The 45° blue bowtie condition is just a visual
statement of the fact that at the fixed point the absolute value of the derivative
is less than 1. |

For the logistic map L(x) = s⋅x⋅(1 - x),
the derivative is L'(x) = s - 2⋅s⋅x. |

For the fixed point _{f} = 0'(x_{f}) = L'(0) = s. |

Consequently, this fixed point is stable for |

For the fixed point _{f} = (s - 1)/s'(x_{f}) = L'((s - 1)/s) |

Consequently, this fixed point is
stable for |

Return to Fixed Points of the Logistic Map.