Lorenz System Bifurcation Diagram- Exploring Parameter Space

What happens if we change the parameters? We take a tour, finding pairs of limit cycles linked through each other, transient chaos, noisy periodicity, as well as strange attractors, and windows of periodicity intermingled within chaos. The Lorenz manifold, the stable manifold of the origin, is also significant. We show some art inspired by the Lorenz system.
► Next, the dynamics on the Lorenz strange attractor
• Dynamics on Lorenz Att...
► Lorenz equations
Simulate it! is.gd/Lorenz
Derivation and chaotic waterwheel • 3D Systems, Lorenz Equ...
Volume contraction and symmetry • Lorenz Equations Prope...
Fixed point analysis • Lorenz Equations Fixed...
Deducing the Lorenz attractor • Lorenz Attractor- How ...
Lorenz' original 1963 paper (PDF) is.gd/lorenzpaper
► Additional background
Definitions of chaos and attractor • Chaotic Attractors: a ...
Lyapunov exponents to quantify chaos • Lyapunov Exponents & S...
Pitchfork bifurcations of fixed points • Bifurcations Part 3- P...
Hopf bifurcations, unstable limit cycles • Bifurcations in 2D, Pa...
Quasiperiodic motion on a torus • Coupled Oscillators, Q...
Trapping region, Poincaré-Bendixson • Limit Cycles, Part 3: ...
► Advanced lecture on the center manifold of the origin in the Lorenz system
• Center Manifolds Depen...
► From 'Nonlinear Dynamics and Chaos' (online course).
Playlist is.gd/Nonlinea...
► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
Subscribe is.gd/RossLabS...
► Follow me on Twitter
/ rossdynamicslab
► Course lecture notes (PDF)
is.gd/Nonlinea...
References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 9: Lorenz Equations
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...
📕PDF of preprint: web.archive.or...
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[@Sinanmmd]

Thank you, professor, this is such a great and beautiful topic.

[@ProfessorRoss]

You are very welcome. Thank you for watching!

[@ingenierohermetico3931]

Thanks! Precisely what I needed for the homework 🔥

[@ProfessorRoss]

Great to hear!

[@enriquealvarado3989]

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?

[@chiragadwani1875]

Yes!

[@LYehiaX]

Plz the paper u used in bifurcation diagram

[@ProfessorRoss]

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

[@LYehiaX]

@@ProfessorRoss Thank you, professor ^^

[@oosmanbeekawoo]

Are we able to extract a diagram similar to the 2D Logistic Map in the Lorenz Attarctor?

[@ProfessorRoss]

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]

@@ProfessorRoss Yes! I mean, what is the closest we can get to the x = rx(1 - x) graph?

[@ProfessorRoss]

@@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.

[@oosmanbeekawoo]

@@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.

[@pradeeshm2570]

Name of the software

[@ProfessorRoss]

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