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Prove the following saddle-node bifurcation theorem:

Suppose that $f_λ$ depends smoothly on the parameter $λ$ and satisfies:

(a) $f_{λ_0}(x_0)=x_0$

(b) $f_{λ_0}'(x_0)=1$

(c) $f_{λ_0}''(x_0) \ne 0$

(d) $\frac{\partial f_λ}{\partial λ} \big|_{λ=λ_0}(x_0) \ne 0$

Then there is an interval Ι about $x_0$ and a smooth function $\mu:Ι\to R$ satisfying $\mu(x_0)=λ_0$ and such that $f_\mu(x) (x)=x$

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The formatting of this problem makes it very difficult to read. – Amzoti Dec 23 '12 at 6:08
There is probably a proof in Elaydi's textbook on discrete dynamical systems. I don't have access to my copy at the moment. It's also possible that typing "saddle node bifurcation" into Google will turn up a proof for you. – Gerry Myerson Dec 23 '12 at 16:58
thanks Gerry-myerson i found it in this book – Maisam Hedyelloo Feb 4 '13 at 18:22
I am also looking for that proof! Where did you find it? I cannot find this book, could you maybe post a link or the exact title? – the.polo Dec 14 '15 at 13:48
Differential equations, dynamical systems and an introduction to chaos by Morris W. Hirsch, Stephen Smale, Robert L. Devaney – Maisam Hedyelloo Dec 17 '15 at 20:09

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