MA 543: Ordinary Differential Equations and Dynamical Systems
Spring 2025, Purdue University
http://www.math.purdue.edu/~yip/543
Course Description:
- This is a beginning graduate level course on dynamical systems.
It covers basic results for
(i) linear systems;
(ii) local theory for nonlinear systems
(existence and uniqueness of solutions,
dependence on parameters, flows, linearization, stable manifold theorem);
(iii) global theory for nonlinear systems (global existence, limit sets,
periodic orbits, Poincare maps);
(iv) further topics (time and interests permitting): bifurcations,
averaging, asymptotics.
Instructor:
- Aaron Nung Kwan
Yip
- Department of
Mathematics
- Purdue University
Contact Information:
- Office: MATH 432
-
Email
Lecture Time and Place:
- 54300-001 (17540) TR 12:00pm - 1:15pm, SCHM 309
- (Zoom room for online meetings)
Office Hours:
-
Wed: 3:45pm-5:00pm, or by appointment.
Prerequisites:
- One (undergraduate) course in each of the following topics:
linear algebra
(for example, MA 265, 351, 511),
differential equation (for example, MA 266, 366),
and some familiarity in analysis
(for example, MA 341, 440, 504).
-
Textbooks and References:
-
Main text:
[M]
Differentiable Dynamical Systems (Revised edition, 2017), J.D. Meiss
References:
[P]
Differential Equations and Dynamical Systems, I. Perko
[B]
Stability Theory of
Differential Equations, R. Bellman
[T]
Ordinary Differential Equations and Dynamical System, G. Teschl
[GH] (Best of the best!)
Nonlinear Oscillations,
Dynamical Systems, and Bifurcations of Vector Fields,
J. Guckenheimer, P. Holmes
[HSD]
Differential Equations, Dynamical Systems: An Introduction to
Chaos, M. W. Hirsch, S. Smale, R. L. Devaney
[S]
Nonlinear
Dynamics and Chaos, with Applications to Physics, Biology, Chemistry, and Engineering,
S. H. Strogatz
[A]
Mathematical
Methods of Classical Mechanics,
V. I. Arnold
Some texts at the undergraduate level:
[Br]
Differential Equations and Their Applications, M. Braun
[BD]
Elementary Differential Equations and Boundary Value Problems,
Boyce and DiPrima (textbook of Purdue MA366, on reserve in libray)
,
Course Policy:
-
Your grade is based on:
- (50%) homeworks (five or six problem sets, roughly once every
two weeks, normally due on Thursdays, in class);
- (5%) abstract of paper:
due Friday, Mar 28th, 2025;
- (25%) paper (roughly ten pages, details TBA):
due Friday, Apr 25th, 2025;
- (20%) presentation (rougly 20-30 min, details TBA)
on the submitted paper during the final exam week (or earlier,
schedule TBA).
Guidelines about submitting work:
- For each of the above course components (homework, paper,
presentation), you have the option of submitting as a group which can
consist of up to three people.
You can of course discuss with more people, but the submitted materials
must be distinct for each individual group.
Each member of a group will receive the same grade.
The group members can change for different homework but those for the
paper and presentation must be the same.
- For the homework, sufficient work must be shown to explain your
answer.
Staple your homework to prevent 5% penalty.
- Progressive deduction will be imposed for late submissions.
- You are allowed to utilize extra information from other (online)
resources, such as Wikipedia, plotting routines and so forth.
However,
getting a solution
completely from online (such as ChatGPT, chegg.com and
so forth) is not permitted.
You are expected to observe academic honesty to the
highest standard. Any form of cheating will automatically
lead to an F grade, plus any other disciplinary action,
deemed appropriate.
Nondiscrimination Statement:
-
This class, as part of Purdue University's educational endeavor, is committed to maintaining a
community which recognizes and values the inherent worth and dignity of
every person; fosters tolerance, sensitivity, understanding, and mutual
respect among its members; and encourages each individual to strive to
reach his or her own potential.
Student Rights:
-
Any student who has substantial reason to believe that another person is
threatening the safety of others by not complying with Protect Purdue
protocols is encouraged to report the behavior to and discuss the next
steps with their instructor. Students also have the option of reporting
the behavior to the
Office of the Student Rights and Responsibilities.
See also
Purdue University Bill of Student
Rights and the
Violent Behavior
Policy under University Resources in Brightspace.
Accommodations for Students with Disabilities and
Academic Adjustment:
- Purdue University strives to make learning experiences accessible to all
participants. If you anticipate or experience physical or academic barriers based
on disability, you are also encouraged to contact the
Disability Resource Center (DRC) at:
drc@purdue.edu or by phone at 765-494-1247.
If you have been certified by the DRC as eligible for accommodations, you should
contact me to discuss your accommodations as soon as possible.
See also Courses: ADA Information for further information from the Department of Mathematics.
Campus Emergency:
-
In the event of a major campus emergency or circumstances beyond the
instructor's control, course requirements, deadlines and grading
percentages are subject to change.
Check your email and this course web page for such information.
See also
Emergency Preparedness and Planning for campus wide updates.
More information on University Policies:
- See your MA543 course homepage in Brightspace.
Content (tab at upper left corner): Student Support and Resources, and
University Policies and Statements.
Course Progress and Announcement:
-
Week 1: Jan 14, 16
[M, Chapter 1]
introduction, notations,
conversion between higher order equation and first order system;
examples (from biology and mechanics);
existence and uniqueness of solutions:
- explicit formula for solutions,
integrating factor, variation of parameters,
- finite time blow-up of nonlinear super-linear equations;
- non-uniqueness of solutions.
Note: What are ODEs?
Note: Examples of ODEs
Note: Basic concepts of ODEs
Ref: Mathematical models : mechanical vibrations, population dynamics, and traffic flow : an introduction to applied mathematics, Haberman
Ref: An excellent paper on
Poincare, celestial mechanics and chaos
(Everytime I read this paper, I get something "more" out of it while at
the same time I forgot most of what I read the last time....)
Homework 1: due Thursday, Jan 30th, in class.
Week 2: Jan 21, 23
[M, Chapter 2][B Chapter 1]
First order linear systems: dX/dt = A(t)X + h(t);
- matrix exponentials and fundamental matrices;
- solution formula: integrating factor and variation of parameters (again).
- invertibility of fundamental matrices, Jacobi's formula
(Wiki),
Abel formula [M, Thm. 2.34];
Diagonalization of matrices:
using eigenvectors as basis vectors, decoupling of linear systems,
[M, p.40, Example 2.10]
Baker-Campbell-Hausdorff (BCH) formula: e^C = e^Ae^B;
Invariant subspaces, (generalized) eigenspaces, Jordan canonical form of a
matrix.
Note: Solving linear systems
Note: Computation of matrix
exponentials
Note: Invariant subspaces
Ref: 19
Dubious ways to compute matrix exponential, Moler-van Loan
Ref: 19
Dubious ways to compute matrix exponential- II, Moler-van Loan
Ref: Matrix exponential - yet another
approach, Harris-Fillmore-Smith (See also [M, p.52])
Week 3: Jan 28, 30
[M, Chapter 2][Bellman, Chapter 2]
Invariant subspaces,
stable (E_s), center (E_c), and unstable (E_u) subspaces;
Linear stability and its perturbative statements;
Gronwall's Inequality [M, p.90, Lemma 3.28].
[M, Chapter 3, Thm 3.19][Bellman, Chapter 3]
Construction (and existence) of solutions:
- Picard's Iteration for linear equations;
- Picard's Iteration for nonlinear equation with
Lipschitz nonlinearity;
- Banach Fixed Point Theorem;
- Time discretization.
Uniqueness of solution under Lipschitz condition.
Note: Linear stability and
perturbation-1
Note: Linear stability and
perturbation-2
Note: Construction of solutions - 1
Homework 2: due Thursday, Feb 13th, in class.
Week 4: Feb 4, 6
[M, Chapter 3]
Norm/length of vectors and matrices;
Function spaces, completeness, compactness,
Cauchy sequence, equi-continuity (Arzela-Ascoli Theorem);
Regularity of functions:
continuity, uniform continuity, Lipschitz continuous, differentiability;
[M, Theorem 3.29]
Continuous/Lipschitz/smooth dependence on initial data;
[M, Theorem 3.30]
Continuous/Lipschitz/smooth dependence on parameters.
Note: Construction of solutions (summary)
Note: Construction of
solutions - 2
Note: Function spaces
Note: Regularity of functions
Note: Dependence on initial
data - formula
Note: Dependence on initial
data - proof
Week 5: Feb 11, 13
[M, Chapter 3.5]
Local vs global solutions;
Maximal interval of existence of solutions;
Behavior of solution at the end of the interval;
[M, Chapter 4.2] Flow map, semi-group property;
[M, Chapter 4.3] (Conditions for) global solutions,
change of time scale;
[M, Chapter 4.5] (Notions of) Stability of equilibrium points;
- Lyapunov vs asymptotic stability;
- linear vs nonlinear;
- hyperbolic vs non-hyperbolic.
Note: Maximal interval of
existence
Note: Stability of equilibrium
points
Homework 3: due Thursday, Feb 27th, in class.
Week 6: Feb 18, 20
[M, Chapter 4.6]
Lyapunov and Hamiltonian functions.
[M Chapter 5]
Linear system: (stable, center, unstable) invariant subspaces;
Nonlinear system: (stable, center, unstable) (nonlinear) invariant manifolds;
Note: Stability
using Lyapunov functions
Note: Second order differential
equations and two dimensional systems
Note: Invariant manifolds - intro
Note: Invariant manifolds -
Examples - I
Week 7: Feb 25, 27
[M Thm. 5.9] Existence of stable manifold (hyperbolic case).
[M Lem. 5.8] Existence and uniqueness of bounded solution.
Note: Invariant manifolds -
outline of proof
Note: Invariant manifolds -
Proof
Note: Invariant manifolds -
Examples - II
Homework 4: due Thursday, Mar 27th, in class.
Week 8: Mar 4, 6
[M Chapter 4.7] Topogical conjugacy, diffeomorphism, equivalence between two flows.
[M Chapter 4.8] Hartman-Grobman Theorem (Hyperbolic case);
[M Chapter 5.6] Hartman-Grobman Theorem (Non-hyperbolic case).
[M Chapter 6.1, 6.2, 6.3]
Two dimensional non-hyperbolic equilibrium points.
Analysis using polar coordinates.
Note: Topological conjugacy
Note: Center manifolds, Examples - III
Note: Non-hyperbolic equilibrium points in R^2
Week 9: Mar 11, 13
[Hartman, ODE, Chapter 12, Thm. 1.1, 2.1, 2.3, 2.4]
Existence of periodic orbits:
- linear homogeneous, inhomgeneous system.
- nonlinear systems and systems with parameters
[M Ch 2.8] Floquet Theorem and monodromy matrix (M);
- [M, Lem 2.35] log of a non-singular matrix;
- [M, Thm 2.36] structure of fundamental solution for linear
periodic system.
Note: Periodic solutions
Note: Floquet Theory
Ref: [Hartman, ODEs
Chapter 12, Part I, p.404-417]
(Periodic solutions, Implicit Function and Banach Fixed Point Theorems)
Week 10: Mar 25, 27
[M Sec. 2.8, 4.12]
M = e^{TB):
characteristic multipliers (for M)
vs characteristic exponent (for B)
Poincare map (P)
Relationship between M (monodromy matrix) and DP(0);
Stability of periodic solutions, orbital stability, time shift.
Note: Poincare Map
Note: Stability of periodic
solutions
Abstract of paper (5% of course grade) due:
Friday, Mar 28th, submit in Brightspace.
- Up to one page
- Submit using pdf file
- Can submit as a group of up to three people
Week 14: Apr 22, 24
Paper (25% of course grade) due on Friday, Apr 25th, submit in Brightspace.
- Twelve pages max (not counting references)
- Submit using pdf file
- Necessary sections: introduction, main body, conclusion/future perspectives,
references
- Must state and prove
one theorem or one significant/meaningful mathematical statement
Week 16: Final Exam Week May 5-9
Final Presentation (20% of course grade): schedule TBA
- 20 minutes for each group
- Zoom presentation, open to the whole class or any anyone you wish to invite
- Slides are required:
typed or hand-written using tablet are both acceptable
- Be prepared to answer questions
- More guidelines to come...