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A fierce differential-delay equation: df/dx = f(f(x))

Consider the following set of equations:$$\begin{array}{l}y = f(x) \\\frac{dy}{dx} = f(y)\end{array}$$These can be written as finding some differentiable function $f(x)$ such that $$f^{\prime} = f(f(x))$$ For example, say $y(0) = 1$. Then $\left. \frac{dy}{dx} \right|_{x=0}$ is determined by the value of $y(1)$. The derivative at the $x=0$ had better be negative, otherwise by the time the function gets to 1, the value will be too great and will contradict the alleged value of hte derivative at $x=0$.Many years ago I tried various techniques to...Read more