EXMAPLE II

• $a <-1$ = $x$ direction is faster than $y$
• $-1 < a < 0$ = $y$ direction faster than $x$

Saddle point

• x unstable

• y stable

• stable manifold

  • “start on $y$ axis ends up at $x^*$, but small deviation on $x$ axis throws this point away

Stability language

• neutrally stable = such as periodic/circle trajectories? - oscillator

Classification

• $\bold{v}$ - is eigenvector?

Example I

• first eigenvectors on the phase portrait
  • that enables us to draw any other line
  • eigenvalues = stabilty
    • negative = stable
    • positive = unstable
  • eigenvectors = direction

Example II

• $\lambda_2 < \lambda_1 < 0$
• $|\lambda_2| > |\lambda_1|$ $\implies$ $\lambda_2$ is the faster direction

Classification of fixed points

• For exam

Existence, uniqueness

• Trajectories can never cross!!
• Trajectories can be trapped in circular trajectories.