English: Period-doubling bifurcation of logistic map x_n+1 = f (x_n) = ax(1 − x_n). The black solid curve shows the graph of two iterated logistic map f ²(x_n). The slope of tangent line at the fixed point x_f 2 equals to 1 when the parameter a = 3. Then the slope at x_f 2 becomes more than 1 when a > 3, the fixed point becomes unstable, and the two periodic points x⁽²⁾_f 1 and x⁽²⁾_f 2 appear near the fixed point.