Normal Forms for Vector Fields

Explore normal forms for vector fields in this comprehensive lecture by Dr. Shane Ross from Virginia Tech. Delve into the simplification of nonlinear terms in dynamical systems, focusing on autonomous vector fields with equilibrium points. Learn how normal forms serve as the simplest representation for nonlinear terms, analogous to Jordan canonical form for linear terms. Discover the process of simplifying 2nd order terms through near-identity nonlinear transformations and examine the conditions for complete simplification. Investigate the Bogdanov-Takens normal form as a specific example arising from systems with a double-zero eigenvalue. Gain insights into determining categories of topological equivalence and apply these concepts to both 1D and 2D vector field examples.

Syllabus

Simplifying nonlinear terms and determining categories of topological equivalence.
1D example of simplifying a 1-dimensional ODE.
Normal form theory.
Simplifying the 2nd order terms.
Example of normal form for 2-dimensional vector field.
Takens normal form.
Bogdanov normal form.