Understanding Positive invariance

In summary, a positive invariant set is a set of points in a system that will remain within that set as time goes on. It is distinguished from a non-positive invariant set by the behavior of the system's trajectories, with positive invariant sets being characterized by trajectories that spiral inward towards the origin. The definition on Wikipedia may seem confusing, but it is essentially stating that any trajectory starting within a positive invariant set will remain within that set for all time. Big-O notation is used in the definition, but it is not relevant to understanding the concept of positive invariance.
  • #1
omaiaa0p
8
0
Hello everyone,

I am new to the forum and was wondering if someone can help explain something to me.

I would like to understand the meaning of a positivly invariant system. I have checked the definition on wikipedia,

http://en.wikipedia.org/wiki/Positive_invariance"

and it is confusing to me. I am not advanced in Calculus so I appreciate some patience and thorough description with an example please.

My question is: What is a positive invariant set, and how can I distinguish it from a non-positive invariant one. Also, in the definition on wikipedia, it states the explanation in Big -O notation, which I am not familiar with.

Thank you in advance
 
Last edited by a moderator:
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  • #2
omaiaa0p said:
Hello everyone,

I am new to the forum and was wondering if someone can help explain something to me.

I would like to understand the meaning of a positivly invariant system. I have checked the definition on wikipedia,

http://en.wikipedia.org/wiki/Positive_invariance"

and it is confusing to me. I am not advanced in Calculus so I appreciate some patience and thorough description with an example please.

My question is: What is a positive invariant set, and how can I distinguish it from a non-positive invariant one. Also, in the definition on wikipedia, it states the explanation in Big -O notation, which I am not familiar with.

Thank you in advance

That isn't big-O notation in the standard sense. For some reason the author decided to name a set of points with a script O. He could just as well have called it A.

To see what a positive invariant set is, consider a 2-D system whose trajectories all spiral into the origin as t → ∞. If you consider a set A that is a disk of radius 1 about the point (10,10), and solution that starts in that disk is not going to stay there because the trajectory will spiral to the origin, so A is not positive invariant for this system. However, if B is a disc of any positive radius r about the origin, it is positive invariant because any trajectory starting there can't get out for t > 0 since it spirals inward toward the origin. Hope that helps.
 
Last edited by a moderator:

Related to Understanding Positive invariance

1. What is positive invariance?

Positive invariance is the concept that certain properties or characteristics of a system remain unchanged or consistent even when the system undergoes changes or disturbances.

2. Why is understanding positive invariance important?

Understanding positive invariance is important because it allows us to predict how a system will behave and how it will respond to external factors. This knowledge can be applied in various fields such as engineering, biology, and economics.

3. How is positive invariance different from negative invariance?

Positive invariance refers to the properties or characteristics that remain consistent despite changes or disturbances, while negative invariance refers to the properties that change in response to these factors. In other words, positive invariance focuses on the stability and resilience of a system, while negative invariance focuses on its vulnerability.

4. What are some real-life examples of positive invariance?

One example of positive invariance is the human body's ability to maintain a constant internal temperature despite changes in the external environment. Another example is the stability of a building's structure, which remains intact even during earthquakes or strong winds.

5. How can we apply the concept of positive invariance in our daily lives?

We can apply the concept of positive invariance by identifying and understanding the stable and consistent aspects of our lives, such as our values, beliefs, and relationships. This can help us to navigate through changes and challenges with resilience and stability.

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