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The Logistic Equation: Computable Chaos

Steven Jaffe
Organization: Mathematics Dept, Vassar College
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This notebook explores the strange dynamics of the logistic equation when r=4: x(t+1)=f(x(t)), with f(x)=4x(1-x). Using the results of James V. Whittaker (An Analytical Description of Some Simple Cases of Chaotic Behaviour, American Mathematical Monthly 98: 489--504 (June--July 1991)), it is possible to produce explicit orbits that are periodic, aperiodic, approach a limit cycle, or have any prescribed behavior whatsoever (e.g. appear periodic for, say, 17 periods, and then collapse to 0). Sensitivity to numerical approximation error is also apparent. The theory is developed and explained in tutorial form. Simple Mathematica routines are written to compute and graph the orbits, which may be used for independent exploration.

*Mathematics > Discrete Mathematics > Cellular Automata

Applied Mathematics, Applied Math, Life Sciences, Biology, Tutorial, Physics, logistics, chaos, logistic equation, Pure Mathematics, Pure Math, Number Theory
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ExactLogistic.nb (412.1 KB) - Mathematica notebook

Files specific to Mathematica 2.2 version:
ExactLogistic.ma (88.5 KB) - Mathematica notebook