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- Relationship between and when . Before the period doubling bifurcation occurs. The orbit converges to the fixed point . (en)
- Relationship between and when . The fixed point becomes unstable, splitting into a periodic-2 stable cycle. (en)
- When , there are infinitely many intersections, and we have arrived at chaos via the period-doubling route. (en)
- When , there are three intersection points, with the middle one unstable, and the two others stable. (en)
- When , there are three intersection points, with the middle one unstable, and the two others having slope exactly , indicating that it is about to undergo another period-doubling. (en)
- When , we have a single intersection, with slope exactly , indicating that it is about to undergo a period-doubling. (en)
- Relationship between and when . The tangent slope at the fixed point . is exactly 1, and a period doubling bifurcation occurs. (en)
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