Dynamical Systems (1) (Online)

This course in some sense covers those topics necessary for a clear understanding of the qualitative theory of  ordinary differential equations and the concept of a dynamical system.

It begins with a study of linear systems of ordinary differential equations. An efficient method for solving any linear system is presented. The first topic in the course is concerned with the computation of the matrix e^{At} in terms of  the eigenvalues and  eigenvectors of the square matrix A. Moreover, Diagonalization, Jordan Forms, and Stability Theory are presented.

The major part of this course is devoted to the study of nonlinear systems of ordinary differential equations and dynamical systems. Since most nonlinear differential equations cannot be solved, this course focuses on the qualitative or geometrical theory of nonlinear systems of differential equations.

In fact, the following topics are discussed:

1.Equilibrium Solutions, Stability, and Linearized stability

2. Liapunov Functions

3. Stable, Unstable, and Center Linear Subspaces

4. Invariant Manifolds

5. Periodic Orbits, Bendexon and Dulac 's Criteria 

6. Index Theory

7. Asymptotic Behavior of Trajectories and the Invariance Principle

8. The Poincare-Bendixon Theorem

9. Center Manifolds