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.
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
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.
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.
