By the end of the course, the student will learn the following components of the theory: 

• Autonomous differential equations as dynamical systems: continuous dependence on the initial state, continuation, and the group property. Limit sets.
• Autonomous systems with periodic motions: Poincaré-Bendixson Theorem; Linearization Near Periodic Orbits; Orbital stability.
• Integral manifolds as a method of dimension reduction. Theorems on the existence of stable and unstable manifolds.  The center manifold theorem. 
• Bifurcation as a change of topological structures: the Saddle-Node Bifurcation, the Transcritical Bifurcation, the Pitchfork Bifurcation, and the Hopf Bifurcation.  
• Method of Small Parameters in Critical and Noncritical Cases.