### Author Topic: differences between proper and improper nodes  (Read 13239 times)

#### Richard Qiu

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##### differences between proper and improper nodes
« on: November 18, 2019, 02:19:33 PM »
Hello guys, could anyone help me to explain the differences between proper and improper nodes? btw, any suggestions on how to remember the types and stability of the critical points?

#### Amanda-fazi

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##### Re: differences between proper and improper nodes
« Reply #1 on: November 18, 2019, 02:35:47 PM »
Since both proper and improper nodes have equal eigenvalues, the differences between these two nodes is that: proper node/star point has two independent eigenvectors, while improper/degenerate node has only one independent eigenvector by (A-rI)x =0, and we create a generalized eigenvector associated with the repeated eigenvalues by letting (A-rI)y = x.

« Last Edit: November 18, 2019, 03:21:26 PM by Amanda-fazi »

#### Amanda-fazi

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##### Re: differences between proper and improper nodes
« Reply #2 on: November 18, 2019, 02:49:15 PM »
There are mainly 5 cases of Eigenvalues(from book Elementary Differential Equations and Boundary Value Problems-11th Edition section 9.1):
as it is mentioned above, the equal eigenvalues case mentioned above is CASE 3.

CASE 1: Real, Unequal Eigenvalues of the Same Sign
CASE 2: Real Eigenvalues of Opposite Sign                            ->saddle point
CASE 3: Equal Eigenvalues
CASE 4: Complex Eigenvalues with Nonzero Real Part
CASE 5: Pure Imaginary Eigenvalues                                     ->center

After memorized there are five cases, CASE 1, CASE 3 and CASE 4 have two branches while the rest of the cases(CASE 2 and CASE 5) only have one:
to be more specific:

CASE 1: Real, Unequal Eigenvalues of the Same Sign separated into:
a)lambda1 >lambda2 >0:
critical point called node/nodal source
a)lambda1 <lambda2 <0:
critical point called node/nodal sink

CASE 3:Equal Eigenvalues separated into:
a)two independent eigenvectors:
critical point called proper node or star point
b)one independent eigenvector:
critical point called improper node or degenerate node

CASE 4:Complex Eigenvalues with Nonzero Real Part separated into:
a)pointing-outward trajectories as lambda > 0:
critical point called spiral source
a)pointing-inward trajectories as lambda < 0:
critical point called spiral sink

For the stability, as long as there is one lambda>0, then it is unstable, and the last one lambda=0 is stable. For the rest of them, asymptotically stable applied.

« Last Edit: November 18, 2019, 03:20:51 PM by Amanda-fazi »

#### anntara khan

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##### Re: differences between proper and improper nodes
« Reply #3 on: November 20, 2019, 04:03:25 PM »
I made this handy color coded guide to help me remember all the cases:

« Last Edit: December 05, 2019, 09:37:34 PM by anntara khan »

#### anntara khan

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##### Re: differences between proper and improper nodes
« Reply #4 on: December 11, 2019, 09:59:34 PM »
Based on the stability near locally linear system I have extended the previously posted table, hope this helps remembering