Mr. Integral

THE BIG IDEA

Systems and Phase Planes

How do coupled variables move together?

Write several evolving quantities as one state vector. Eigenmodes and matrix exponentials explain linear motion; phase portraits make the trajectories visible, and Jacobians connect the picture to nonlinear systems locally.

Begin the first lesson →See the lesson route ↓

5 lessons · No deadlines or locked lessons

SEE THE CONNECTIONS

How the ideas fit together.

A map of the main ideas. Follow a node to its lesson.

  1. Start with01Eigenmodes
  2. their time behavior shapes02Phase portraits
  3. a general propagator is03Matrix exponentials
  4. local Jacobians connect to04Nonlinear neighborhoods

YOUR LEARNING ROUTE

One idea at a time.

Read the explanation, use the visualization, then try the check before opening its reasoning.

Keep the unit cheat sheet handy →
Lessons in this unit5 lessons
  1. 01Linear Systems and EigenmodesConvert higher-order equations to first-order systems and use eigenvectors to solve them.
  2. 02Phase Portraits and StabilityClassify planar linear equilibria and follow time-oriented trajectories.
  3. 03Matrix Exponentials and Repeated ModesSolve a constant system even when the matrix lacks an eigenvector basis.
  4. 04Forced Systems and Variation of ConstantsCombine the free state with the accumulated response to a vector input.
  5. 05Nonlinear Systems and Local LinearizationUse a Jacobian near an equilibrium and recognize inconclusive eigenvalues.