Unlocking Chaos: Bifurcation Analysis of the Logistic Map 🌟

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
https://youtu.be/PM6fSdhcw4M

► Logistic map
Introduction https://youtu.be/PVo1mHnU7WU
Bifurcation diagram https://youtu.be/2nEBSyMsQE8

► Additional background
Introduction to mappings https://youtu.be/-vV5A4HullY
Logistic equation (1D ODE) https://youtu.be/iOumaIR5gzA
Lorenz map on strange attractor https://youtu.be/P4tjxOFnGNo
Lorenz equations introduction https://youtu.be/fIG2jtOhW0U
Definitions of chaos and attractor https://youtu.be/uDpYU01dhk0

► Ghosts and bottlenecks
In 1D differential equations https://youtu.be/Q_0oB1DHyQU
In 2D differential equations https://youtu.be/pl3byZQkVd8

► From 'Nonlinear Dynamics and Chaos' (online course).
Playlist https://is.gd/NonlinearDynamics

► Dr. Shane Ross, Virginia Tech professor (Caltech PhD)
Subscribe https://is.gd/RossLabSubscribe​

► Follow me on Twitter
https://twitter.com/RossDynamicsLab

► Course lecture notes (PDF)
https://is.gd/NonlinearDynamicsNotes

► Advanced lecture on maps from another of my courses
https://youtu.be/NYoA5B2qsdc

► Robert May's 1976 article introducing the logistic map (PDF)
https://is.gd/logisticmappaper

► Courses and Playlists by Dr. Ross

📚Attitude Dynamics and Control
https://is.gd/SpaceVehicleDynamics

📚Nonlinear Dynamics and Chaos
https://is.gd/NonlinearDynamics

📚Hamiltonian Dynamics
https://is.gd/AdvancedDynamics

📚Three-Body Problem Orbital Mechanics
https://is.gd/SpaceManifolds

📚Lagrangian and 3D Rigid Body Dynamics
https://is.gd/AnalyticalDynamics

📚Center Manifolds, Normal Forms, and Bifurcations
https://is.gd/CenterManifolds


References:
Steven Strogatz, "Nonlinear Dynamics and Chaos", Chapter 10: One-Dimensional Maps






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