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The Schwarzian Derivative of Iterated Functions

This Demonstration graphs the Schwarzian derivative for iterates of a few different functions of a single variable. Of particular interest in the study of one-dimensional dynamical systems is if a function or some iterate of that function has a Schwarzian derivative that is strictly negative. You can determine if a particular function has this property for some parameter values and by varying the number of iterates taken and examining the graphs of their Schwarzian derivatives.

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If a function of an interval to that interval has a continuous third derivative then its Schwarzian derivative is defined by . It may happen that a function's Schwarzian derivative may not be negative on this interval but that it has an iterate with this property, in which case it is said to have an eventual negative Schwarzian derivative.
The study of maps with an eventual negative Schwarzian derivative was originally suggested by L. Bunimovich in relation to some maps arising in neuroscience.
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