Understanding Positive invariance

[omaiaa0p]

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

 

[LCKurtz]

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.