Logistic Map, Part 3: Bifurcation Point Analysis | Bottlenecks in Maps, Intermittency Chaos

The logistic map bifurcation diagram can be analytically explained. We calculate the value of first few bifurcation points, where the non-zero fixed point emerges and stable cycles of period 2 and 4 emerge via a period-doubling bifurcation (or flip bifurcation). We see a map version of fixed point ghosts and bottlenecks, regions of high residence time, related to the intermittency route to chaos.
► Next, the universality of features in the logistic map
• Period-Doubling Route ...
► Logistic map
Introduction • Logistic Map, Part 1: ...
Bifurcation diagram • 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 ...
► Ghosts and bottlenecks
In 1D differential equations • Flows on the Circle | ...
In 2D differential equations • Bifurcations in 2D, Pa...
► From 'Nonlinear Dynamics and Chaos' (online course).
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► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
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► Course lecture notes (PDF)
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► Advanced lecture on maps from another of my courses
• Center Manifold Theory...
► Robert May's 1976 article introducing the logistic map (PDF)
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► Courses and Playlists by Dr. Ross
📚Attitude Dynamics and Control
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📚Nonlinear Dynamics and Chaos
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📚Hamiltonian Dynamics
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📚Three-Body Problem Orbital Mechanics
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📚Lagrangian and 3D Rigid Body Dynamics
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📚Center Manifolds, Normal Forms, and Bifurcations
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References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 10: One-Dimensional Maps
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[@jonasoliveira1143]

Thank you very much for this masterpiece of a lecture, prof. Ross!

[@JAYMOAP]

Great chanmel

[@ProfessorRoss]

Thanks Jay! I'm happy to contribute to learning.

[@JAYMOAP]

@@ProfessorRoss much appreciated Ross. I gone through 5 of your lecture today all of them beautifully presented and broke down the way almost anyone can understand it. I am looking to learn more on the dimensional representation of these systems especially in correspondence to physics as gravity related to chaos as far as I see from dynamic perspective. I did some simulations and lattice models related to string theory but tend to think there is something fishy going with correlations between fix points. Seems to me black holes behave like attractors and in my modelling I see some dimensional reduction. Looking forward if you have any lectures on these relationships.
The duffing oscillator also a nice analogue, especially to the big bang scenario but these are just my speculative assumptions :)