CDS - Examples of Phase Space drawn from Eigenvalues and Eigenvectors • Nodes • Degenerate Node • Spirals • Circles

Remember:

Suppose to have the following system:$\dot{x}=Ax$And $x\in {\mathbb{R}}^2$, for simplicity Let’s see how the phase space and stabilty and type of the steady steate changes, with the eigenvalues:

• 2 real-negative eigenvalues ⇒ stable node (or star if ${\lambda }_1={\lambda }_2$).
• 2 real-positive eigenvalues ⇒ unstable node (or star if ${\lambda }_1={\lambda }_2$).
• 2 real-positive eigenvalues but with eigenvectors $v_1=v_2$ ⇒ degenerate node.
• 1 real-positive and 1 real-negative eigenvalues ⇒ saddle.
• 2 complex-conjugate eigenvalues but with negative real part ⇒ stable spiral.
• 2 complex-conjugate eigenvalues but with positive real part ⇒ unstable spiral.
• 2 complex-conjugate eigenvalues but with zero real part ⇒ center.

Stable node:
Unstable node:
Degenerate node:
Spiral:
Center:

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Let’s make an example: where we take 2 negative eigenvalues ${\lambda }_1$ and ${\lambda }_2$:

We can draw the flow, suppose $|{\lambda }_1|>|{\lambda }_2|$:

Also if instead we took the two eigenvalues both positives, the geometry would have been the same, but the flow would have been reversed (unstable ss):

Example of “degenerate node”, with both ${\lambda }_1<0$ and ${\lambda }_2<0$, and $v_1=v_2$:

Again if we change: ${\lambda }_1>0$ and ${\lambda }_2>0$, we obtain the same geometry with inverted flow:

• We can say that degenerate nodes lie in-between nodes and spiral.

Complex eigenvalues:

• For $a<0$:
  
• Remeber that if we where to draw the evolution of $x$ over $t$ (the “phase plane”) we would have:
  
• Like before for $a<0$ we would invert the flow.
• For $a=0$: