Linear EPS in 2D and higher

Summary --- Linear EPS in 2D and higher

  • Simple EPs in 2D. definition of EP in 2D
  • Stability of 2D is tricky. Can use test points, but not always clear where they are pointing. Need lots of test points with very small \( \Delta_t \)
  • Is there a principle of Linear Stability Analysis in 2D? Yes, but we haven’t learned how to take the derivative (the linear approximation) of a 2D function. Will learn that next quarter.
  • Still true in 2D that all EPs are linear
  • How to make a 2D EP: mix and match 2 1D systems. Purely stable, purely unstable, saddle point.
  • Truly 2D EPs (not decomposable). Recall the example of the spring from ch 1. Derive dif eqs and EPs for stable spiral, unstable spiral, neutral center.
  • EPs in higher dimensions. Principle of mix and match still works. Case in 3D with a 2D unstable spiral and a 1D stable EP