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.
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 $b$.
Interactive cobweb diagram: https://www.desmos.com/calculator/8socaf7aaw
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 $[0, 1]$.
Interactive cobweb diagram: https://www.desmos.com/calculator/dotttzyypy
[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.