the way you motivate the properties of the lyapunov function is so natural, and puts this lesson leaps and bounds beyond the others i've seen. now, the lesson will stick. thank you
It's so funny when your University teachers make theorems SOOOO damn difficult to understand and you find a 10 min video that explain a 2 hour Lecture+ better,simpler and much easier to understand. Perfect explanation,thx for saving lives here!!
Thanks so much for the feedback, I really appreciate it. My experience is overwhelmingly that most things do have simple explanations. So keep doing what you're doing - if you find something confusing, go explore and study for yourself, there is likely a simple explanation out there!
How do you approach to a general solution for stability from a stand point of a LYAPUNOV functions for a class of nth order nonlinearar differential equations?
Sort of a tangential question (?). Does the eigenvalues of a system correlate to the lyapunov function somehow? Because I notice that negative eigenvalues means that the system is stable, a positive eigenvalue means that a system's exponents diverge, and a 0 eigenvalue means that the system is constant somehow and doesn't diverge to infinity or converge to 0. I feel like I'm drawing lines but I can't seem to generalize it.
No. It can only exist around an asymptotically stable equilibrium. (We can prove that an equilibrium is asymptotically stable by showing that a Lijapunov function exists there.)
Brilliant question - and it sounds like you found the answer already. There are very strong senses in which a Lyapunov function is guaranteed to exist. For example, if I have a system \dot{x}=f(x) which is globally asymptotically stable, there will always be a Lyapunov function that proves it. So it is rather like a very precise generalisation of the concept of potential energy, but for general dynamic systems as you say - very cool stuff! A slight note of caution though - I'm a bit fuzzy on the precise details myself, but I believe you can also prove things like: finding a Lyapunov function is in general as difficult as solving the original differential equation for all initial conditions. And if you can do this, you wouldn't need the Lyapunov function anymore. And to make things worse, lots of the usual tricks you might think of to approximate them - for example approximating the set of all Lyapunov functions by some nice function classes (for example sum of squares polynomials) - also might not work. That is given a globally asymptotically stable dynamical system, the Lyapunov function that proves it might not be differentiable or smooth. So while it is very nice to know that a function exists, don't expect any free lunches!
Hello, can you help me please, I am working on bifurcations and solving a system of 3 equations with 6 variables and solving them using a Local Method of Lyapunov - Schmidt, I need some help please
the way you motivate the properties of the lyapunov function is so natural, and puts this lesson leaps and bounds beyond the others i've seen. now, the lesson will stick. thank you
It is like I'm being taught by a floating head and hands. This is great!
hahaha, yes indeed! I've finally found my true calling :)
This is an amazingly intuitive explanation, especilaly the part about the dot product towards the end, thank you!
The energy level correlation to the lyapunov function is beautiful and so intuitive
Great. I have been searching for Lyapunov function properties. Especially to understand the intuitive idea about Vdot(x) = Grad(x)f(x)
The best explaining video (and material) I have found on internet... Great Job and keep going
You explained in 10 minutes what my university couldn't explain in 60.
Beautifully explained!
Thanks!!
Thanks for the explanation, literature and websites had me worried but it's much simpler than what I thought.
Glad it helped!
Made things super clear
I can't thank you enough
very happy to hear that. Good luck with your studies!
Very very nice and well explained :thumbsup:
perfect explanation
Great! An amazing explanations!!!
Very well, thanks for this great explanation!
Great explanation
It's so funny when your University teachers make theorems SOOOO damn difficult to understand and you find a 10 min video that explain a 2 hour Lecture+ better,simpler and much easier to understand. Perfect explanation,thx for saving lives here!!
Thanks so much for the feedback, I really appreciate it. My experience is overwhelmingly that most things do have simple explanations. So keep doing what you're doing - if you find something confusing, go explore and study for yourself, there is likely a simple explanation out there!
fantastic explanation, thank you!
Wow, awesome video, thank you!
i wish all professors taught this way instead of reading slides
my man is the real hero
How do you approach to a general solution for stability from a stand point of a LYAPUNOV functions for a class of nth order nonlinearar differential equations?
Sort of a tangential question (?). Does the eigenvalues of a system correlate to the lyapunov function somehow? Because I notice that negative eigenvalues means that the system is stable, a positive eigenvalue means that a system's exponents diverge, and a 0 eigenvalue means that the system is constant somehow and doesn't diverge to infinity or converge to 0. I feel like I'm drawing lines but I can't seem to generalize it.
sir, do you really write from right to left (in reverse)??
It would be an awesome party trick - but the truth is unfortunately more dull. I'm just mirroring in post production...
Awsome
for an arbitrary dynamical system, does Lyapunov function always exist?
No. It can only exist around an asymptotically stable equilibrium.
(We can prove that an equilibrium is asymptotically stable by showing that a Lijapunov function exists there.)
Thank you for sharing.
My pleasure - hopefully it was useful!
huh it almost seems like Lyapunov function applies generally to dynamical systems and "potential energy" is just the one used in classical mechanics
Brilliant question - and it sounds like you found the answer already. There are very strong senses in which a Lyapunov function is guaranteed to exist. For example, if I have a system \dot{x}=f(x) which is globally asymptotically stable, there will always be a Lyapunov function that proves it. So it is rather like a very precise generalisation of the concept of potential energy, but for general dynamic systems as you say - very cool stuff! A slight note of caution though - I'm a bit fuzzy on the precise details myself, but I believe you can also prove things like: finding a Lyapunov function is in general as difficult as solving the original differential equation for all initial conditions. And if you can do this, you wouldn't need the Lyapunov function anymore. And to make things worse, lots of the usual tricks you might think of to approximate them - for example approximating the set of all Lyapunov functions by some nice function classes (for example sum of squares polynomials) - also might not work. That is given a globally asymptotically stable dynamical system, the Lyapunov function that proves it might not be differentiable or smooth. So while it is very nice to know that a function exists, don't expect any free lunches!
Omg ur the Son of the Red Dwarf computer 👏🏽
hahaha - a blast from the past, but spot on!
had that figured out before I had even heard of lyapunov. Isn't this obvious?
Hello, can you help me please, I am working on bifurcations and solving a system of 3 equations with 6 variables and solving them using a Local Method of Lyapunov - Schmidt, I need some help please
Thankyou
Thanks a lot. Voice needs to be more clear.
its definitely black magic hehehe
Audio quality could be better
Akıl mantık işi değil, Barış hocaya selamlar.
I am outsider of the topic, I have a homework, and I understand nothing about it