MA 543: Ordinary Differential Equations and Dynamical Systems
Spring 2018, 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 linear systems, local theory for nonlinear systems (existence and uniqueness, dependence on parameters, flows and linearization, stable manifold theorem) and their global theory (global existence, limit sets and periodic orbits, Poincare maps). Some further topics include bifurcations, averaging techniques and applications to mechanics and mathematical biology.

Prerequisites:

One undergraduate course in each of the following topics:
linear algebra (for example, MA 265, 351),
differential equation (for example, MA 266, 366),
analysis (for example, MA 341, 440, 504).

Textbooks and References: (All are available online through Purdue library page.)

Main text:
[M] Differentiable Dynamical Systems, J. D. Meiss

References:
[HSD] Differential Equations, Dynamical Systems: An Introduction to Chaos, M. W. Hirsch, S. Smale, R. L. Devaney

[T] Ordinary Differential Equations and Dynamical System, G. Teschl, available online

[S] Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry, and Engineering, S. H. Strogatz

[GH] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, J. Guckenheimer, P. Holmes

[A] Mathematical Methods of Classical Mechanics, V. I. Arnold

Some texts at the undergraduate level:
[BP] Elementary Differential Equations and Boundary Value Problems, Boyce and DiPrima (textbook of Purdue MA266, on reserve in libray)

[B] Differential Equations and Their Applications, M. Braun

Course Announcement:

Your grade is based on class participation (though I do not intentionally take attendance), typing up of class notes (shared among the whole class), a short (say 5 to 6 pages) paper, and a short (30 min) presentation about the paper toward the end of the semester, probably during the final exam week or earlier.

Please submit a title and abstract (with just regular text, no mathematical formula) of your paper by e-mail, by Mar 30.


Week 1:

Tue Jan 9:
[M, Chapter 1]
introduction, notations,
conversion between higher order equation and first order system,
examples;



Week 2:

Tue Jan 16:
introduction:
explicit solution, integrating factor, finite time blow-up of nonlinear equations, non-uniqueness of solutions.

Thur Jan 18:
``invariant domain'', estimation of long time behavior of solutions,
matrix exponential, solution of linear, inhomogeneous system with constant matrix, dX/dt = AX + h(t).


Week 3:

Tue Jan 23:
norm(s) of vector and matrices;
uniqueness of solution for linear inhomogeneous system, dX/dt = A(t)X + h(t), Gronwall's inequality,
successive approximation, Picard's iteration.

Thur Jan 25:
existence of solution for linear inhomogeneous system, dX/dt = A(t)X + h(t), Picard's iteration,
fundamental solution, dPhi(t)/dt = A(t)Phi(t), Phi(0)=I,
invertibility of fundamental solution,
variation of parameters and solution formula for linear inhomogeneous systems, dX/dt=A(t)X + h(t)


Week 4:

Thur Feb 1: (M, Chapter 2)
Abel Theorem for det(Phi(t));
(computation of) exponential of a matrix, diagonalization, eigenvectors and generalized eigenvectors for repeated eigenvalues, Jordan canonical form,
invariant subspaces, stable, unstable, and center subspaces.


Week 5:

Tue Feb 6: (M, Chapter 3)
Existence and uniqueness of nonlinear differential equations,
continuity and Lipschitz condition for functions,
existence and uniqueness of dX/dt=F(X) when F is Lipschitz, by Picard's iteration and Gronwall's lemma.

Thur Feb 8: (M, Chapter 3)
Existence and uniqueness of nonlinear differential equations (cont'), space of continuous functions, compactness (Arzela-Ascoli Theorem),
contraction mapping (Banach Fixed Point Theorem), explicit Euler scheme, Schauder Fixed Point Theorem,
Gronwall's Inequality (Lemma 3.13), Lipshitz dependence of solution on initial data (Theorem 3.14).


Week 6:

Tue Feb 13: (M, Chapter 3)
Differentiability of vector valued functions, linear approximation (D_X F(X)),
smooth (C^1) dependence of solution on initial data (Theorem 3.15), smooth (C^1) dependence of solution on initial data and parameter (Theorem 3.16),
local Lipschitz condition, local in time existence of solution (Lemma 3.12), maximal interval of existence (Theorem 3.17).

Thur Feb 15:
[M 3.5] limit of solution near the end of the maximal interval of existence,
[M 4.3] global existence of solutions with linear growth of F(X),
(Bellman, Chapter 2) exponential decay of solutions and eigenvalues, perturbation results for stability of solutions.


Note Taking Activity:
(Each week's typed note is due in two Tuesdays afterward.)

Jan 23, 25: Dustin Enyeart
Jan 30, Feb 1: Fouad Khoury
Feb 6, 8: Yang Mo
Feb 13, 15: Vivek Muralidharan
Feb 20, 22: Nik Prljevic
Feb 27, Mar 1: Alex Van Anderlecht and Yangjia Xie
Mar 6, 8: Dustin Enyeart
(Spring Break)
Mar 20, 22: Fouad Khoury
Mar 27, 29: Yang Mo
Apr 3, 5: Vivek Muralidharan
Apr 10, 12: Nik Prljevic
Apr 17, 19: Alex Van Anderlecht and Yangjia Xie
(Last week of class)