In case it helps anyone with some timestamps: Saddle Node Bifurcation - 2:24 Transcritical Bifurcation - 8:50 Pitchfork Bifurcation - 13:23 Hopf Bifurcation - 16:50 AMATH502 was by far the most influential class that made me fall in love with how beautiful math is when it comes to attempt to explain (or not as with chaos) applications in nature. Thank you, and keep up the good work! :-)
Hello Professor I have a system of three equations and I want to review the distinguished group, but my calculator is not very efficient. I ask you to help me, please.
5:12 How do you differentiate the top equation to get the result at the bottom? The -2 coefficient seems like it came out of nowhere, when I differentiate the first by the x perturbation, assuming x0 and x vanish I get a contradictory result of 0 = 1???? Help would be appreciated
I had the same problem, I think I figured it out. If we have x = x_0 + \tilde{x}, and we differentiate, we get \dot{x} = \dot{\tilde{x}}, since x_0 is a constant. We then know \dot{x} = mu - x^2, where we can fill in x = x_0 + \tilde{x}. We thus get \dot{x} = mu - (x_0 + \tilde{x})^2 = mu - x_0^2 - 2*x_0*\tilde{x} - \tilde{x}^2. Now, since mu - x_0^2 = 0 (equilibrium), the first two terms disappear, and since \tilde{x} \approx 0, we can surely neglect \tilde{x}^2, and set it to zero. This is then the result he arrives at.
Hello Mr Kutz . Tnx for the great lectures . Making some playlists will be a great help for us learners to find the specific subjects we're looking for . We'll be appreciated if you do that . Best wishes
Really fantastic explanation. I feel like I got much more intuition from looking at the specific cases. That transcritical bifurcation looks like ReLU...hmmm. It would be cool if somehow stability is associated with why ReLU is so affective.
This reply is a bit late, but I think this is just a coincidence. When you say "looks like ReLU" keep in mind the bifurcation plot (at 11:29) is made by sweeping a parameter, which is not something you'd typically do (or at least not to that extent). I think the real reason that it looks like ReLU is that ReLU looks like 2 lines and a lot of things look like 2 lines (e.g., the Landau Zener formula), although I will be enthusiastic if I am wrong.
Looks nice, but not sure if true. Still you have to define mu which is why it's a little frustrating... as it makes it very complex. I believe our brain calculator is more accurate than that.
In case it helps anyone with some timestamps:
Saddle Node Bifurcation - 2:24
Transcritical Bifurcation - 8:50
Pitchfork Bifurcation - 13:23
Hopf Bifurcation - 16:50
AMATH502 was by far the most influential class that made me fall in love with how beautiful math is when it comes to attempt to explain (or not as with chaos) applications in nature. Thank you, and keep up the good work! :-)
Hello Professor I have a system of three equations and I want to review the distinguished group, but my calculator is not very efficient. I ask you to help me, please.
Kutz is an amazing teacher. I knew he did a lot of great work, but I didn’t know he was also a great teacher.
This channel is waaaaay too underrated!
Fantastic Prof., very easy approach to explain the bifurcation theory .
Who needs sleep when Nath uploads a new video??!
no body
Right? New discovery for me. I'm totally addicted.
@@insightfool if you have not yet, please do yourself the favour and discover Steve Brunton 🙂
@@Music_Engineering real
humans
5:12 How do you differentiate the top equation to get the result at the bottom? The -2 coefficient seems like it came out of nowhere, when I differentiate the first by the x perturbation, assuming x0 and x vanish I get a contradictory result of 0 = 1???? Help would be appreciated
I had the same problem, I think I figured it out. If we have x = x_0 + \tilde{x}, and we differentiate, we get \dot{x} = \dot{\tilde{x}}, since x_0 is a constant. We then know \dot{x} = mu - x^2, where we can fill in x = x_0 + \tilde{x}. We thus get \dot{x} = mu - (x_0 + \tilde{x})^2 = mu - x_0^2 - 2*x_0*\tilde{x} - \tilde{x}^2. Now, since mu - x_0^2 = 0 (equilibrium), the first two terms disappear, and since \tilde{x} \approx 0, we can surely neglect \tilde{x}^2, and set it to zero. This is then the result he arrives at.
Separation of variables: dx/x = -2 x_0 dt -> ln |x| = -2 x_0 t + c -> x = c e^(-2 x_0 t)
I don't know how to thank you enough... God bless you Professor
Hello Mr Kutz . Tnx for the great lectures . Making some playlists will be a great help for us learners to find the specific subjects we're looking for . We'll be appreciated if you do that . Best wishes
Thanks a lot for this nice presentation
Nice explanation professor
Great explanation luv u sir.....
Really fantastic explanation. I feel like I got much more intuition from looking at the specific cases. That transcritical bifurcation looks like ReLU...hmmm. It would be cool if somehow stability is associated with why ReLU is so affective.
This reply is a bit late, but I think this is just a coincidence. When you say "looks like ReLU" keep in mind the bifurcation plot (at 11:29) is made by sweeping a parameter, which is not something you'd typically do (or at least not to that extent). I think the real reason that it looks like ReLU is that ReLU looks like 2 lines and a lot of things look like 2 lines (e.g., the Landau Zener formula), although I will be enthusiastic if I am wrong.
Amazing, this was crystal clear. Thanks!
why for Hopf bifurcation allways have to put stability in origin, while some system have their stability in other values? in this case what happen?
Excellent teaching
Great lecture. Thank you!
nice intro(city shot morphing to the circuit board) 👍
very well explained! thanks!
Truly amazing!
Amazing video!
Good lectrure, even I understood it !! Thanks
great vid
Looks nice, but not sure if true. Still you have to define mu which is why it's a little frustrating... as it makes it very complex. I believe our brain calculator is more accurate than that.
Amazing! 🤓
I'll probably need to rewatch this. lol...
this all look an awful lot like renormalization group flow in theoretical physics