C. Hinsley

27 June 2025


This is a collection of piecewise-continuous multimodal (i.e., having strictly more than 2 monotone continuous pieces) endomorphisms of an interval along with the associated kneading diagrams, computed by a new generalized version of the algorithm introduced in [Bar1]. To accompany the kneading diagrams, I will try to give interactive Desmos visualizations of the map so one can study the various kneading contours appearing in the diagrams.

Cubic map

This is the monic cubic polynomial map $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x(x-a)(x-b)$, where $a \leq 0 \leq b$. Note that this map may not have a compact globally attracting set, depending upon the parameters $a$ and $b$.

kneading_diagram_Matrix_50.png

Interactive cobweb diagram: https://www.desmos.com/calculator/8socaf7aaw

Double tent map

The double tent map $f: [0, 1] \to [0, 1]$ is piecewise linear with nonlinearities $\frac13$ and $\frac23$, satisfying $f(0) = 0, f(1/3) = a, f(2/3) = b,$ and $f(1) = 1$, where $a$ and $b$ are parameters in the interval $[0, 1]$.

kneading_diagram_Matrix_50.png

Interactive cobweb diagram: https://www.desmos.com/calculator/dotttzyypy


References

[Bar1] Barrio, Roberto, Andrey Shilnikov, and Leonid Shilnikov. "Kneadings, Symbolic Dynamics and Painting Lorenz Chaos." International Journal of Bifurcation and Chaos 22, no. 4 (2012): 1230016. https://doi.org/10.1142/S0218127412300169.