Mathematical Mountains – Steve Brunton (GS)
The logistic map
The hat map
The cosine map
These images are excerpts from the bifurcation diagrams of various one-dimensional maps, including the logistic map, the hat map, and the cosine map. Each of these dynamical systems model various physical phenomena in the real world. For example, the logistic map is a crude model of population dynamics with reproduction and limited resources, and it is often used as an example of the period-doubling route to chaos. Typical of chaotic systems, many regions in these figures exhibit self-similarity and reflect the order that emerges out of chaos.
These images were generated numerically by iterating the discrete-time maps above as a bifurcation parameter is varied. The bifurcation parameter is plotted as the y-axis (elevation), and at each elevation, the stratified layer represents the attracting set of the dynamical system for that particular choice of bifurcation parameter. Bifurcation refers to a qualitative change in the behavior or topology of a dynamical system as a parameter is varied.