At a=1 the map becomes noninvertible, but since the orbit does not visit the region of the inflection point, this is irrelevant to the dynamics. However for a>1 the map has quadratic turning points, and chaos can develop through the period doubling cascade.
a | b | Orbit | Spectrum | Dynamics |
1.0 | 0.5311 | Locked | ||
1.4 | 0.5311 | Period Doubled | ||
1.5 | 0.5311 | Period 4 | ||
1.65 | 0.5311 | Period 8 | ||
1.69 | 0.5311 | 4 chaotic bands |
so that the onset of chaos is between 1.65 and 1.69
Using the idea of functional composition e.g. f^{ 2}=f(f(x)) etc. and enlarging the region near the center of the range convinces us that the universal period doubling cascade is to be expected, e.g. for the value of a=1.5 giving the period four and looking at f^{ 2}:
.
The function f^{ 2} is complicated, but enlarging just the region 0.475<x<0.6 with the mouse yields a curve that is quite reminiscent of the quadratic map (and the approach gets closer as the order of the period increases).
Some of the behavior for this single frequency ratio is shown by the bifurcation diagram:
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