Schedule:
Week 1 (January 11-15) Overview and first examples
- T: What is an ODE? What is a solution?
Recording (passcode bg@+h1?4z&)
- R: The logistic model. Slope field, phase portrait.
Basic discussion of existence and uniqueness.
Recording (passcode Npp3*O^xZB)
-
F: Tutorial. Examples
of (non-) uniqueness and local existence.
(Work sheet)
Read: Chapter 1 Section 1-3; Chapter 2.
Exercises: Chapter 1 # 2, 3, 4, 5, 13. Chapter 2
# 2, 3, 4, 5, 7.
Week 2 (January 18-22) Planar linear systems: phase portraits
- T: First-order linear systems.
Brief comment on matrix exponentials.
Recording (passcode m@@Qq!8ASS)
- R: Superposition Principle.
The role of eigenvalues and eigenvectors.
Recording (passcode &bj1m7#!ov)
Quiz 1
-
F: Tutorial. Examples of matrix exponentials.
(Work sheet, Petr's notes)
Read: Chapters 2 and 3.
Exercises: Chapter 2 # 8, 9, 10, 14. Chapter 3 # 1, 2, 4, 8, 9, 13.
Week 3 (January 25-29)
Planar linear ssystems: Phase portraits; classification
- M:
Problem Set 1
- T: Complex eigenvalues; multiplicities.
Recording (passcode Xb0RygD@3V)
- R: Saddle points, nodes, spirals, and centers.
Stability; sources and sinks.
Recording (passcode &EKwGo53QB)
Quiz 2
-
F: Tutorial. Higher-order linear equations.
(Work sheet, Petr's notes)
Read: Chapters 3 and 4.
Exercises: Chapter 3 # 3, 5, 6, 7, 10, 16.
Chapter 4 # 1, 3. Chapter 5 # 2, 3.
Week 4 (February 1-5) Higher-dimensional linear systems.
Eigenvalues and eigenvectors
- M: Problem Set 1 due 11:59pm
(Comments and sketch of solutions)
- T: Diagonalization and canonical forms.
Recording (passcode 3c0HLzt*@$)
- R: Jordan canonical form. Matrix exponential.
Recording (passcode A1!aw.8@jR)
Quiz 3
- F: Tutorial. Review of Chapters 1-4.
(Petr's notes)
Read: Chapters 5 (as review) and 6.
Exercises: Chapter 5 # 5, 6, 8, 9, 14, 15. Chapter 6 # 1, 3, 4, 5
Week 5 (February 8-12)
Higher-dimensional linear systems, cont'd. Matrix exponentials,
Duhamel's formula
-
M: Midterm 1
- T: Properties of the matrix exponential.
Recording (passcode m^gW6b1$BT)
- W Problem Set 2
- R: Solving x'=Ax+f(t). Existence and uniqueness.
Recording (passcode zb40cF2Bk*)
- F: Tutorial. Vector fields and diffeomorphisms.
(Work sheet)
Reading week (February 15-19)
Week 6 (February 22-24) Nonlinear systems
- M: Problem Set 2 due 11:59pm
- T: The dynamical system generated by a system of ODE.
Recording (passcode tck76i^Rvp)
- R: Existence, uniqueness, and continuous dependence.
Recording (passcode E?EzKNiP9F)
Quiz 4
- F: Tutorial. Banach's Contraction Mapping Theorem.
(Work sheet)
Read: Chapter 7 Sections 1-3; Chapter 17 Sections 1-3
Exercises: Chapter 6 # 8, 9, 12. Chapter 7 # 2, 3, 5, 9.
Chapter 17 # 4, 8, 10
Week 7 (March 1-5) Existence and uniqueness
-
- M: Problem Set 3
- T: The Picard map.
Recording (passcode eiQw#1x@&$)
- R: Proof of the existence and uniqueness theorem.
Recording (passcode rS4bgW?wMU)
Quiz 5
- F: Tutorial. Global existence and uniqueness.
(Work sheet)
Read: Chapter 17 Sections 3-4; Chapter 7 Section 3
Exercises: Chapter 7 # 1, 6, 8; Chapter 17 # 1, 11, 12;
Chapter 8 # 3; Chapter 6 # 15
Week 8 (March 8-12) Existence and uniqueness (wrap-up)
- T: Global existence and uniqueness theorem
on a domain.
Recording (passcode o9F3=d!Q04)
- W:Problem Set 3 due 11:59pm
- R: Continuity and differentiability w.r.t. initial values; linearization and the variational equation.
Recording (passcode 1*BSX++E=N)
- F: Tutorial. Review of Chapters 5, 6
Review: Chapters 5, 6, 7, 17
Week 9 (March 15-19) Equilibria in nonlinear systems.
Linearization and stability
- M: Midterm 2
- T: Examples
Recording (passcode ZET!TA@Z5d)
-
- W: Problem Set 4
- R: Some linearization theorems. Stability and asymptotic stability
Recording (passcode Z?KTFvr4@y)
- F: Tutorial. Linearization about equilibria
(Work sheet)
Read: Chapter 8 Sections 1-4.
Exercises: Chapter 8 # 1, 2, 3, 5, 6, 7, 8, 10, 11, 12
Week 10 (March 22-25) Linearization, continued. Nonlinear
sources, sinks, and saddles
- T: Lyapunov functions; stability
Recording (passcode H7y?4.M.T*)
- R: Topological conjucacy for sinks.
Stable and unstable manifolds.
Recording
Part 1 (passcode +2y8U^Y7z6),
Part 2 (Q3kz+!VY8V)
Quiz 6
- F: Tutorial. The Lotka-Volterra system
(Work sheet)
Read: Chapter 8 Sections 4-5; Chapter 9 Sections 1-2.
Exercises: Chapter 9 # 1, 2, 3, 4, 6.
Problem 1 from the
(March 19 tutorial work sheet)
Week 11 (March 29-April 2) Global nonlinear techniques
- M: Problem Set 4 due 11:59pm
- T: Planar systems. Nullclines
Recording (passcode A6.+qG0p#U)
- R:
Gradient flows and Hamiltonian systems
Recording (passcode P.Vdh8#Eeo)
Quiz 7
- F: No tutorial (University is closed)
Read: Chapter 9 Sections 2-4.
Exercises: Chapter 9 # 7, 8, 9, 10, 11, 12, 14, 16.
Week 12 (April 5-9) Closed orbits and limit sets
- T:
Positive and negative invariance; α- and ω-limit sets.
Recording (passcode Tj^Lh$uB72)
Problem Set 5
- R: The Poincaré-Bendixson theorem
Recording (passcode tsW.@g.P62)
Final assessment: Wednesday April 14, 9-12 am (online)
(Material covered by assessment)
F April 23: Problem Set 5 due 11:59pm