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\begin{document}
\hfill [Lecture Note 11]
\bigskip

%\centerline{\Large \sffamily }
\centerline{\Large \sffamily Saddle-Node and Period-Doubling Bifurcations}
\medskip

Let $\bar q$ be a nonhyperbolic fixed point of a diffeomorphism $f$ in $\mathbb R^d$.
Let $J=Df(\bar q)$, and denote
\[
\sigma^s=\sigma(J)\cap\{|z|<1\}, \sigma^c=\sigma(J)\cap\{|z|=1\},\ \hbox{ and }\ \sigma^u=\sigma(J)\cap\{|z|>1\}
\]
the set of stable eigenvalues, center eigenvalues, unstable eigenvalues, respectively,  of
the linearizatoin $Df(\bar q)$, counting multiplicity. Consider two cases
\[
\hbox{either $\sigma^c=\{1\}$ or  $\sigma^c=\{-1\}$.}
\]
for which the former is called a saddle-node bifurcation problem 
and the latter a period-doubling bifurcation problem.
Specifically, consider a one-parameter family of map $f:\mathbb R\times\phasespace\to \phasespace$
such that 
\[
f(\fp,\bar q)=\bar q. 
\]
By augmenting the map to $\tilde f(\lambda,p)=(\lambda, f(\lambda, p))$, the parameter dimension
is a trivial center direction for the augmented map. Then by the Center Manifold Theorem, 
for a sufficiently smooth $f$, the dynamics  
of $\tilde f$ is reduced to a two-dimensional local center manifold with $\lambda$ being a trivial 
center dimension. The reduced map is just a parameterized one-dimensional mapping, 
denoted by $F(\lambda, x)$ for $(\lambda, x)\in \mathbb R\times\mathbb R$. Therefore, the 
dynamics of $f$ near the fixed point $\bar q$ is completely determined by its center manifold 
reduction $F$. Here are application-friendly versions for these two elementary bifurcations.

\begin{theorem}[Saddle-Node Bifurcation]\label{thmsnb}
Let $F(\lambda, x), (\lambda, x)\in \mathbb R\times\mathbb R$ be a $C^2$ function satisfying 
\begin{itemize}
\item[(i)] $F(\fp,\fx)=\fx$ for some $\fp$ and $\fx$ for some $\fp$ and $\fx$. 
\item[(ii)] $\frac{\partial F}{\partial x}(\fp,\fx)=1$.
\item[(iii)] $a:=\frac{\partial^2 F}{\partial x^2}(\fp,\fx)\ne 0$.
\item[(iv)] $b:=\frac{\partial F}{\partial\lambda}(\fp,\fx)\ne 0$.
\end{itemize}
Then 
\begin{itemize}
\item[(a)] If 
\[
\hbox{$ab(\lambda-\fp)=\frac{\partial^2 F}{\partial x^2}(\fp,\fx)\cdot\frac{\partial F}{\partial\lambda}(\fp,\fx)\cdot(\lambda-\fp)<0$,}
\] 
then there are two fixed points of $F(\lambda, x)$
near $\fx$ for $\lambda$ near $\fp$, one is asymptotically stable, and the other is asymptotically unstable.
\item[(b)] If
\[
\hbox{$ab(\lambda-\fp)=\frac{\partial^2 F}{\partial x^2}(\fp,\fx)\cdot\frac{\partial F}{\partial\lambda}(\fp,\fx)\cdot(\lambda-\fp)>0$,}
\] 
then there is no fixed points for $F(\lambda, x)$.
\end{itemize}
\end{theorem}

\begin{theorem}[Period-Doubling Bifurcation]\label{thmpdb}
Let $F(\lambda, x), (\lambda, x)\in \mathbb R\times\mathbb R$ be a $C^3$ function satisfying
\begin{itemize}
\item[(i)] $F(\fp,\fx)=\fx$ for some $\fp$ and $\fx$ for some $\fp$ and $\fx$.
\item[(ii)] $\frac{\partial F}{\partial x}(\fp,\fx)=-1$.
\item[(iii)] $c:=\frac{\partial^3 F^2}{\partial x^3}(\fp,\fx)\ne 0$.
\item[(iv)] $d:=\frac{d}{d\lambda}[\frac{\partial F}{\partial x}(\fp,x(\fp))]:=\frac{d}{d\lambda}[\frac{\partial F}{\partial x}(\lambda,x(\lambda))]|_{\lambda=\fp}\ne 0$,
where $x(\lambda)$ is the fixed point of $F(\lambda, x)$ parameterized by $\lambda$: $F(\lambda, x(\lambda))=x(\lambda),\ x(\fp)=\fx$.
\end{itemize}
Then
\begin{itemize}
\item[(a)] If 
\[
\hbox{$cd(\lambda-\fp)=\frac{d}{d\lambda}[\frac{\partial F}{\partial x}(\fp,x(\fp))]\cdot \frac{\partial^3 F^2}{\partial x^3}(\fp,\fx)\cdot (\lambda-\fp)>0$,}
\]
then there is a unique period-2 orbit
of $F(\lambda, x)$ near $\fx$ for $\lambda$ near $\fp$. Moreover, the period-2 orbit
is asymptotically stable (respectively, unstable) if the corresponding fixed point $x(\lambda)$ is asymptotically unstable (respectively, stable).
\item[(b)] If
\[
\hbox{$cd(\lambda-\fp)=\frac{d}{d\lambda}[\frac{\partial F}{\partial x}(\fp,x(\fp))]\cdot \frac{\partial^3 F^2}{\partial x^3}(\fp,\fx)\cdot (\lambda-\fp)<0$,}
\]
then there is no period-2 orbits near $\fx$.
\end{itemize}
\end{theorem}

\medskip\noindent
\textit{Remark}: For $F(\lambda, x(\lambda))=x(\lambda)$, by implicit differentiation,
\[
\hbox{$\frac{dx}{d\lambda}(\lambda)=\frac{\partial F}{\partial\lambda}(\lambda,x(\lambda))
/[1-\frac{\partial F}{\partial x}(\lambda,x(\lambda))]$ and $\frac{dx}{d\lambda}(\fp)=\frac{\partial F}{\partial\lambda}(\fp,\fx)/2$.}
\]
As a result,
\[
\hbox{$\frac{d}{d\lambda}[\frac{\partial F}{\partial x}(\fp,x(\fp))]=\frac{\partial^2 F}{\partial\lambda\partial x}(\fp,\fx)+\frac{\partial^2 F}{\partial x^2}(\fp,\fx)\frac{\partial F}{\partial x}(\fp,\fx)/2$.}
\]

To keep the presentation simple enough, we will present a proof for an equivalent form 
of the second theorem. Similar treatment can be applied to the saddle-node bifurcation. 


\end{document}