Carlos J. García-Cervera.
South Hall, 6707.
Office Phone: (805) 893 3681
Office Fax: (805) 893 2385
Office Hours: Tuesday and Thursday 2:00-3:30pm

• Differential Equations and Dynamical Systems, by Lawrence Perko.
• Ordinary Differential Equations, by Fritz John.
• Differential Equations, Dynamical Systems, and Linear Algebra, by Morris W. Hirsch and Stephen Smale.
• Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, by John Guckenheimer and Philip Holmes.
• Mathematical Methods of Classical Mechanics, by V.I. Arnold.
• Classical Mechanics, by Herbert Goldstein.
• Introduction to Dynamical Systems, by D.K. Arrowsmith and C.M. Place.
• Ordinary Differential Equations, by Philip Hartman.

• Vector calculus, by Jerrold E. Marsden and Anthony J. Tromba.

Course description: This is the first part of a graduate course in Ordinary Differential Equations (ODEs) and Dynamical Systems. In this course we do not assume previous knowledge in ODEs. We will begin by studying some techniques for finding exact solutions to some ODEs, such as separation of variables and integrating factors. We will develop the general existence and uniqueness theory for ODEs (Peano's Theorem and Picard's Theorem, global solutions and extensions, dependence on initial data and parameters), and then study linear systems of equations. For this we will need some results from Linear Algebra, which will be used to construct the Exponential Matrix. We finish studying the stability of systems of equations with respect to small perturbations.

Prerequisites: Mathematics 118A-B-C.

Assignments and grading: Homework will be assigned every two weeks. It will be handed out on Thursdays, and will be collected at the beginning of the class on the Thursday of the second following week. Late homeworks will not be accepted. The homework will generally consist of both theoretical and practical questions.

In addition to the homework, there will be a final exam. Your final grade for the course will be decided according to the following formula:

Final Grade = 60% Homework + 40% Final

Syllabus: During this course we will try to follow the following schedule. However, much like everything said earlier, this is subject to change.

Weeks 1 & 2: Preliminaries: Elementary methods.

1. Generalities and basic terminology.
2. Separation of variables.
3. Integrating factors.
4. Change of variables: Bernouilli's equation and Euler's equation.
5. Geometric interpretation: flow of an ODE.

Weeks 3 & 4: Local existence theorems.

1. Peano's theorem.
2. Picard's theorem.

Week 5: Global solutions.

1. Maximal interval of existence.

Weeks 6 & 7: Linear Ordinary Differential Equations.

1. Generalities: Fundamental solution matrix.
2. Linear equations with constant coefficients: Exponential matrix.
3. Non-homogeneous linear equations.

Weeks 8 & 9: Dependence on initial data and parameters.

1. Continuity with respect to initial data and parameters.
2. Differentiability with respect to initial data and parameters.

Weeks 10 & 11: Perturbation theory and structural stability.

1. Linear case and small perturbations.
2. Direct method of Liapunov.