** 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).

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.

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.

APPROXIMATE SYLLABUS

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. |

files_example.m

files_exampleII.m

ode_example.m

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.