Thank you for your enthusiasm! I wanted to highlight one point: I find that locating fixed points becomes more straightforward and easier to understand when we set the derivative of the potential to zero, essentially making the slope zero on the potential diagram. Thank you.
Thanks for the enthousiasm :) I have an exam on topics like this and found it quite difficult and boring before, but since you explain it so well I start to enjoy it too.
Great Explanation and drawings! :) I have one questions about linearization and fixed points: Why don't we linarize about other points than the fixed points? Theoretically it should, because I see no problem with the mathematical machinery of linearization. Am I missing something?
If you don't start in a fixed point (generally it can be a periodic orbit or invariant manifold) then your perturbation and the point where you're trying to linearize will change in time due to the flow of your system of differential equations. In other words, you need a fixed reference (relative to your flow) frame to analyze how your perturbation will change, and that change is just a linear systems of differential equations for your perturbation
Only near the fixed points is it possible to linearize, by shifting the (linearized) expansion point in the coordinates if not at the origin. Analogous to Taylor vs Maclaurin series expansions.
Mathematical induction is something very natural for us. For double well potential, the phase portrait is like this. So for the triple well potential, it's easy to imagine the phase portrait, ans so on. But nature really hates this approach. Nature will agree to "think like this" till around 10 quasi-periodic orbits, but no more. Then, nature gonna change its strategy. And its new strategy is so amazing, so beautiful, one of the most beautiful thing in the world : the self-similarity...
Hello, your lectures are very interesting! Thanks for these. But you can not use a linearization method for studying nonlinear dynamics even locally, if your linearization is not hyperbolic (if eigenvalues belong to imaginary axis, like in this video and not only).
Additional Code by Eugenio Sainz Ortiz: github.com/GenioSainz/Dynamical-Systems
Never seen phase portraits so well explained
Phase Portraits can be gnarly; Dr. Brunton's video does an excellent point of explaining them in detail
Thank you for your enthusiasm! I wanted to highlight one point: I find that locating fixed points becomes more straightforward and easier to understand when we set the derivative of the potential to zero, essentially making the slope zero on the potential diagram. Thank you.
Thanks for the enthousiasm :) I have an exam on topics like this and found it quite difficult and boring before, but since you explain it so well I start to enjoy it too.
Amazing stuff!
So where can I get a phase space T-shirt?
Yeah
I made one and ordered it
Nice stuff. Thanks!
Must put in the dot... need to put in the dot... Thank you for your cooperation.
4th order polynomial ! plus color markers! nice!
Awesome. Great content.
Nice stuff! Great explanation
incredibly excellent !
Seriously tho... where can we get a shirt?
Just in time for Halloween hahaha
Love this lectures
Great Explanation and drawings! :) I have one questions about linearization and fixed points: Why don't we linarize about other points than the fixed points? Theoretically it should, because I see no problem with the mathematical machinery of linearization. Am I missing something?
If you don't start in a fixed point (generally it can be a periodic orbit or invariant manifold) then your perturbation and the point where you're trying to linearize will change in time due to the flow of your system of differential equations. In other words, you need a fixed reference (relative to your flow) frame to analyze how your perturbation will change, and that change is just a linear systems of differential equations for your perturbation
Only near the fixed points is it possible to linearize, by shifting the (linearized) expansion point in the coordinates if not at the origin. Analogous to Taylor vs Maclaurin series expansions.
Mathematical induction is something very natural for us. For double well potential, the phase portrait is like this. So for the triple well potential, it's easy to imagine the phase portrait, ans so on.
But nature really hates this approach. Nature will agree to "think like this" till around 10 quasi-periodic orbits, but no more. Then, nature gonna change its strategy. And its new strategy is so amazing, so beautiful, one of the most beautiful thing in the world : the self-similarity...
I found that strangely attractive.
Hello, your lectures are very interesting! Thanks for these. But you can not use a linearization method for studying nonlinear dynamics even locally, if your linearization is not hyperbolic (if eigenvalues belong to imaginary axis, like in this video and not only).
You don't know a priori if the fixed point is going to be hyperbolic, you linearize the system locally around a fixed point to determine that
12:02 This picture must have been looking funny on your wife's T-shirt :J
Spooky Spooky