This series of videos helped me so much to understand this Thm. Thank you so much for the clear lecture. If I may ask a couple of questions, at 1:58 is it the case that you only modified r-dot just to show an example of how Poincare-Bendixson Thm helps create a trapping region whenever you have an r-dot that's coupled with theta? Because, otherwise, if you have an r-dot with only r's and not coupled with theta, it's simpler to see where the limit cycle is just by graphing and determining if trajectories point toward the equilibrium solution on each side?l, correct? Thanks!
I'm glad the videos are helping you. "is it the case that you only modified r-dot just to show an example of how Poincare-Bendixson Thm helps create a trapping region whenever you have an r-dot that's coupled with theta?" Yes. "
I know i'm posting it late, but my question is what if you can't find the poicare orbit, does it prove it is not there or just that you simply can't find it? ____________________________ I know poicare method is used to prove limit cycles, can poicare be used to disprove limit cycles or no?
There is a limit cycle iff each of the 4 supposed statements in the Poincare-Bendixson Theorem are shown to be true. i.e. if any of of those statements can be proven false, we have shown that there is no limit cycle. Please correct me if I am wrong but this is how I understand it.
Have an exam tomorrow, this proved extremely helpful! Wish I found it sooner 😅
Glad to help. Good luck on the test!
SAME LMAO
Your video help me lot to clear my doubts...thank you so much..
This series of videos helped me so much to understand this Thm. Thank you so much for the clear lecture. If I may ask a couple of questions, at 1:58 is it the case that you only modified r-dot just to show an example of how Poincare-Bendixson Thm helps create a trapping region whenever you have an r-dot that's coupled with theta? Because, otherwise, if you have an r-dot with only r's and not coupled with theta, it's simpler to see where the limit cycle is just by graphing and determining if trajectories point toward the equilibrium solution on each side?l, correct? Thanks!
I'm glad the videos are helping you. "is it the case that you only modified r-dot just to show an example of how Poincare-Bendixson Thm helps create a trapping region whenever you have an r-dot that's coupled with theta?" Yes. "
Thank you so much for these video. You help me a lot!
0:50 - if you can not establish it analytically, you can only assign some probability to such a situation.
I'm still confused about showing the vector field direction in that slope. do we just need to compare dx and dy there?
This lecture seems inspired from Prof. Steven Strogatz’s lectures on dynamical systems.
ruclips.net/video/nWO74rlr58Y/видео.html
He is teaching out of Strogatz's book on nonlinear dynamics and chaos
I would imagine that you could write a region given by 0.9 sqrt(1+mu cos theta)
I know i'm posting it late,
but my question is what if you can't find the poicare orbit, does it prove it is not there or just that you simply can't find it?
____________________________
I know poicare method is used to prove limit cycles, can poicare be used to disprove limit cycles or no?
There is a limit cycle iff each of the 4 supposed statements in the Poincare-Bendixson Theorem are shown to be true.
i.e. if any of of those statements can be proven false, we have shown that there is no limit cycle. Please correct me if I am wrong but this is how I understand it.
@@kylejohnson8447 Does it prove that there are no limit cycles or does it prove that you can't determine if it does or doesn't?
@@presidentevil9951 if any of of those statements can be proven false, we have shown that there is no limit cycle.
@@kylejohnson8447 then whats the point of the lapinov and dulac when we can just use this one?
@@presidentevil9951 that’s a good question
Omg is that pplot
There are only a few living who still remember pplot…