Understanding the Logistic Map: The Path to Chaos Through Period Doubling 📈

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
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
Lyapunov exponents to quantify chaos https://youtu.be/22VVVn1zPdM

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

► 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 X
https://x.com/RossDynamicsLab

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

► Advanced lecture on maps from another course of mine
https://youtu.be/NYoA5B2qsdc

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

► *Related Courses and Series Playlists by Dr. Ross*

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

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

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

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

📚3-Body Problem Orbital Dynamics
https://is.gd/3BodyProblem

📚Space Manifolds
https://is.gd/SpaceManifolds

📚Space Vehicle Dynamics
https://is.gd/SpaceVehicleDynamics



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