Logistic Map Exploration using Wolfram Mathematica

How to use Wolfram Mathematica to start the construction of a bifurcation diagram of period doubling bifurcations for the logistic map f(x)=μ*x*(1-x) under iteration. When μ is between 1 and 3, there is one attracting (stable) fixed point. When μ is between 3 and 1+√6, there is an attracting 2-cycle. When μ is between 1+√6 and about 3.54409, there is an attracting 4-cycle. The key for detecting bifurcation points is when the derivative of higher-order iterates is equal to -1. The first Feigenbaum constant is also discussed. Burden, Faires, Burden "Numerical Analysis": https://amzn.to/3DBvCWn

Numerical Analysis, Class 7A

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#PeriodDoubling #ChaosTheory #mathematica #Wolfram

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(0:00) Goal: use both human skills and computer abilities
(1:02) Logistic map
(3:12) Fixed points of the logistic map f(x)=μx(1-x) in terms of the parameter μ
(7:49) Stability analysis for the fixed points
(12:29) Wolfram Mathematica NestList confirmation
(20:15) Observe stable (attracting) 2 cycle
(21:43) Second iterate
(29:22) Two cycles of f are fixed points of the second iterate
(31:38) Solve for 2 cycles in terms of μ using Mathematica
(32:36) Start plotting the bifurcation diagram
(35:46) The next bifurcation value (to a stable 4 cycle)
(40:32) Plot the function and the second iterate
(44:47) Solve for 4 cycles in terms of μ using Mathematica
(46:44) Extending the bifurcation diagram
(49:51) The next bifurcation value (to a stable 8 cycle)
(51:49) Logistic map bifurcation values online
(52:53) Feigenbaum constant 4.669...
(58:56) It is a universal constant (universality)

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