MA 543: Ordinary Differential Equations and Dynamical Systems
Spring 2023, Purdue University
http://www.math.purdue.edu/~yip/543

Course Description:

This is a beginning graduate level course on ordinary differential equations. It covers basic results for
(i) linear systems,
(ii) local theory for nonlinear systems (existence and uniqueness, dependence on parameters, flows and linearization, stable manifold theorem), and
(iii) global theory for nonlinear systems (global existence, limit sets and periodic orbits, Poincare maps).
Some further topics include bifurcations, averaging techniques and applications to mechanics and mathematical biology.

Instructor:

Aaron Nung Kwan Yip
Department of Mathematics
Purdue University

Contact Information:

Office: MATH 432
Email

Lecture Time and Place:

54300-001 (17540) TR 10:30am - 11:45am, UNIV 303

Office Hours:

T: 3:30-4:30pm, 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),
analysis (for example, MA 341, 440, 504).

Textbooks and References:

Main text:
[M] Differentiable Dynamical Systems , 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
[BP] Elementary Differential Equations and Boundary Value Problems, Boyce and DiPrima (textbook of Purdue MA366, on reserve in libray) ,

Course Announcement:

Your grade is based on:
(i) class participation (though I do not intentionally take attendance). I do hope you can occasionally come talk to me about the course and your interests and projects. I am certainly curious how this course benefits your study.
(ii) typing up two weeks of class notes, one before and one after the Spring Break (workload shared among the whole class). It is preferably to use Latex (but not absolutely necessarily).
(iii) a short (5 to 10 pages) paper, and
(iv) a short (20 min) presentation about the paper toward the end of the semester, probably during the final exam week or earlier.

Due dates: (submit all of the following in pdf format in Brightspace)
(1) Each week's lecture write-up is due in two weeks. (For example, the write-up for 2/7 and 2/9 is due on 2/24 (Friday).)
(2) Title and abstract of paper, due Friday, Mar 10 (right before the Spring Break)
(3) Paper due Friday, Apr 21
Week 1: Jan 10, 12
[M, Chapter 1]
introduction, notations, conversion between higher order equation and first order system;
examples (from biology and mechanics);
explicit solution, integrating factor, variation of parameters,
finite time blow-up of nonlinear super-linear equations.

Note: Examples of ODEs
Ref: An excellent paper on Poincare, celestial mechanics and chaos
(Everytime I read it, I get something "more" out of it while at the same time I forgot most of what I read the last time....
Hopefully by the end of this semester, parts of the paper will make more sense for us....)

Week 2: Jan 17, 19
(Note typing, due Feb 3: Ashkarian, Fahey, Huang, Richmond, Zhang)

[M, Chapter 1]
Explicit solution, invariant domain, the use of differential inequality;
[M, Chapter 2]
First order linear systems: dX/dt = A(t)X + h(t);
matrix exponentials and fundamental matrices;
solution formula: integrating factor and variation of parameters.

Week 3: Jan 24, 26
(Note typing, due Feb 10: Binder, Fitch, Kumagai, Sasin)

[M, Chapter 2][B, Chapter 1]
invertibility of fundamental matrices, Jacobi's formula (Wiki), Abel formula [M, Thm. 2.11];
using eigenvectors as basis vectors, decoupling of linear systems, diagonalization of matrices;
(generalized) eigenspaces, Jordan form of a matrix, Schur's Canonical Form [B, p.21, Thm. 6] [see also Strang];
"Formula" for matrix exponential, e^(At);
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

Week 4: Jan 31, Feb 2
(Note typing, due Feb 17: Brady, Haug, Leyva, Lotlikar, Sathiyamoorthy)

[M, Chapter 2][Bellman, Chapter 2]
Invariant subspaces, stable (E_s), center (E_c), and unstable (E_u) subspaces;
Gronwall's Inequality [M, Lemma 3.13, p. 95; Ex. 9, p. 103];
linear stability and its perturbative statements;

Note: Linear stability analysis.

Week 5: Feb 7, 9
(Note typing, due Feb 24: Charla, Khilnani, Morton, Schmitt)

[M, Chapter 3][Bellman, Chapter 3]
Norm for vectors, matrices, and funtions;
Convergence of Cauchy sequences and power series for exponential functions;
e^A*e^B = e^(A+B) if AB = BA; e^A*e^B = e^C, Baker-Campbell-Hausdorff Thm [M, p.43];
Construction (and existence) of solutions:
- Picard's Iteration for linear equations;
- Picard's Iteration for nonlinear equation with Lipschitz nonlinearity;
Uniqueness of solution under Lipschitz condition.

Week 6: Feb 14, 16
(Note typing, due Mar 3: Cummings, Iannamorelli, Naik)

General methods of constructing solutions:
(i) iterations, (ii) Banach Fixed Point Theorem, (iii) time-stepping, (iv) Schauder Fixed Point Theorem;
Local vs global in time solutions: locally vs globally bounded and Lipschitz condition;
Finite time blow-up for superlinear nonlinearity vs global in time solution for nonlinearity with linear growth;
Necessity of Lipschitz condition for uniqueness of solution.

Regularity of functions: continuous, Lipschitz continuous and differentiable functions;
Lipschitz dependence [M, Thm 3.14] and differentiability [M, Thm 3.15] of solutions with respect to initial condition and parameters.

Note: Construction of Solutions
Note: Regularity of Functions
Note: Dependence of Solutions on Initial Conditions and Parameters.

Week 7: Feb 21, 23
(Note typing, due Mar 10: Das, Kapeles, Parthipan, Williams)

Concatenation of solution, maximal interval of existence, change of time scale;
Flow map and its semi-group property.

[M Chapter 4]
Linearization around an equilibrium point;
Hyperbolic vs non-hyperbolic equilibrium point;
Asymptotic stability for nonlinear system (with all eigenvalues having negative real parts).

Week 8: Feb 28, Mar 2
(Note typing, due Mar 17: Ashkarian, Fahey, Khilnani, Richmond, Zhang)

[M Chapter 4]
Stability of an equilibrium point:
linear vs nonlinear (asymptotic) stability and Lyapunov stability vs asymptotic stability
[M Chapter 5]
Linear system: (stable, center, unstable) invariant subspaces
Nonlinear system: (stable, center, unstable) (nonlinear) invariant manifolds
Uniqueness of (un)stable manifolds vs nonuniqueness of center manifolds.
[M Thm. 5.3] Existence of stable manifold (hyperbolic case).

Note: Invariant Manifolds
Note: Existence of (local) Stable Manifolds

Week 9: Mar 7, 9 (Title and abstract due, Mar 10)
(Note typing, due Mar 24: Binder, Kumagai, Sasin)

[M Thm. 5.3, p.175] Existence of stable manifold (hyperbolic case).
[M Thm. 5.8, p.186] Existence of center manifold (nonhyperbolic case).
[M p.189-192] Examples of using Taylor expanion to find invariant manifolds.

Note: Examples of invariant manifolds
(Excerpts from [P], [M], [Carr: Applications of Centre Manifold Theory])

Spring Break: Mar 14, 16

Week 10: Mar 21, 23
(Note typing, due Apr 7: Kapeles, Haug, Leyva, Lotlikar, Sathiyamoorthy)

[M Chapter 6.1-6.3]
Two dimensional non-hyperbolic equilibrium points.
Analysis using polar coordinates.

[M Chapter 4.7]
Topological conjugacy, diffeomorphism, equivalence between two dynamical systems
[M Theorem 4.13, p.138] Hartman-Grobman Theorem (hyperbolic case);
[M Theorem 5.9, p.189] Hartman-Grobman Theorem (non-hyperbolic case);

Note: Non-hyperbolic equilibrium points in R^2
Note: Topolgical Conjugacy

Week 11: Mar 28, 30
(Note typing, Apr 14: Charla, Huang, Morton, Schmitt)

[M Chapter 4.9, 4.10]
Omega and alpha limits sets, invariant sets, and attractors.
[Hartman, ODE, Chapter 12, Lem 1.1 and Thm 1.1]
Existence of periodic orbits: linear homogeneous, inhomgeneous system.

Note: Maximal Interval of Existence
Note: Limit sets, invariant sets, attractors
(Excerpts from [M]: Examples of Limit Sets)
Note: Existence of Periodic Orbits (revised)
Ref: [Hartman, Chapter 12, Part I, p.404-417]

Week 12: Apr 4, 6
(Note typing, due Apr 21: Cummings, Fitch, Iannamorelli, Naik)

[Hartman, ODE, Chapter 12, Thm. 2.1, Thm. 2.3, and Thm. 2.4]
Existence of periodic orbits: nonlinear systems and systems with parameters
Periodic solutions of autonomous systems, concept of time shift
Stability of periodic orbits, orbital stability;
[M 2.8] Floquet Theorem and monodromy matrix (M);
[M 4.11, 4.12] Poincare Map (P).

Note: Stability of periodic orbits

Week 13: Apr 11, 13
(Note typing, due Apr 28: Das, Brady, Parthipan, Williams)

Stability of periodic orbits in terms of M and P
Relationship between M and P
characteristic exponent vs characteristic multipliers;
Examples of computing characteristic multipliers.
Formula of P'(0) (in R^2), examples

Week 14: Apr 18, 20 (Paper due, Apr 21)

Existence and persistence of periodic case
Autonomous vs non-autonomous system
Hamiltonian vs non-hamiltonian case
Example from biology: predator and prey system
Nonlinear oscillator, Van der Pol Oscillator
[M 6.7] Lienard System

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

Week 15: Apr 25, 27

[M 6.6] Poincare-Bendixson Theorem and Jordan Curve Theorem in R^2
Nonexistence of Periodic Orbits: Bendixson and Dulac Criterior
[T, Ch 12] Introduction to Melnikov integral

Ref: Sanders: Melnikov's Method and Averaging

Week 16: Final Exam/Presentation Week: May 1-5

All presentations will be in zoom. (For zoom link, go to Brightspace, MA543 page, Content/Course Contents/Zoom.)

Sunday, Apr 30, 2023
10:00am: Joshua Fitch, Stability Analysis of Spacecraft Relative Motion via the Thrust-Augmented Clohessy-Wiltshire Equations
10:30am: Andrew Binder, Sensing Periodic Solutions in the CR3BP When Starting from Existing Solutions - the Broucke Stability Diagram
11:00am: Liam Fahey, Applying Dynamic Systems Theory to Understand the Evolution of Equilibrium Points in the Circular Restricted 3 Body Problem with Augmented Low-Thrust Terms
11:30am: Jack Iannamorelli, Stability of Periodic Orbits in the Earth-Moon Elliptic Restricted Three-Body Problem
12:00pm: Justin Cummings, Further Investigation of Numerical Examples for Nesterov's Accelerated Gradient Method

Tuesday, May 2, 2023
3:00pm: Rodrigo Schmitt, Applications of Manifold Theory
3:30pm: Cody Kapeles, Bifurcations and the Search for Periodic Orbits About Lagrange Points
4:00pm: Simon Sasin, Mathematical Background and Computations of Unstable and Stable Manifolds In CR3BP
4:30pm: Jonathan Richmond, Detecting Bifurcations in Families of Periodic Solutions of the Circular Restricted Three-Body Problem

Wednesday, May 3, 2023
9:00am: Daniel Leyva, Flow Curvature Method and Slow Invariant manifolds
4:00pm: Abhimanyu Das, Trajectory optimization for optimal control
4:30pm: Estepan Ashkarian, Control Theory, Dynamic Programming and Hamilton- Jacobi-Bellman Equation
5:00pm: Sesha Charla, Describing Function Method: Application to Nonlinear Control
5:30pm: Yuanhanqing Huang, On the Asymptotic Behavior of Second Order Differential Equations for Solving Monotone Inclusions

Friday, May 5, 2023
4:00pm: Deerajkumar Parthipan, Stability Analysis of a Modified SIR Model for COVID-19 Transmission Dynamics
4:30pm: Dale Williams, A Brief Introduction to Lagrangian Coherent Structures and their Applications
5:00pm: Anvesh Naik, Dynamical systems approach to understanding human movement
5:30pm: Naoya Kumagai, Control of chaos and perturbations in the circular restricted three-body problem

Saturday, May 6, 2023
11:00am: Brian Morton, Dynamics of ABC Flows
11:30am: Rohan Lotlikar, Dynamical Systems in Composite Materials
12:00pm: Akshay Sathiyamoorthy, Utilizing the Carleman Linearization Method to Solve Nonlinear Aeronautical Problems
12:30pm: Ethan Brady, Motivation and examples for the Smale-Birkhoff homoclinic theorem

Sunday, May 7, 2023
11:30am: John Huag, Lyapunov Functions to Solve ODEs
12:00pm: Qian Zhang, The stability of stochastic dynamical systems
12:30pm: Aneesh Khilnani, Methods of Estimation for Dynamical Systems