Deterministic Chaos

6.K. Tangent Bifurcations

A tangent bifurcation occurs when the graph of L(x) crosses the diagonal as s increases.
The crossings occur where the graph becomes tangent to the diagonal, hence the name tangent bifurcation.
As s continues to increase, the graph crosses the diagonal at two points, initially nearby, but moving apart.
All this happens because as s increases, the maxs of the graph of L(x) increase and the mins decrease.
For the graph of L3(x), three lobes become tangent to the diagonal at the same s-value, so this tangent bifurcation gives rise to six new fixed points of L3(x), hence to two 3-cycles of L(x). See the left picture.
Looking carefully at the crossing, we see one of the new fixed points is stable (the one toward the extremum in the lobe of the graph crossing there), the other is unstable. See the right picture.
Click the picture to animate. Click the picture to animate.
The stable fixed points of L3(x) constitute one 3-cycle (light blue below) of L(x), the unstable fixed points another 3-cycle (dark blue).

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