2x2 Systems of ODEs: Imaginary Eigenvalues and Center Fixed Points

This video investigates a 2-dimensional linear system of ordinary differential equations with a pair of purely imaginary complex conjugate eigenvalues. These solutions are known as neutrally stable center fixed points. We investigate the solutions using eigenvalues and eigenvectors, as well as with phase portrait pictures. Playlist: https://www.youtube.com/playlist?list=PLMrJAkhIeNNTYaOnVI3QpH7jgULnAmvPA Course Website: http://faculty.washington.edu/sbrunton/me564/ @eigensteve on Twitter eigensteve.com databookuw.com This video was produced at the University of Washington %%% CHAPTERS %%% 0:00 Overview 3:09 Examples of physical systems with complex eigenvalues 5:12 Quick recap of basic properties of complex numbers 9:50 Computing the eigenvectors 16:34 Writing the full solution 25:04 Geometric intuition: The solution is a rotation matrix 31:15 Adding small friction: Center becomes spiral sink