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Feigenbaum universal transition of a dynamical system (like the logistic map) to chaotic behaviour through period-doubling cascade involves the functional equation $$g(x)=-\frac1\lambda g(g(\lambda x))$$ with the boundary conditions $$g(0)=1,\qquad g'(0)=0,\qquad g''(0)<0.$$ The parameter $\lambda$ for which a solution exists near $x=0$ is the inverse of the Feigenbaum constant.