Course web page: Available on CMU Canvas

Lectures: MWF 3:20 -4:10 pm online via Zoom

Professor: Robert Pego     Email: rpego AT cmu.edu

• Office hours after 1st week: M 8:30-9:30pm, TTh 5-6pm, or just arrange a time

• Ordinary Differential Equations with Applications by Carmen Chicone, 2nd ed., Springer-Verlag, 2006.
  Has a modern selection and treatment of topics close to the approach we will take in the course.
  The classic sources below are more comprehensive within their scope, however.
• Differential Equations and Dynamical Systems by Lawrence Perko, 3rd ed., Springer-Verlag, 2001.
  Treats most of the topics we will cover in roughly the same order, but at a too-slow pace and with too-limited applicability for us. May be a good source of basic exercises.

Classic reference books

• Theory of Ordinary Differential Equations by E. Coddington and N. Levinson, McGraw-Hill, 1955.
• Ordinary Differential Equations by P. Hartman, Birkhauser, 1982 (reprint of the 2nd ed., 1973).
• Dichotomies in Stability Theory by W. A. Coppel, Springer Lec. Notes in Math. 629, 1978.

Additional/supplementary texts (less advanced)

• Differential Equations, Dynamical Systems, and an Intro. to Chaos by M. Hirsch, S. Smale and R. Devaney, Elsevier 2004
• Nonlinear dynamics and Chaos by S. Strogatz, Perseus, 2nd ed. 2014.

Course Description:  

Outline of topics (see separate pdf for details)   The course is a fundamental introduction to the qualitative theory of ordinary differential equations (ODE), particularly emphasizing a dynamical systems approach and a geometric viewpoint. We develop basic infrastructure for analyzing solutions of ODE (theorems on existence and uniqueness, dependence on parameters, continuation), study questions of stability and instability (using Lyapunov functions, linearization, Floquet theory, dichotomies), develop geometric methods (Poincare-Bendixson index theory, attractors and flows, invariant manifold theorems) and study generic structure (normal forms, basic bifurcation theory). Much motivation comes from applications in many areas of science and engineering.

Prerequisites: real analysis (21-355) and basic linear algebra (21-241) are essential, but no other ODE or measure theory.

Learning objectives:   Throughout this course, students will learn to apply integral inequalities to the understanding and analysis of continuous-time dynamical systems. They will master a variety of methods to obtain the existence of solutions on a maximal domain and study their stability with respect to time and parameters. They will gain knowledge of concepts, techniques, and structures that facilitate the analysis of long-time behavior and stability. They will learn to characterize the structure of subspaces of solutions for linear systems, and manifolds of solutions in nonlinear systems.

Grading: Based on approx. 6 homework sets and a midterm and final exam.

Homework will be posted on the course Canvas page, and solutions should be uploaded into Gradescope.

Academic integrity requires that your tests and homework solutions are your independent work and not copied from other sources. The CMU policy is at http://www.cmu.edu/policies/student-and-student-life/academic-integrity.html . On homework you are encouraged to discuss with others and consult other resources to improve your understanding, but it is important to develop your independent capacity to problem-solve.

Make-up tests are only possible in the case of a documented medical excuse, a university-sanctioned absence (e.g., participation in a varsity sporting event), or a family emergency. Please contact me at the earliest time possible to schedule a make-up.

Health and wellness: Graduate study is time-consuming and can be stressful. Make your health a priority, be smart about time management, seek social support, and do ask for help when needed.

Quote: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions. A simple system representing cellular convection is solved numerically. All of the solutions are found to be unstable, and almost all of them are nonperiodic. The feasibility of very-long-range weather prediction is examined in the light of these results ---Edward Lorenz, Abstract from his landmark paper introducing Chaos Theory in relation to weather prediction, 'Deterministic Nonperiodic Flow', Journal of the Atmospheric Science (Mar 1963), 20, 130.