This is more like the quadratic mapping in that the parameter raises and lowers the function without changing its shape. As expected, the orbits gave a bifurcation diagram nearly identical to that for the quadratic map, but with a bit of a twist. Again, we see an abrupt change from behavior characteristic of the quadratic map to a broad ergodic region. As with the previous sinusoidal map, this extra ergodic region has windows of even periodicity. The full diagram runs diagonally through the origin across the parameter space from negative infinity to positive infinity Some more mundane bifurcation diagrams are shown below.
All functions are smooth and non-monotonic.
Note that each one undergoes a period-doubling route to chaos and that there are always windows of odd periodicity within the chaotic regime. What look like single lines turn out to be, on closer examination, pairs of lines. When these are examined in more detail, they also turn out to be pairs of lines and so on. Orbits in the non-ergodic regions are not periodic but tend to cluster together and thus appear to have even periodicity. There is no bifurcation with this map.