Logistic Map, Part 1: Period Doubling Route to Chaos
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- Опубликовано: 18 сен 2024
- The logistic map is a simple discrete model of population growth with very complicated dynamics. It depends on a growth rate parameter r. We consider the dynamics at various values of the parameter and find that there’s a branch of stable fixed points which bifurcates into stable attractor cycles of period 2, 4, 8, 16, .... The period-doubling cascade. The bifurcation diagram shows chaos intermingled with periodic windows.
► Next, the bifurcation diagram and self-similarity
• Logistic Map, Part 2: ...
► Additional background
Introduction to mappings • Maps, Discrete Time Dy...
Logistic equation (1D ODE) • Population Growth- The...
Lorenz map on strange attractor • Dynamics on Lorenz Att...
Lorenz equations introduction • 3D Systems, Lorenz Equ...
Definitions of chaos and attractor • Chaotic Attractors: a ...
Lyapunov exponents to quantify chaos • Lyapunov Exponents & S...
► Robert May's 1976 article introducing the logistic map (PDF)
is.gd/logistic...
► From 'Nonlinear Dynamics and Chaos' (online course).
Playlist is.gd/Nonlinea...
► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
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► Follow me on X
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► Course lecture notes (PDF)
is.gd/Nonlinea...
► Advanced lecture on maps from another course of mine
• Center Manifold Theory...
References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 10: One-Dimensional Maps
► Related Courses and Series Playlists by Dr. Ross
📚Nonlinear Dynamics & Chaos
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📚Hamiltonian Dynamics
is.gd/Advanced...
📚Lagrangian & 3D Rigid Body Dynamics
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📚Center Manifolds, Normal Forms, & Bifurcations
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📚3-Body Problem Orbital Dynamics
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📚Space Manifolds
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📚Space Vehicle Dynamics
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Professor, Ross, I need to stop and say thank you. This was probably the most "easy to understand, yet so dense information-wise" lecture I have ever seen on RUclips. Phenomenal teaching style. You even showed us the original paper (never seen someone show us the original paper in the lecture at my university) and made a couple of jokes along the way. This is almost like how I imagine an ideal lecture. I wish you can cover all the topics of my computer science program. I know this is impossible, so I am at least happy with those bits I can use. This one for example is useful for my course/class named "Modeling and Simulation" at my university in the Czech Republic. Greetings from there and I'm looking forward to more great content!
What are the attractors? Are they the values that xn converges to / fixed points?
Ex:
at r = 0.5, x = 0;
thus 0 is the attractor
r = 2.8 some xn = c = attractor;
thus c is the attractor
at r= 3.3 there are 2 attractors, xn = a and xn = b ; which is normalized a,b e [0,1].
thanks for the series! I can't wait to complete it!
For each value of r, the attractor is the set of points shown. So for examples, at r=3.3, there are not two attractors. We would say the attractor is that set of 2 points. For some r's, it appears like the attractor isn't an integer number of points, but rather a continuum -- for that r, we'd say the attractor is chaotic.
How long can the length of thr growth parameter be? Won't it depend on the OS used?
Note that 4x*(1-x) is chaotic with respect to seed values sin^2(k * Pi/N), with Pi = 180 deg and N, odd. = x and f(x) = 4x*(1-x). Example: say N = 11, then our seed value can be k = 1 , i.e sin^2(Pi/11) = .079746....Then further iterates are .292292...= sin^2(2*Pi/11)....-> ,8274303....= sin^2(4*Pi/11)......-> ..571157....= sin^2(3 Pi/11), -> .979696...= sin^2 (5 Pi/11)...then the cycle repeats with period 5 for N = 11. We can calculate the results quicker by extracting the coefficients k in sin^2(k *Pi/N), by using a doubling algorithm. For N=11, we double mod 11 with the abs value of every term except the last being (N - 1)/2 = 5. Our orbit is (1, 2, 4, -3, 5). (1, 2, 4 are straightforward, but the next term would be 8 (greater than 5), so we select -3. (ignore signs for multiplication). Then 2 * 3 = 6, but 6 is greater than 5, so we select -5. Delete the (-) signs, getting 1, 2, 4, 3, 5, our k coefficients for the sin^(k * Pi/11) orbit. Brief list of periods for N, odd.: For N = 3, 5, 7, 9, 11, 13, 15, 17....the periods are respectively 1, 2, 3, 3, 5, 6, 4, 4.
That’s fascinating. I wasn’t aware of this about this case of the logistic map. Think you for sharing!
very well explained thank you
You are welcome!
It's very interesting, thank you!
Thankyou
You are amazing! thank you
Anyone else notice Minecraft’s cameo?
Give the pdf
The link to the PDF lecture notes is here is.gd/NonlinearDynamicsNotes
Where possible, I give the link the PDF lecture notes in the description of my videos.