Stability and Asymptotic Stability of Critical Pts

• Real, Distinct, Same Sign
  • Both negative: nodal sink (stable, asymtotically stable)
  • Both positive: nodal source (unstable)
• Real, opposite sign: saddle point (unstable)
• Both Equal
  • 2 linearly independent eigenvectors (e.g. [[1,0],[0,1]]): proper node
    • the eigenvalue is negative: sink, stable, asymptotically stable
    • the eigenvalue is positive: source, unstable
  • 1 linearly independent eigenvector (e.g. [[1,0],[1,1]]): improper node
    • the eigenvalue is negative: sink, stable, asymptotically stable
    • the eigenvalue is positive: source, unstable
• Complex, real part not 0: spiral point
  • real part positive: spiral source [unstable]
  • real part negative: spiral sink [stable, asymptotically stable]
• Complex, real part equals 0 (purely imaginary): center [stable, not asymptotically stable]