MHB What are the different types and stabilities of critical points?

[ineedhelpnow]

identify the type of critical point as spiral, center, saddle, proper node or improper node. in addition, identify the stability of the critical point as stable, unstable, or attractive and stable.

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In order: proper node, saddle point, improper node, center
But I don't understand how to figure out the stability

 

[Dethrone]

I'm assuming that "attractive and stable" is the same as asymptotically stable. I'm not sure how they're defined in your book, but these are how I define them:
Asymptotically stable: all trajectories near the equilibrium solution (or points or critical points) approaches it.
Stable: a given trajectory doesn't get closer or farther away from the equilibrium solution...it stays equidistant.
Unstable: all trajectories near the equilibrium solution depart from it.

Given these definition, how would you classify them?

 

[ineedhelpnow]

Well I figured them out to be unstable, unstable, stable+attractive, stable. Not sure if it's correct

 

[Dethrone]

Yep, those seem right to me :)