of the dynamics of the Lorenz attractor and homoclinic phenomena.
Grading:
A takehome final, 100% of the final grade.
Schedule
 20170118
 Introduction, course information.
 Review of planar linear systems.
Reading: "Current adgenda", 6.4 HSD.
 20170119
 Existence and uniqueness of solutions.
 Picard's method of successive approximations.
Reading: 17.1 through 3. HSD.
 20170123
 Picard's method continued.
 CauchyPeano existence theorem.
Reading: a shorter version of the proof given in class (proof 2)
 20170125
 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
 20170131
 Discrete dynamical systems. Quadratic family.
 Continous dynamics. ODE's as flows.
Reading: HSD 15.115.4,7.1, 8.2.
 20170202
 Equilibria. Linearization of sinks.
Reading: HSD 8.2, 8.3.
 20170208
 Stable/unstable manifold theorem.
Reading: HSD 8.3, 9.2, 9.3
 20170210
 Stable/unstable manifold theorem.
 Stability.
Lyapunov function.
Reading: HSD 9.4, 10.1,10.2
 20170214
 Lyapunov function.
 Limit set.
Reading: HSD 10.1,10.2
 20170217
Reading: HSD 10.2, 10.3, 10.4, 10.5, 10.6
 20170222
 Flow boxes. Poincare map.
Reading: HSD 10.3, 10.4, 10.5, 10.6
 20170224
 Flow boxes, PoincareBendixson Theorem
Reading: HSD 10.3, 10.4, 10.5, 10.6
 20170228
 PoincareBendixson Theorem
 Selkov system
Reading: HSD 10.3, 10.4, 10.5, 10.6
 20170302
 Selkov system
 Lorenz system.
Reading: chapter 14.
 20170306
Reading: HSD chapter 14.
