Hi great video! Question -- is there transient chaos in the windows of periodicity after around r = 28, or is it only in that first region where we have two fixed points?
We get a bifurcation diagram, so that's similar. But maybe you're asking if there are windows of periodic behavior within the chaotic region? Is that what you're asking?
@@oosmanbeekawoo Okay, I think I see what you mean. As I discuss in this other video, th-cam.com/video/P4tjxOFnGNo/w-d-xo.html, we're able to make a 1-D map from the dynamics on the Lorenz attractor called the Lorenz map.
@@ProfessorRoss Very fair. Thank you! Side info: This question arises because Veritasium showed there is the Logistic Map embedded into the Mandelbrot Cardioid, that if the Cardioid is the top view, then the Logistic Map is the side view. Furthermore, the Henon Map for varying values of 'b' at when a = -1.176 approx. shows we are moving through the Logistic Map again although cross-section by cross-section. Although this time the Logistic Map is 3D and appears folded. I am starting to believe the Logistic Map is in all Chaotic systems, also given we have to have the Feigenbaum constant a.k.a. period doubling in all Chaotic systems. That is why.. anyway, thank you and have a good day/night.
Yes, it is: Doedel, Eusebius J., Bernd Krauskopf, and Hinke M. Osinga. "Global organization of phase space in the transition to chaos in the Lorenz system." Nonlinearity 28, no. 11 (2015): R113. 📕PDF of paper: iopscience.iop.org/article/10.1088/0951-7715/28/11/R113/pdf?casa_token=424PwMEitGkAAAAA:BK2T3hBqO69Ku1y645tdNcx3eEqCkLE1QsBReDU3UkzqwWtw2lVdzHdkT-nA3GbeNSEFYs_pKnX5DahI_1-9 📕PDF of preprint: web.archive.org/web/20160211184135id_/www.math.auckland.ac.nz/~berndk/transfer/dko_tochaos_prep.pdf
The software that is simulating the Lorenz system live during the lecture is a simulation by Hendrick Wernecke at Uni Frankfurt: itp.uni-frankfurt.de/~gros/Vorlesungen/SO/simulation_example/ For the detailed bifurcation calculations, you can look at the following papers: Doedel, Eusebius J., Bernd Krauskopf, and Hinke M. Osinga. "Global organization of phase space in the transition to chaos in the Lorenz system." Nonlinearity 28, no. 11 (2015): R113. 📕PDF of paper: iopscience.iop.org/article/10.1088/0951-7715/28/11/R113/pdf?casa_token=424PwMEitGkAAAAA:BK2T3hBqO69Ku1y645tdNcx3eEqCkLE1QsBReDU3UkzqwWtw2lVdzHdkT-nA3GbeNSEFYs_pKnX5DahI_1-9 📕PDF of preprint: web.archive.org/web/20160211184135id_/www.math.auckland.ac.nz/~berndk/transfer/dko_tochaos_prep.pdf
Thank you, professor, this is such a great and beautiful topic.
You are very welcome. Thank you for watching!
Thanks! Precisely what I needed for the homework 🔥
Great to hear!
Hi great video! Question -- is there transient chaos in the windows of periodicity after around r = 28, or is it only in that first region where we have two fixed points?
Yes!
Are we able to extract a diagram similar to the 2D Logistic Map in the Lorenz Attarctor?
We get a bifurcation diagram, so that's similar. But maybe you're asking if there are windows of periodic behavior within the chaotic region? Is that what you're asking?
@@ProfessorRoss Yes! I mean, what is the closest we can get to the x = rx(1 - x) graph?
@@oosmanbeekawoo Okay, I think I see what you mean. As I discuss in this other video, th-cam.com/video/P4tjxOFnGNo/w-d-xo.html, we're able to make a 1-D map from the dynamics on the Lorenz attractor called the Lorenz map.
@@ProfessorRoss Very fair. Thank you!
Side info: This question arises because Veritasium showed there is the Logistic Map embedded into the Mandelbrot Cardioid, that if the Cardioid is the top view, then the Logistic Map is the side view.
Furthermore, the Henon Map for varying values of 'b' at when a = -1.176 approx. shows we are moving through the Logistic Map again although cross-section by cross-section. Although this time the Logistic Map is 3D and appears folded.
I am starting to believe the Logistic Map is in all Chaotic systems, also given we have to have the Feigenbaum constant a.k.a. period doubling in all Chaotic systems.
That is why.. anyway, thank you and have a good day/night.
Plz the paper u used in bifurcation diagram
Yes, it is:
Doedel, Eusebius J., Bernd Krauskopf, and Hinke M. Osinga. "Global organization of phase space in the transition to chaos in the Lorenz system." Nonlinearity 28, no. 11 (2015): R113.
📕PDF of paper: iopscience.iop.org/article/10.1088/0951-7715/28/11/R113/pdf?casa_token=424PwMEitGkAAAAA:BK2T3hBqO69Ku1y645tdNcx3eEqCkLE1QsBReDU3UkzqwWtw2lVdzHdkT-nA3GbeNSEFYs_pKnX5DahI_1-9
📕PDF of preprint: web.archive.org/web/20160211184135id_/www.math.auckland.ac.nz/~berndk/transfer/dko_tochaos_prep.pdf
@@ProfessorRoss Thank you, professor ^^
Name of the software
The software that is simulating the Lorenz system live during the lecture is a simulation by Hendrick Wernecke at Uni Frankfurt: itp.uni-frankfurt.de/~gros/Vorlesungen/SO/simulation_example/
For the detailed bifurcation calculations, you can look at the following papers:
Doedel, Eusebius J., Bernd Krauskopf, and Hinke M. Osinga. "Global organization of phase space in the transition to chaos in the Lorenz system." Nonlinearity 28, no. 11 (2015): R113.
📕PDF of paper: iopscience.iop.org/article/10.1088/0951-7715/28/11/R113/pdf?casa_token=424PwMEitGkAAAAA:BK2T3hBqO69Ku1y645tdNcx3eEqCkLE1QsBReDU3UkzqwWtw2lVdzHdkT-nA3GbeNSEFYs_pKnX5DahI_1-9
📕PDF of preprint: web.archive.org/web/20160211184135id_/www.math.auckland.ac.nz/~berndk/transfer/dko_tochaos_prep.pdf