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:

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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.

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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])

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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.

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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

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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.

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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

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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.

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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

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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)

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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

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Week 11: Apr 1, 3

Note: Existence of Periodic Solutions
Perturbation from existing/known periodic solution;
Case study (using level set/curve): periodic solution of predator-prey (Lotka-Volterra) model.
Case study (using level set/curve): Hamilontian system (with one-degree of freedom)

Note: Non-existence of Periodic Solutions
Gradient flow, system with strict Lyapunov function;
Bendixson criterior [P, p.264, Thm 1]; Dulac criterior [P, p.265, Thm 2];

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Week 12: Apr 8, 10

Note: Poincare-Bendixson Theorem [M, Thm 6.32]
Note: Lienard System [M, Thm 6.42]
Case study: van der Pol oscillator.

Ref: MacCluer-Bourdon-Kriete, Diff. Eqns., Chapter 10
(Nonlinear systems with many examples of biological models)

Homework 5: due Thursday, Apr 24th, in class.

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Week 13: Apr 15, 17

[M 4.9, 4.10] Global description
Omega, alpha limit sets, invariant sets, attractors.
[S, Ch 7, 8] Examples and applications of periodic orbits.
Failure of regular series expansion

Note: Limits sets
Note: Weakly perturbed (nonlinear) oscillators - I

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Week 14: Apr 22, 24

[S, Ch. 7.6] Weakly nonlinear oscillators
Multiple time scale expansions, Poincare-Lindsted method
[S, Ch. 7.5] Relaxation oscillators
Fast/slow dynamics: van der Pol oscillator

Note: Weakly perturbed (nonlinear) oscillators - II
Note: Fast/Slow dynamics (van der Pol oscillator)
Ref: Witelski-Bowen: Methods of Math. Modelling, Ch. 9, Weakly Nonlinear Oscillators
Ref: Witelski-Bowen: Methods of Math. Modelling, Ch. 10, Fast/slow Dynamical Systems
Ref: Kutz: Advanced Diff. Eqns.: Asymptotics and Perturbations (2025)

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

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Week 16: Final Exam Week May 5-9

Final Presentation (20% of course grade): schedule TBA

Choose by Sunday Apr 20th, 11:59pm at least three 30-minute time slots in Doodle Poll
(check email from Brightspace for links)

20 (+ 5 Q&A + 5 intermission) minutes for each group.
Each presentation can be of single- or multiple-presenters.
Preferably all the group members are present.

Zoom presentation, open to the whole class or any anyone you wish to invite

Slides are required (submit in Brightspace the day before your presentation):
typed or hand-written using tablet are both acceptable

Be prepared to answer questions