MAP 6336: Ordinary Differential Equations.

Instructor: J.D. Mireles James.

Textbook: The course will follow the lecture notes wirtten by J.P. Lessard, Konstantin Mischaikow, Marcio Gameiro and myself.

Notes: (The notes will be updated often. I will make announcements in class).

Course Description, Objectives, Learning Outcome Goals: This is a graduate course in ordinary differential equations. The main goal of the course is to cover in parallel the classical results of the subject as well as more modern constructive methods which lead to mathematically rigorous computational techniques. This is possible because of the strong resonance between traditional theoretical developments in the theory of ODEs and the so called a-posteriori methods for computer assisted proof in analysis.

Assignments and Grades: There will be weekly homework assignments. Students will choose (with help from the instructor, and subject to instructor approval)an end of term project. Students may choose to work in small groups. Final grades in the course will be based on the homework and project grades. The grades will weighted as: 60 percent homwork, 30 percent projects, 10 percent class participation. There will be no midterm or final exams. We may use the final exam time slot for some student presentations.

IFP General Education Outcomes:
  • Knowledge in several different disciplines;
  • The ability to think critically;
  • The ability to communicate effectively;
  • An appreciation for how knowledge is discovered, challenged, and transformed as it advances; and
  • An understanding of ethics and ethical behavior.
  • Information available at Computers: We do not assume that students have a strong background in computer programing. Our aim is that students will learn as we proceede by first running our programs, then modifying them, and then finally writing their own. Programing will be done MatLab. Later we will talk about the IntLab library for interval arithmetic in MatLab.

    Academic Honesty:
    Students at Florida Atlantic University are expected to maintain the highest ethical standards. Academic dishonesty is considered a serious breach of these ethical standards, because it interferes with the university mission to provide a high quality education in which no student enjoys an unfair advantage over any other. Academic dishonesty is also destructive of the university community, which is grounded in a system of mutual trust and places high value on personal integrity and individual responsibility. Harsh penalties are associated with academic dishonesty. For more information, see University Regulation 4.001.

    Students with Disabilities:
    In compliance with the Americans with Disabilities Act (ADA), students who require special accommodation due to a disability to properly execute coursework must register with the Office for Students with Disabilities (OSD) -- in Boca Raton, SU 133 (561-297-3880) and follow all OSD procedures.

    Chapter Topics
    Ch 1 Motivation
    1.1 Lorenz Equations
    1.2 FitzHugh-Nagumo
    1.3 Swift-Hohenberg
    1.4 Gray-Scott
    1.5 Kuramoto-Sivashinsky/Michelson
    1.6 Ginzburg-Landau
    Ch 2 Existence and Uniqueness Theory
    2.1 Review of Calculus
    2.2 Contraction Mapping Theorem.
    2.3 Existence and Uniqueness of ODEs
    2.4 Flows and Topological Equivalence.
    Ch 3 Equilibria
    3.1 Newton's Method
    3.2 Implementations: Newton-Kantorovich and Newton-Lessard
    Ch 4 Continuation of Equilibria
    4.1 Classical finite dimensional continuation
    4.2 Uniform Contraction Theorem.
    4.3 Bifurcations: Saddle-Node, Transcritical, and Pitchfork.
    Ch 5 Periodic Orbits
    5.1 Peroidic Orbits
    5.2 Formal Solution: Fourier Series, Banach Spaces of fast decaying coefficients, Fixed Point Problem and Frechet Differentials
    5.3 Convolution estimates
    5.4 Bootstrap regularity
    5.5 Finite Dimensional Projections
    5.6 Radii Polynomials
    5.7 Continuation of periodic solutions
    Ch 6 Initial and Boundary Value Problems
    6.1 Chebyshev Polynomials
    6.2 Initial Value Problems
    6.3 Boundary Value Problems
    Ch 7 Linear Theory (Equilibria)
    7.1 Stability
    7.2 Hyperbolicity
    7.3 Hartman-Grobman
    7.4 Hopf Bifurcation
    7.5 Existence and Computation of Stable and Unstable Manifolds: Linear Approximation, Higher Order Approximation
    Ch 8 Linear Theory (Periodic Orbits)
    8.1 Floquet Theory
    8.2 Linear stability and linear bundles
    8.3 Stable and Unstable Manifolds
    Ch 9 Connecting Orbits
    9.1 Operator Equation: free boundary problem
    9.2 Transversality
    9.3 Continuation
    Ch 10 Chaos
    10.1 Smale Tangle Theorem and applications rigorous computation

    TOPICS: To be decided by instructor based on interest and progress of the students.


    Short MatLab Tutorial: Download these files and put them in a folder (there is a zipped folder below containing all the files). Then start MatLab either (a) from that folder or (b) start MatLab and then navigate to the folder. The main example files are:
    To run any of these programs just type its name (without the .m extension) at the MatLab command prompt. I hope that they are easy to understand and play with.

    Basic MatLab Operations: (matlabCalculator.m) This file contains some very simple matlab commands. Rather than running this file I just suggest entering the instructions by hand and even varying them/experimenting a little
    Use of M-Files: (files_example.m) This file looks at the use of M-File function calls by studying the dynamics of the logistic map.
    The Logistic Map File: (logisticMap.m) This is the file needed in order to run the program files_example.m
    More Use of M-Files and loops: (files_exampleII.m) This program illustrates the use of loops and function calls and some plotting.
    Logistic Orbit File: (logisticsOrbit.m) Program needed in order to run files_exampleII.m.
    Differential Equations: (ode_example.m) This program illustrates how MatLab is used to compute numerical solutions of ordinary differential equations with some plotting.
    Lorenz Vector Field File: (lorenzField.m) The file containing the Lorenz vector field. This file is needed in order to run ode_example.m.

    Zipped Version: (matLabFiles.tar.gz) Zipped folder containing all of the matLab files above. This is the best way to get the files.

    Simple MatLab Exercises: (Problems File) These are some problems you can try in order to get the hang of computing using MatLab.