What are the different types and stabilities of critical points?

In summary, the types of critical points discussed in the conversation are proper node, saddle point, improper node, and center. The stability of these critical points is categorized as unstable, unstable, stable and attractive, and stable. The definitions of these stability types are described as follows: asymptotically stable, all trajectories approach the equilibrium solution; stable, trajectories remain equidistant from the equilibrium solution; unstable, trajectories depart from the equilibrium solution.
  • #1
ineedhelpnow
651
0
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.

View image: 20151215 084536 1

In order: proper node, saddle point, improper node, center
But I don't understand how to figure out the stability
 
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  • #2
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?
 
  • #3
Well I figured them out to be unstable, unstable, stable+attractive, stable. Not sure if it's correct
 
  • #4
Yep, those seem right to me :)
 

FAQ: What are the different types and stabilities of critical points?

What is the purpose of identifying critical points?

Identifying critical points allows scientists to better understand the behavior of a system or process. These points represent key moments where significant changes or shifts occur, and can help inform decision making and problem solving.

How do you identify critical points?

Critical points can be identified through careful observation, data analysis, and experimentation. Scientists may also use mathematical models and simulations to identify and predict critical points.

Can critical points change over time?

Yes, critical points can change as the system or process being studied changes. This could be due to external factors, such as environmental changes, or internal factors, such as the introduction of new variables or conditions.

What is the difference between a critical point and a turning point?

A critical point is a specific moment or value where a significant change occurs in a system or process. A turning point is a more general term that can refer to any point where a change in direction or outcome occurs.

Why is it important to accurately identify critical points?

Accurately identifying critical points is important because it allows scientists to better understand and predict the behavior of a system or process. This information can be crucial for making informed decisions and developing effective solutions to problems.

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