CDS - Examples of Phase Space drawn from Eigenvalues and Eigenvectors • Saddle • Marginally Stable Steady State

Remember:

Saddle, if we have one positive and one negative eigenvalue:

Centers, If we have complex conjugate eigenvalues with zero real part:

Cloud of steady states, for$\{\begin{array}{l}{\dot{x}}_1=0\\ {\dot{x}}_2=0\end{array}$So for all eigenvalues nullnot-sure-about-this
:

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• ${\lambda }^2-\tau \lambda +\Delta$ is called the “charateristic polyomial”

• This is the plot, and with this we can define all the “types of nodes” or flow, for a 2 dimensional system.
• Remember that:
  • $\tau ={\lambda }_1+{\lambda }_2$
  • $\Delta ={\lambda }_1\cdot {\lambda }_2$ Example of nodes (stable - black flow, and unstable - red flow):
    

Example of a saddle:

Analizing the initial conditions on the $x$ and the initial conditions on the $y$ axis, we have made example also in the phase plane, and said that: The eigenvectors are invariant for the motion.

• However this is not very clear/useful i think (Lecture 6 27:00)not-sure-about-this

Stable (in red) and unstable (in black) spiral:

• In the phase plane, stable spiral:
  
• unstable spiral:
  

While in case of “centers” (for $a=0$):

• In the phase plane:
  

Somenthing i skipped, example of stable and unstable ss:

Somenthing i skipped, example of marginally ss: