Systems of ODEs: Imaginary Eigenvalues and Center Fixed Points

Explore the concept of 2x2 systems of ordinary differential equations with imaginary eigenvalues and center fixed points in this 38-minute lecture. Delve into examples of physical systems with complex eigenvalues and review basic properties of complex numbers. Learn how to compute eigenvectors, write full solutions, and gain geometric intuition about rotation matrices. Investigate the effect of adding small friction, transforming center fixed points into spiral sinks. Enhance your understanding of neutrally stable center fixed points through eigenvalue and eigenvector analysis, as well as phase portrait visualizations.

Overview
Examples of physical systems with complex eigenvalues
Quick recap of basic properties of complex numbers
Computing the eigenvectors
Writing the full solution
Geometric intuition: The solution is a rotation matrix
Adding small friction: Center becomes spiral sink