text:
\section{Introduction}\label{sec:intro}

A \emph{slow-fast system} (SFS) is a singularly perturbed ordinary differential equation of the form
\eq{\label{intro:sf1}
  \dot x &= f(x,z,\varepsilon)\\
  \varepsilon\dot z &= g(x,z,\varepsilon),
}

where $x\in\mathbb{R}^m$, $z\in\mathbb{R}^n$ are local coordinates and where $\varepsilon>0$ is a small parameter. The over-dot denotes the derivative with respect to the time parameter $t$. Throughout this text, we assume that the functions $f$ and $g$ are of class $\mathcal{C}^{\infty}$. In applications (e.g \cite{Zeeman1}), $z(t)$ represents states or measurable quantities of a process while $x(t)$ stands for control parameters. The parameter $\varepsilon$ models the difference of the rates of change between the variables $z$ and $x$. That is why systems like \cref{intro:sf1} are often used to model phenomena with two time scales. Observe that the smaller $\varepsilon$ is, the faster $z$ evolves with respect to $x$. Therefore we refer to $x$ (resp. $z$) as the \emph{slow} (resp. \emph{fast}) variable. The time parameter $t$ is known as the \emph{slow time}. For $\varepsilon\neq 0$, we can define a new time parameter $\tau$ by the relation $t=\varepsilon\tau$. With this time reparametrization \cref{intro:sf1} can be written as
\eq{\label{intro:sf2}
x' &= \varepsilon f(x, z,\varepsilon)\\
z' &= g(x,z,\varepsilon),
}

where now the prime denotes the derivative with respect to the rescaled time parameter $\tau$, which we call \emph{the fast time}. Since we consider only autonomous systems, we often omit to indicate the time dependence of the variables. In the rest of this document, we prefer to work with slow-fast systems presented as \cref{intro:sf2}.\smallskip

Observe that as long as $\varepsilon\neq 0$ and $f$ is not identically zero, systems \cref{intro:sf1} and \cref{intro:sf2} are equivalent. A first approach to understand the qualitative behavior of slow-fast systems is to study the limit $\varepsilon\to0$. The slow equation \cref{intro:sf1} restricted to $\varepsilon=0$ reads as
\eq{\label{intro:cde1}
\dot x &= f(x,z,0)\\
0 &= g(x,z,0).
}

A system of the form \cref{intro:cde1} is called \emph{constrained differential equation} (CDE) \cite{Jardon1,Takens1}. On the other hand, in the limit $\varepsilon\to0$, a system given by \cref{intro:sf2} becomes
\eq{\label{intro:layer}
x' &= 0\\
z' &= g(x,z,0),
}

which is called \emph{the layer equation}. Associated to both systems, \cref{intro:cde1} and \cref{intro:layer}, the slow manifold $S$ is defined by
\eq{
S=\left\{ (x,z)\in\mathbb{R}^m\times\mathbb{R}^n\, | \, g(x,z,0)=0 \right\},
}
which serves as the phase space of the CDE \cref{intro:cde1} and as the set of equilibrium points of the layer equation \cref{intro:layer}. In the latter context, it is useful to recall the concept of Normally Hyperbolic Invariant Manifold (NHIM).

\begin{definition}[Normally Hyperbolic Invariant Manifold] Consider a slow-fast system given by a vector field of the form
\eq{X_\varepsilon=\varepsilon f(x,z,\varepsilon)\parc{x}+g(x,z,\varepsilon)\parc{z}.
}

The associated slow (invariant) manifold $S=\left\{ g(x,z,0)=0\right\}$ is said to be normally hyperbolic if each point of $S$ is a hyperbolic equilibrium point of $X_0$.
\end{definition}

NHIMs are relevant in the context of the geometric study of slow-fast systems, see for example \cite{Fenichel}. It is known that compact NHIMs persist under $\mathcal C^1$ small perturbation of the vector field \cite{Jones,Kaper}. In the particular context presented above, a normally hyperbolic compact subset of the slow manifold $S$  persists as an invariant manifold of the slow-fast system $X_\varepsilon$. We show in \cref{fig:intro1}  a schematic of the previous description.

\begin{figure}[htbp]\centering
  \includegraphics{qual1.pdf}
  \caption{A schematic representation of the persistence of a NHIM under the perturbation of the corresponding vector field. $S$ denotes the slow manifold. Left-above: $S$ is a set of hyperbolic equilibrium points of the layer equation. Left-below: $S$ is the phase space of the constrained equation. Right: since $S$ is a NHIM, it persists as an invariant manifold $S_\varepsilon$ under small perturbations of the vector field. }
  \label{fig:intro1}
\end{figure}

After this intruduction, we turn into the subject of this paper. Our goal is to understand the dynamics of a particular slow-fast system which has one fast and two slow variables given as

\eq{\label{intro:eqcusp}
X_\varepsilon=\varepsilon(1+\varepsilon f_1)\parc{x_1}+\varepsilon^2f_2\parc{x_2}-\left( z^3+x_2z+x_1+\varepsilon f_3 \right)\parc{z},
}

where the functions $f_i=f_i(x_1,x_2,z)$, for $i=1,2,3$, are smooth and vanish at the origin. The corresponding slow manifold is defined by
\eq{
  S=\left\{ (x_1,x_2,z)\in\mathbb{R}^3 \,| \, z^3+x_2z+x_1=0  \right\}.
}

\begin{remark}
  The slow manifold $S$ can be regarded as the critical set of the cusp (or $A_3$) catastrophe, which is given as \rm\cite{Arnold_singularities,Brocker}\eq{\label{eqV}
  V(x_1,x_2,z)=\frac{1}{4}z^4+\frac{1}{2}x_2z^2+x_1z.
  }
\end{remark}

 We denote by $\Delta$ the set of points in $S$ at which $S$ is tangent to the fast direction, that is
 \eq{\label{eqD}
 \Delta=\left\{ (x_2,z)\in S \, | \, 3z^2+x_2=0\right\}.
 }

In other words, $\Delta$ is the set of degenerate critical points of \cref{eqV}. See figure \cref{fig:qual1} for a description of the slow manifold and the set $\Delta$.

\begin{figure}[htbp]\centering
  \begin{tikzpicture}
    \pgftext{\includegraphics{cusp1.pdf}}
    \node at (1,1.4) {$S$};
    \node at (-2,0.25) {$\Delta$};
    \node at (1.6,.75) {$C$};
    \node at (2.2,.25) {$z$};
    \node at (1.2,-1) {$x_2$};
    \node at (-1.2,-1) {$x_1$};
  \end{tikzpicture}
  \caption{The manifold $S$ is two dimensional and can be defined as the critical set of the potential function $V(x_1,x_2,z)=\tfrac{1}{4}z^4 + \tfrac{1}{2}x_2z^2+x_1z$. The curve $\Delta$ is defined by the set of degenerate critical points of $V$. Geometrically, $B$ is the set of point of $S$ where $S$ is tangent to the fast direction, and $C$ denotes the cusp point.}
  \label{fig:qual1}
\end{figure}

Our interest in studying \cref{intro:eqcusp} is due to the fact that the origin $(x_1,x_2,z)=(0,0,0)$ is a \emph{non-hyperbolic equilibrium point} of $X_0$. This implies that a compact subset, around the origin, of the slow manifold $S$ is not a NHIM of $X_0$, and therefore, the Geometric Singular Perturbation Theory \cite{Fenichel,Jones,Kaper} is not enough.

\subsection{Motivation}

There have been several studies, e.g. \cite{Krupa3,Krupa20102841}, dealing with a SFS of the form
\eq{
X_\varepsilon=\varepsilon(1+f_1)\parc{x_1}-\left( z^2+x_1+ \varepsilon h \right)\parc{z},
}

whose slow manifold is the critical set of the fold catastrophe. The next natural step is to consider the following case in the Thom list \cite{Stewart1}, i.e., a slow-fast system induced by the cusp catastrophe. That is
\eq{\label{mot:eqcusp}
X_\varepsilon=\varepsilon(1+f_1)\parc{x_1}+\varepsilon f_2\parc{x_2}-\left( z^3+x_2z+x_1+\varepsilon f_3 \right)\parc{z}.
}

In \cite{BKK}, the system \cref{mot:eqcusp} is studied in a qualitative way. Here, however, we aim to refine the results by heavily using techniques from normal form theory. Moreover, we remark that the methods presented here are applicable to a larger class of slow-fast system given by
\eq{
  X_\varepsilon=\varepsilon(1+f_1)\parc{x_1}+\sum_{i=2}^{k-1}\varepsilon f_i\parc{x_i}-\left( z^k+\sum_{j=1}^{k-1}x_jz^{j-1}-\varepsilon f_k \right)\parc{z},
}

which is called (regular) $A_k$-SFS, see \cite{JardonThesis}.

\subsection{Statement}

We shall study the SFS

\eq{\label{s1}
  X_\varepsilon=\varepsilon (1+f_1)\parc{x_1}+\varepsilon f_2\parc{x_2}-\left( z^3+x_2z+x_1 + \varepsilon f_3 \right)\parc{z},
}

where the functions $f_i=f_i(x_1,x_2,z,\varepsilon)$ are smooth. To avoid working with an $\varepsilon$-parameter family of vector fields as \cref{s1}, it is customary to extend \cref{s1} by adding the trivial equation $\varepsilon'=0$, and thus consider a smooth vector field in $\mathbb{R}^4$ which reads as
\eq{\label{s2}
  X=\varepsilon (1+ f_1)\parc{x_1}+\varepsilon f_2\parc{x_2}-\left( z^3+x_2z+x_1 + \varepsilon f_3 \right)\parc{z} + 0\parc{\varepsilon}.
}

We regard \cref{s2} as a perturbation of ``the principal part'' $F$ which is given as
\eq{\label{s3}
  F=\varepsilon \parc{x_1}+0\parc{x_2}-\left( z^3+x_2z+x_1  \right)\parc{z} + 0\parc{\varepsilon}.
}

 Note that in a qualitative sense, $F$ contains the essential elements of $X$. To state our main result, we first define the sections

\eq{
  \Sigma^-=\left\{ (x_1,x_2,z,\varepsilon)\in\mathbb{R}^4\, |\, x_1=-x_1^{i}  \right\}\\
  \Sigma^-=\left\{ (x_1,x_2,z,\varepsilon)\in\mathbb{R}^4\, |\, x_1=x_1^{f}  \right\},
}

where $x_1^{i}>0$ and $x_1^{f}>0$ are arbitrarily large constants. For $\varepsilon>0$ but sufficiently small, the sections $\Sigma^{-}$ and $\Sigma^+$ are transversal to the flow of $X_\varepsilon$. Next, let $\Pi:\Sigma^{-}\to\Sigma^{+}$ be the Poincaré map induced by the flow of $X_\varepsilon$. We shall prove the following.

\paragraph{{\bfseries{Transition along the cusp}} ({\rm see \cref{teo:main}})} {
Consider a slow-fast system given by \cref{s2}. Let $\Sigma^-$, $\Sigma^+$ and $\Pi:\Sigma^-\to\Sigma^+$ be defined as above. Then, we can choose coordinates in $\Sigma^-$ and in $\Sigma^+$ such that the map $\Pi$ reads as
\eq{
\Pi(X_2,Z,\varepsilon)=(\tilde X_2,\tilde Z,\tilde \varepsilon),
}

where $\tilde X_2=X_2+H(X_2,\varepsilon)$ (with $H$ flat at $(X_2,\varepsilon)=(0,0)$), $\tilde\varepsilon=\varepsilon$ and where

\eq{\label{s4}
  \tilde Z=\Phi(X_2,\varepsilon)+Z\exp\left( -\frac{1}{\varepsilon}(A(X_2,\varepsilon)+\varepsilon\Psi(X_2,Z,\varepsilon)) \right),
}

where $A(X_2,0)>0$. Details of the functions $\Phi$, $A$, and $\Psi$ are given in \cref{teo:main}. In an heuristic way, this result is described in \cref{figcusp}.

\begin{figure}\centering
  \begin{tikzpicture}
    \pgftext{\includegraphics[scale=1.5]{cusp_qual2.pdf}}
    \node at (1,2) {$S_\varepsilon$};
    \node at (-3.3,-0.1) {$S$};
    \node at (-2,2.5) {$\Sigma^-$};
    \node at (3,-0.1) {$\Sigma^+$};
  \end{tikzpicture}
  \caption{Description of our main result. We may choose appropriate coordinates at the sections $\Sigma^-$ and $\Sigma^+$ under which the invariant manifold $S_\varepsilon$ is given by $Z=0$. Moreover form \cref{s3} we have that all other trajectories starting at $\Sigma^-$ are exponentially attracted to the invariant manifold $S_\varepsilon$. In this paper we provide quantitative information regarding this exponential contraction.}
  \label{figcusp}
\end{figure}

}

\subsection{Idea of the proof}

Our proof consists of two main steps.

\begin{enumerate}
  \item From \cite{Jardon2}, it is known that there exists a formal transformation bringing \cref{s2} into
\eq{\label{s5}
  F=\varepsilon \parc{x_1}+0\parc{x_2}-\left( z^3+x_2z+x_1  \right)\parc{z} + 0\parc{\varepsilon}.
}

Then, by Borel's lemma \cite{Brocker}, the vector field $F$ can be realized as a smooth normal form $X^N=F+R$ of \cref{s2} and where $R$ is flat at $(x_1,x_2,z,\varepsilon)=(0,0,0,0)$. See more details in \cref{sec:formal_nf}.

\item Based the previous normalization, next we use the geometric desingularization or blow up method (as introduced in \cite{DumRou2}) to study the flow of the normal form $X^N=F+R$. This is detailed in \cref{sec:GeomDes}.
\end{enumerate}

\begin{remark} With this document we aim at two goals:

\begin{enumerate}
  \item To refine the results of \cite{BKK}. This is, we do not only provide a qualitative description of the transition $\Pi$, but details on the differentiability of such a map is also presented.
  \item To prepare a framework for the geometric desingularization of $A_k$ slow-fast systems. These are a generalization of \cref{s2} given as
  \eq{
  X=\varepsilon(1+ f_1)\parc{x_1} + \sum_{i=1}^{k-1}\varepsilon f_i\parc{x_i} - \left( z^k+\sum_{j=1}^{k-1}x_jz^{j-1}+\varepsilon f_k \right)\parc{z}+0\parc{\varepsilon}.
  }
\end{enumerate}

\end{remark}

The rest of this document is arranged as follows: in \cref{sec:preliminaries} we provide a brief recollection of preliminary results that will simplify our later studies. Next, in \cref{sec:GeomDes} we pose our result and prove it by means of the geometric desingularization method and the results of  \cref{sec:preliminaries}. For readability purposes, many technicalities have been put in the appendix.

\section{Preliminaries of slow-fast systems}\label{sec:preliminaries}
In this section, we provide a number preliminary results that will be used later in \cref{sec:GeomDes}. First of all, we consider slow-fast systems along normally hyperbolic regions of the slow manifold. Afterwards, we recall a result from \cite{Jardon2} dealing with the normal form of \cref{s2}. We remark that we only consider SFS with one fast variable. Let us be more precise with the type of SFS that we shall study first.

\begin{definition} A slow-fast system is said to be (locally) regular around a point $p_0$, if its corresponding slow manifold is normally hyperbolic in a some neighborhood of $p_0$.
\end{definition}

\subsection{The slow vector field}

Let us consider a slow-fast system given by
\eq{\label{sdi1}
  X_\varepsilon = \sum_{i=1}^m \varepsilon f_i(x,z,\varepsilon)\parc{x_i}+H(x,z,\varepsilon)\parc{z},
}

where $x\in\mathbb{R}^m$, $z\in\mathbb{R}$, and as usual $0<\varepsilon\ll 1$. Furthermore, assume that $f(0,0,0)\neq 0$, $H(0,0,0)=0$ and $\tparcs{H}{z}(0,0,0)<0$. Thus $X_\varepsilon$ is regular around $0\in\mathbb{R}^{m+2}$. The slow manifold associated to \cref{sdi1} is defined by
\eq{
  S=\left\{ (x,z)\in\mathbb{R}^{m+1}\, | \, H(x,z,0)=0 \right\}.
}

From the defining assumptions of \cref{sdi1}, we have that $S$ is a NHIM in a neighborhood of the origin. By looking at the Jacobian of $X_\varepsilon$ at $0$, it follows that there exists an $m+1$ dimensional a center manifold. Since $X$ is smooth, we can choose a $\mathcal{C}^{\ell}$  center manifold $\mathcal{W}^{^C}$ for any $\ell<\infty$. The manifold $\mathcal{W}^{^C}$ is given as a graph $z=\phi(x,\varepsilon)$ where $\phi$ is a $\mathcal{C}^{\ell}$ function.

\begin{remark} Along the rest of the document we frequently make use of a finite class of differentiability. As it is customary in the present context, when we say that a manifold (or a map) is $\mathcal{C}^{\ell}$, we mean that such a manifold (or map) is $\ell$-differentiable for $\ell$ as large as necessary.
\end{remark}

The slow manifold $S$ is naturally given by the restriction $\mathcal{W}^{^C}|_{\varepsilon=0}=S$. Next, let us consider the vector field $\frac{1}{\varepsilon}X_\varepsilon(x,\phi,\varepsilon)$. Since $\mathcal{W}^{^C}$ is locally invariant, it follows that $\frac{1}{\varepsilon}X_\varepsilon$ is tangent to $\mathcal{W}^{^C}$. Therefore the vector field
\eq{
  X^{slow}=\lim_{\varepsilon\to 0}\frac{1}{\varepsilon}X_\varepsilon(x,\phi,\varepsilon),
}

is tangent to $S$ at each point of $S$, and we call it \emph{the slow vector field}. We remark that the slow vector field $X^{slow}$ is only well defined whenever $\phi$ is invertible.

\subsubsection{The slow divergence integral}\label{sec:sdi}

Associated to a regular slow-fast system and the corresponding slow vector field, the \emph{slow divergence integral} is defined here. For this, let $\Sigma^-$ and $\Sigma^+$ be two sections which are transversal to the flow of $X_\varepsilon$ given by \cref{sdi1}. For $\varepsilon\neq 0$ but sufficiently small, these sections are also transversal to the slow manifold $S$. Let $\gamma_\varepsilon$ be a solution curve of $X_\varepsilon$ chosen along a center manifold $\mathcal{W}^{^C}$, thus $\gamma_\varepsilon$ is transversal to the sections $\Sigma^-$ and $\Sigma^+$. In the limit $\varepsilon=0$, the curve $\gamma_0$ is a curve along the slow manifold $S$. The idea now is to borrow the well-known divergence theorem \cite{Spivak} to get some sense on how the trajectories of $X_\varepsilon$ are attracted to $S$ (recall that we made the assumption $\tparcs{H}{z}<0$). The divergence of $X_\varepsilon$ (given by \cref{sdi1}) reads as
\eq{
  \Div X_\varepsilon &= \parcs{H(x,z,\varepsilon)}{z}+O(\varepsilon).
}

We can now take the integral of $\Div X_\varepsilon$ along the orbit $\gamma_\varepsilon$ of $X_\varepsilon$ parametrized by the fast time $\tau$, we have
\eq{\label{sdi2}
  \int_{\gamma_\varepsilon}\Div X_\varepsilon  \, d\tau=\int_{\gamma_\varepsilon} \left(  \parcs{H(x,z,\varepsilon)}{z}+O(\varepsilon) \right)d\tau.
}

The \emph{slow divergence integral} is defined by
\eq{
  I(t)=\int_{\gamma_0} \Div X_0 \, dt,
}

where $t$ is the slow time defined by the slow vector field $X^{slow}$. Our goal then is to relate the divergence integral \cref{sdi2} with $I$.

\begin{proposition} Under the assumptions made in this section, we have that
\eq{
\int_{\gamma_\varepsilon}\Div X_\varepsilon \, d \tau=\frac{1}{\varepsilon}\left( I(t)+o(1)\right),
}

where $I(t)$ is the slow divergence integral.

\end{proposition}

\begin{proof}
  Recall that the slow vector field reads as $X^{slow}=\lim_{\varepsilon\to 0}\frac{1}{\varepsilon}X_\varepsilon(x,\phi,\varepsilon)$, where $\phi=\phi(x,\varepsilon)$ is a $\mathcal{C}^{\ell}$ function. By our assumptions, the curve $\gamma_\varepsilon$ is transversal to the sections $\Sigma^-$ and $\Sigma^+$ for $\varepsilon$ small enough. Without loss of generality we can assume that $\gamma_\varepsilon$ is parametrized by $x_1$. Then let $x_1^-$ and $x_1^+$ be defined by $\gamma_\varepsilon(x_1^{-})=\gamma_\varepsilon\cap\Sigma^-$ and $\gamma_\varepsilon(x_1^{+})=\gamma_\varepsilon\cap\Sigma^+$. Next,  the integral of the divergence of $X_\varepsilon$ along $\gamma_\varepsilon$ from $\Sigma^-$ to $\Sigma^+$ reads as

  \eq{
  \int_{\gamma_\varepsilon}\Div X_\varepsilon  \,  d\tau &=\frac{1}{\varepsilon}\int_{x_1^-}^{x_1^+} \left(\parcs{H(x,z,0)}{z}+O(\varepsilon)\right) \frac{dx_1}{f_1(x,z,0)+o(1)}\\
                    &=\frac{1}{\varepsilon} \left(\int_{x_1^-}^{x_1^+} \parcs{H(x,z,0)}{z}\frac{dx_1}{f_1(x,z,0)} + o(1)\right)\\
                    &=\frac{1}{\varepsilon} \left(\int_{\gamma_0}\Div X_0\, dt+o(1)\right),
  }

  where $t$ is the slow time induced by $X^{slow}$, which in coordinates means that $\frac{dx_1}{dt}=f_1$.
\end{proof}

Observe that the slow divergence integral is a first order approximation of the divergence along orbits of $X_\varepsilon$. This will be useful when presenting our main result in \cref{sec:GeomDes}.

\subsubsection{Normal form and transition of a regular slow-fast system}\label{sec:nf_reg}

Now we consider the problem of finding a suitable normal form of a regular SFS. The following is a well-known result but we recall it here for completeness.

\begin{proposition}\label{prop:nf_reg}
  Consider a regular slow-fast system on $\mathbb{R}^{m+3}$ given by
\eq{\label{eqr}
  X_\varepsilon=\varepsilon(1+ f_1)\parc{u}+\sum_{j=1}^m\varepsilon g_j\parc{v_j}+H\parc{z},
}
where $(u,v_1,\ldots,v_m,z,\varepsilon)\in\mathbb{R}^{m+3}$; where the functions $f_1=f_1(u,v,z,\varepsilon)$ and $g_j=g_j(u,v,z,\varepsilon)$, for $2\geq j\geq k-1$, are smooth and where the function $H=H(u,v,z,\varepsilon)$ is smooth with $H(0,0,0,0)=0$ and $\tparcs{H}{z}(0,0,0,0)<0$. Then, the vector field $X$ is $\mathcal{C}^{\ell}$-equivalent to a normal form given by
\eq{
  X_{\varepsilon}^N=\varepsilon\parc{U} + \sum_{j=1}^m 0\parc{V_j}-Z\parc{Z},
}

where $\left\{ Z=0\right\}$ corresponds to a choice of the center manifold $\mathcal{W}^{^C}$ of $X_\varepsilon$.
\end{proposition}

\begin{proofof}{\cref{prop:nf_reg}}
The first step is to divide the vector field $X$ by $1+f_1$. In a sufficiently small neighborhood of the origin this is a smooth equivalence relation. That is $Y=\tfrac{1}{1+f_1}X$ reads as
\eq{
Y= \varepsilon\parc{u}+\sum_{j=1}^m\varepsilon \tilde g_j\parc{v_j}+\tilde H\parc{z},
}

where $\tilde g_j$, for $2\geq j\geq k-1$, and $\tilde H$ are smooth with $\tilde H(0)=0$ and $\tparcs{\tilde H}{z}(0)<0$. Now we note that the origin of $\mathbb{R}^{m+3}$ is a semyhyperbolic equilibrium point with $(u,v,\varepsilon)$ being center coordinates and $z$ being the hyperbolic coordinate. We can now use Takens-Bonckaert results on normal forms of partially hyperbolic vector fields \cite{Bonckaert1,Bonckaert2,Takens_partially}. Thus, there exists a $\mathcal{C}^{\ell}$ change of coordinates (maybe respecting some constraints if required) under which $Y$ is conjugated to
\eq{
  \bar Y=\varepsilon\parc{U}+\sum_{j=1}^m\varepsilon \bar G_j\parc{V_j}+\bar H Z\parc{Z},
}

where $\bar G_j=\bar G_j(U,V,\varepsilon)$, for $2\geq j\geq k-1$, and $\bar H=\bar H(U,V,\varepsilon)$ are $\mathcal{C}^{\ell}$ functions, and where $\left\{ Z=0\right\}$ corresponds to a choice center manifold which we denote by $\mathcal{W}^{^C}$. We remark that in the vector field $\bar Y$, the functions $\bar G_j$ and $\bar H$ are independent of $Z$. Furthermore we have
\eq{
  \bar H(0,0,0)=\parcs{\tilde H}{z}(0,0,0,0)<0.
}

This means that in a small neighborhood of the origin $\bar Y$ can be divided by $|\bar H|$. In other words, $\bar Y$ is $\mathcal{C}^{\ell}$-equivalent to
\eq{
  \mathcal Y=\varepsilon\mathcal G \parc{U}+\sum_{j=1}^m\varepsilon \bar K_j\parc{V_j}-Z\parc{Z},
}

where $\mathcal G(0,0,0)\neq 0$ and $\bar K_j=\bar K_j(U,V,\varepsilon)$, for $2\geq j\geq k-1$, are $\mathcal{C}^{\ell}$. Next, since $\mathcal{W}^{^C}=\left\{ Z=0 \right\}$ is invariant under the flow of $\mathcal Y$, we can study the restriction $\mathcal Y|_{Z=0}$. This is
\eq{
  \mathcal Y|_{Z=0}=\varepsilon\mathcal G \parc{U}+\sum_{j=1}^m\varepsilon \bar K_j\parc{V_j}.
}

For $\varepsilon\neq 0$, the vector field $\mathcal Y|_{Z=0}$ is regular because $\mathcal G(0,0,0)\neq 0$. Thus, by the flow-box theorem, there exists a change of coordinates, depending in a $\mathcal{C}^{\ell}$ way on $\varepsilon$, under which $\mathcal Y|_{Z=0}$ can be written as
\eq{
  \varepsilon\parc{U}+\sum_{j=1}^{m}0\parc{V_j}.
}

This implies that $\mathcal Y$ is $\mathcal{C}^{\ell}$-equivalent to
\eq{
  X_{reg}^N=\varepsilon \parc{U}+\sum_{j=1}^m0\parc{V_j}-Z\parc{Z},
}

as stated in the proposition.

\end{proofof}

Motivated by \cref{prop:nf_reg} let us now discuss the dynamics of the vector field
\eq{\label{r1} X_{reg}^N=\varepsilon \parc{U}+\sum_{j=1}^m0\parc{V_j}-Z\parc{Z}.
}

The slow manifold $S$, corresponding to the normal form \cref{r1}, is given by
\eq{
  S=\left\{ \varepsilon=0, \, Z =0 \right\}.
}

Furthermore, we can parametrize the solution of \cref{r1} by $U$. Let us define the sections
\eq{\label{sec_reg}
  \Sigma^{-} &= \left\{ (U,V,Z,\varepsilon)\in\mathbb{R}\times\mathbb{R}^m\times\mathbb{R}\times\mathbb{R}\, | \, U=U^- \right\}\\
  \Sigma^{+} &= \left\{ (U,V,Z,\varepsilon)\in\mathbb{R}\times\mathbb{R}^m\times\mathbb{R}\times\mathbb{R}\, | \, U=U^+ \right\},
}

where $U^-<U^+$. The sections $\Sigma^{-}$ and $\Sigma^{+}$ are transversal to the manifold $S$ and therefore, for $\varepsilon\neq 0$, are also transversal to the flow of \cref{r1}. Associated to these sections, we define the transition
\eq{\label{regtr}
  \Pi &:\Sigma^-\to\Sigma^+\\
  &(V,Z,\varepsilon)\mapsto (\tilde V,\tilde Z,\tilde\varepsilon).
}

To compute the component $\tilde Z$ we only need to integrate $\tfrac{dZ}{dU}=-\tfrac{1}{\varepsilon}Z$. Then it follows that $\tilde Z=Z(T)$, where $T$ is the time to go from $\Sigma^-$ to $\Sigma^+$, which is $T=U_f-U_i$. Then it follows that
\eq{
  \tilde V&= V\\
  \tilde Z&=Z\exp\left(-\frac{1}{\varepsilon}(U_f-U_i)\right)\\
  \tilde\varepsilon&=\varepsilon.
}

Observe the particular format of the transition $\Pi$. The $Z$ component is an \emph{exponential} contraction towards the center manifold $\left\{ Z=0 \right\}$. Maps with this characteristic appear frequently in our text and also in several other cases where slow-fast systems are studied. Therefore, in \cref{sec:Exp_trans} we discuss in a rather general way, the properties of such maps.

\subsection{Formal normal form of $A_k$ slow-fast systems}\label{sec:formal_nf}
In this section we recall a normal form of the so-called $A_k$ slow-fast systems. A proof can be found in \cite{Jardon2}. This normalization is important since it eliminates many unwanted terms from the system being studied here.

\begin{definition}\label{def:AkSFS} Let $k\in\mathbb{N}$ with $k\geq 2$. An $A_k$ slow-fast system ($A_k$-SFS) is an ODE of the form
\eq{\label{formal1}
x_1' &= \varepsilon(1+ f_1)\\
x_j' &= \varepsilon f_j\\
z' &= -\left( z^k+\sum_{i=1}^{k-1} x_iz^{i-1} \right) + \varepsilon f_k\\
\varepsilon' &= 0,
}

where $j=2,\ldots,k-1$, and where the functions $f_i=f_i(x_1,\ldots,x_{k-1},z,\varepsilon)$, for $1\leq i\leq k$, are smooth.
\end{definition}

\begin{remark}\leavevmode
\begin{itemize}
  \item The system investigated in this work is an $A_3$-SFS.

  \item The slow manifold associated to an $A_k$-SFS is defined by
  \eq{
  S=\left\{ (x,z)\in\mathbb{R}^k\, | \, z^k+\sum_{i=1}^{k-1} x_iz^{i-1}=0  \right\}.
  }

  The manifold $S$ can equivalently be defined as the critical set of an $A_k$ catastrophe \cite{Arnold_singularities}. Hence the name $A_k$-SFS.
\end{itemize}

\end{remark}

Locally, we can regard \cref{formal1} as $X=F+P$ where $F$ and $P$ are smooth vector fields of the form

\eq{\label{formalF}
F &= \varepsilon\parc{x_1}+\sum_{j=2}^{k-1}0\parc{x_j}+g\parc{z}+0\parc{\varepsilon}
}

and
\eq{\label{formalP}
P &=\sum_{i=1}^{k-1}\varepsilon f_i\parc{x_i}+\varepsilon f_k\parc{z}+0\parc{\varepsilon},
}

respectively and where $g=-\left( z^k+\sum_{i=1}^{k-1} x_iz^{i-1} \right)$. We refer to $F$ as the ``principal part'' and to $P$ as the ``perturbation''. Briefly speaking we want to eliminate, via a change of coordinates, the perturbation. The procedure of normalizing the vector field $X$ is motivated by \cite{Sto10}, where normal forms of analytic perturbations of quasihomogeneous vector fields are investigated. The relevant result is the following

\begin{theorem}[Formal normal form \cite{Jardon2}]\label{prop:formal_nf} Let $k\geq 2$ and let $X=F+P$ be a smooth vector field where
\eq{
F=\varepsilon\parc{x_1}+\sum_{i=2}^{k-1}0\parc{x_i}-\left( z^k+\sum_{j=1}^{k-1}x_jz^{j-1} \right)\parc{z}+0\parc{\varepsilon}.
}

and where
\eq{
  P=\sum_{i=1}^{k-1}P_i\parc{x_i}+P_k\parc{z}+0\parc{\varepsilon},
}

where each $P_i=P_i(x_1,\ldots,x_{k-1},z,\varepsilon)$ is a smooth function. Assume that the following conditions are satisfied
\begin{enumerate}
  \item $P_i(x_1,\ldots,x_{k-1},z,0)=0$,
  \item $\rho(\hat P_i)\geq 2k-i+1$,
\end{enumerate}

where $\hat P_i$ denotes the Taylor expansion of $P_i$ and $\rho(\hat P_i)$ is the quasihomogeneous order of the polynomial $\hat P_i$.
Then, there exists a formal diffeomorphism $\hat\Phi$ such that $\hat\Phi_*\hat X=F$.
\end{theorem}

In words, \cref{prop:formal_nf} shows that $\hat X$ and $F$ are conjugated via $\hat\Phi$. It follows that, by Borel's lemma \cite{Brocker}, the formal vector field $\hat X^N=F$ can be realized as a smooth vector field $X^N=F+\tilde P$ where $\tilde P$ is \emph{flat} at $(x,z,\varepsilon)=(0,0,0)$. This has important consequences in the geometric desingularization of an $A_3$-SFS, presented in the following section.

\section{Geometric desingularization of a slow-fast system near a cusp singularity}\label{sec:GeomDes}
\renewcommand{\a}{a}
\renewcommand{\b}{b}
\newcommand{z}{z}

In this section we study an $A_3$ slow-fast system based on: a) the techniques introduced in \cref{sec:preliminaries} and in \cref{sec:Exp_trans}, and b) the blow up method. To simplify the notation, let us now write the $A_3$-SFS  as
\eq{\label{eq:cusp1}
X &=\varepsilon(1+f_1)\parc{\a}+\varepsilon f_2\parc{\b}-(z^3+\b z+\a+\varepsilon f_3)\parc{z} + 0\parc{\varepsilon},
}

where thanks to \cref{prop:formal_nf}, the smooth functions $f_i=f_i(a,b,z,\varepsilon)$ are flat at the origin of $\mathbb{R}^4$. We invetigate the transition associated to \cref{eq:cusp1} between the sections
\eq{\label{def:cuspSectionsfar}
\Sigma^{-} &= \left\{ (\a,\b,z,\varepsilon)\in\mathbb{R}^4\, | \, \a=-a^-, \, z>0 \right\}\\
\Sigma^{+} &= \left\{ (\a,\b,z,\varepsilon)\in\mathbb{R}^4\, | \, \a=a^+, \, z<0 \right\},
}

where $a^->0$ and $a^+>0$ are arbitrarily large constants. However, since the trajectories of $X$ spend a long time along regular parts of $S$, it will be useful to define the ``entry'' and ``exit'' sections
\eq{\label{def:cuspSections}
\Sigma^{\en} &= \left\{ (\a,\b,z,\varepsilon)\in\mathbb{R}^4\, | \, \a=-\a_0, \, z>0 \right\}\\
\Sigma^{\ex} &= \left\{ (\a,\b,z,\varepsilon)\in\mathbb{R}^4\, | \, \a=\a_0, \, z<0 \right\},
}

where $\a_0$ is a positive but sufficiently small constant, for reference see \cref{fig:rr}.

\begin{figure}[htbp]\centering
  \begin{tikzpicture}
    \pgftext{\includegraphics[scale=1.5]{cusp_qual.pdf}}
    \node at (-3.3,-.25) {$S$};
    \node at (-1.3,2.5) {$S_{\varepsilon}^{-}$};
    \node at (3,0.4) {$S_{\varepsilon}^{+}$};
    \node at (1.8,1.4) {$\mathcal{M}_{\varepsilon}$};
    \node at (2.1,0.75) {$\Sigma^{\ex}$};
    \node at (.5,2.5) {$\Sigma^{\en}$};
  \end{tikzpicture}
  \caption{ Qualitative representation of the investigation performed in this section. The sections $\Sigma^{\en}$ and $\Sigma^{\ex}$ are arbitrarily close to the cusp point. On the other hand the sections $\Sigma^{-}$ and $\Sigma^{+}$ (not shown) are parallel to $\Sigma^{\en}$ and $\Sigma^{\ex}$ but far away from the cusp point. In a qualitative sense, we will construct an invariant manifold $\mathcal{M}_\varepsilon$ and then extend it all the the way up to the sections $\Sigma^{-}$ and $\Sigma^{+}$. Our analysis aims for simplicity and thus depends extensively on the usage of normal forms. This, of course, makes our results coordinate-dependant.}
  \label{fig:rr}
\end{figure}

It will be clear from our analysis in the blow up space \cref{Ken} that the section $\Sigma^-$ needs to be partitioned as follows.

\begin{definition}[The inner layer and the lateral regions]\label{def:layers} Let $0<L<M<\infty$ be constants. The inner layer $\Sigma^{\inn}\subset\Sigma^-$ is defined as
  \eq{
    \Sigma^-\supset\Sigma^{\inn}=\left\{ (b,z,\varepsilon)\in\Sigma^-\, | \, |b|<M\varepsilon^{2/5} \right\}.
  }

  On the other hand, the lateral regions are defined as
  \eq{
    \Sigma^-\supset\Sigma^{+b}&=\left\{ (b,z,\varepsilon)\in\Sigma^-\, | \, b>L\varepsilon^{2/5} \right\}\\
    \Sigma^-\supset\Sigma^{-b}&=\left\{ (b,z,\varepsilon)\in\Sigma^-\, | \, -b>L\varepsilon^{2/5} \right\}.
  }
  Note that the set $\left\{\Sigma^{\inn},\Sigma^{+b}, \Sigma^{-b}  \right\}$ is an open cover of $\Sigma^-$, see \cref{fig:part0}

  \begin{figure}[htbp]\centering
  \begin{tikzpicture}
    \pgftext{\includegraphics[scale=1.5]{region_partition0.pdf}}
    \node at (1.25,2.9) {$\Sigma^{\inn}$};
    \node at (4.1,-2.1) {$\Sigma^{+b}$};
    \node at (-4.1,-2.1) {$\Sigma^{-b}$};
    \node at (0,-3) {$b$};
    \node at (-3,0) {$\varepsilon$};
  \end{tikzpicture}\hfill

  \caption{ The section $\Sigma^{-}$ needs to be partitioned into three subsections: the inner layer $\Sigma^{\inn}$ and the lateral regions $\Sigma^{+b}$, $\Sigma^{-b}$. From a qualitative point of view, these three layers correspond to three different types of trajectories: 1. Trajectories starting at $\Sigma^{\inn}$ pass close to the cusp point. Observe that $\lim_{\varepsilon\to 0}(\Sigma^{\inn})=\left\{ b=0 \right\}$ and then corresponds to a solution of the associated $CDE$ passing exactly through the cusp point. 2. Trajectories starting at $\Sigma^{+b}$ pass sufficiently away from the cusp point along the regular side of the manifold $S$. 3. Trajectories starting at $\Sigma^{-b}$ pass sufficiently away from the cusp point along the folded side of the manifold $S$.  }
  \label{fig:part0}
\end{figure}
\end{definition}

We are now in position to present our main result. In the following theorem, we characterize the transition $\Pi:\Sigma^-\to\Sigma^+$ under a suitable choice of coordinates at the section $\Sigma^-$ and $\Sigma^+$. Furthermore, we give details on the differentiability of this map according to the cover of $\Sigma^-$, see \cref{def:layers}.

\begin{theorem}[Transition map of an $A_3$-SFS]\label{teo:main} Let $X$ be an $A_3$ slow-fast system. This is, $X$ is a vector field defined by
\eq{\label{eq:mainteo}
X=\varepsilon(1+ f_1)\parc{\a}+\varepsilon f_2\parc{\b}-\left(z^3+\b z+\a+\varepsilon f_3\right)\parc{z} + 0\parc{\varepsilon},
}

where each $f_i=f_i(\a,\b,z,\varepsilon)$, $i=1,2,3$, is smooth. Let the sections $\Sigma^{-}$, $\Sigma^{+}$ be defined as above. Then we can choose suitable $\mathcal{C}^{\ell}$-coordinates $(B,Z,\varepsilon)$ in $\Sigma^{-}$  and $\mathcal{C}^{\ell}$-coordinates $(\tilde B,\tilde Z,\tilde\varepsilon)$ in $\Sigma^{+}$ such that the transition $\Pi:(B,Z,\varepsilon)\mapsto (\tilde B,\tilde Z,\tilde\varepsilon)$ is an exponential type map of the form
  \eq{\label{eq:expmainteo}
     \Pi(B,Z,\varepsilon)=\left( B+h, \, \phi(B,\varepsilon)+Z\exp\left( -\frac{A(B,\varepsilon)+\Psi(B,Z,\varepsilon)}{\varepsilon} \right),\varepsilon \right),
    }
    where $h$ is flat at the origin,  $A>0$ is $\mathcal{C}^{\ell}$, $\phi$ is $\mathcal{C}^{\ell}$-admissible with $\phi(B,0)=0$, and  $\Psi$ is $\mathcal{C}^{\ell}$-admissible with $\Psi(B,Z,0)=0$, see \cref{sec:Exp_trans} for the definition of $\mathcal{C}^{\ell}$-admissible. Moreover, we have the following properties of the function $A$, $\phi$ and $\Psi$.

\begin{enumerate}
  \item $-A(B,0)=I(B)$ where $I$ is the slow divergence integral associated to \cref{eq:mainteo}.
  \item Restricted to $(B,Z,\varepsilon)\in\Sigma^{\inn}$, there are functions $\tilde \phi$ and $\tilde \Psi$ such that
   \eq{
   \phi(B,\varepsilon) &=\tilde\phi\left(\mu,\varepsilon^{1/5}\right)\\
      \Psi(B,Z,\varepsilon) &= \tilde\Psi\left(|B|^{1/2},\varepsilon^{1/5}, \varepsilon\ln\varepsilon, \mu,Z\right),
   }
   where $\tilde \phi$ and $\tilde \Psi$ are $\mathcal{C}^{\ell}$-functions with respect to monomials (see \cref{def:mon}) with $\mu=B\varepsilon^{-2/5}$. Note that in this domain, $\mu$ is well defined in the sense that $\mu$ is bounded by a constant as $\varepsilon\to 0$.
  \item Restricted to $(B,Z,\varepsilon)\in\Sigma^{+b}$, there is a function $\tilde\Psi$ such that
   \eq{
   \phi(B,\varepsilon) &= 0\\
      \Psi(B,Z,\varepsilon) &= \tilde\Psi\left(|B|^{1/2},\varepsilon^{1/5}, \varepsilon\ln(|B|), \sigma,Z\right),
   }
   where $\tilde\Psi$ is a $\mathcal{C}^{\ell}$-function with respect to monomials (see \cref{def:mon}) with $\sigma=\varepsilon|B|^{-5/2}$. Note that in this domain, $\sigma$ is well defined since $|B|>0$.
  \item  Restricted to $(B,Z,\varepsilon)\in\Sigma^{-b}$, there are functions $\tilde \phi$ and $\tilde \Psi$ such that
   \eq{
   \phi(B,\varepsilon) &=\tilde\phi\left(|B|^{1/2},\sigma\right)\\
      \Psi(B,Z,\varepsilon) &= \tilde\Psi\left(|B|^{1/2},\varepsilon^{1/5}, \varepsilon\ln(|B|), \sigma\right),
   }
   where $\tilde \phi$ and $\tilde \Psi$ are $\mathcal{C}^{\ell}$-functions with respect to monomials (see \cref{def:mon}) with $\sigma=\varepsilon|B|^{-5/2}$. Note that in this domain, $\sigma$ is well defined since $|B|>0$.
\end{enumerate}
\end{theorem}

\paragraph{Sketch of the proof.} { The first step is to recall \cref{prop:formal_nf}, which shows that $X$ is formally conjugate to
\eq{
  F=\varepsilon\parc{\a}+0\parc{\b}-\left(z^3+\b z+\a\right)\parc{z} + 0\parc{\varepsilon}.
}

Next, by means of the Borel's lemma \cite{Brocker}, the vector field $F$ can be realized as a smooth vector field $X^N=F+\varepsilon H$ where $H$ is flat at $(a,b,z,\varepsilon)=(0,0,0,0)$. Thus, from now on, we only treat an $A_3$-SFS given as

\eq{
X=\varepsilon(1+\varepsilon \tilde f_1)\parc{\a}+\varepsilon^2 \tilde f_2\parc{\b}-\left(z^3+\b z+\a+\varepsilon \tilde f_3\right)\parc{z} + 0\parc{\varepsilon},
}

where each $\tilde f_i=\tilde f_i(a,b,z,\varepsilon)$ is flat at $(a,b,z,\varepsilon)=(0,0,0,0)$.

Another important ingredient of the proof is the blow up technique, which is described in \cref{sec:blowup}. This method provides several local vector fields whose corresponding transitions are of exponential type, refer to \cref{sec:Exp_trans}. Later  all these local transitions are composed to produce an exponential type transition between the sections $\Sigma^-$ and $\Sigma^+$. Along the analysis of the local vector fields (in the blow up space) we will take advantage of the flatness of the higher order terms of $X$. The complete proof follows \cref{sec:blowup,Ken,Ke,Kex,Kpm} and is given in \cref{sec:proofofmain}.

Now, assuming that the transition $\Pi$ is of the form \cref{eq:expmainteo}, we can show that $A(B,0)$ is given by the slow divergence integral of $X$. For this, let us recall the Poincaré-Leontovich-Sotomayor formula \cite{DeMaesschalck2005}, which in general is given as follows.

\begin{proposition} Let $X$ be a vector field on a manifold $M^n$ with a volume form $\Omega$. Let $\Sigma^-$ and $\Sigma^+$ be two open sections of $M$ and transverse to the flow of $X$. Let $\gamma_\varepsilon$ be an orbit of $X$ along a center manifold $\mathcal{W}^{^C}$ of $X$, starting at $p=\gamma_\varepsilon\cap\Sigma^-$ and reaching $q=\gamma_\varepsilon\cap\Sigma^+$ in finite time. Let $\Pi:\Sigma^-\to\Sigma^+$ be the transition map defined in a neighborhood of $p$. If $\psi^-:U\to\Sigma^-$ and $\psi^+:V\to\Sigma^+$, with $U\subset\mathbb{R}^{n-1}$ and $V\subset\mathbb{R}^{n-1}$, are coordinates in $\Sigma^-$ and in $\Sigma^+$ respectively, then
\eq{\label{aux:sdi}
  \det\left( D\left( (\psi^{+})^{-1}\circ\Pi\circ\psi^- \right) \right)(s^-)=\frac{\langle \Omega(p), D\psi^-(s^-)\times X(p) \rangle}{\langle \Omega(q), D\psi^+(s^+)\times X(p) \rangle}\exp\left( \int_{\gamma_\varepsilon}\Div_{\Omega}X  \, d\tau \right),
}%

where $s^-=(\psi^-)^{-1}(p)$ and $s^+=(\psi^+)^{-1}(q)$. The integral is taken along the orbit $\gamma_\varepsilon$ from $p$ to $q$ parametrized by the fast time $\tau$.
\end{proposition}

So we have the following.

\begin{proposition} \label{prop:slowdiv} Consider an $A_3$-SFS and assume that the transition $\Pi:\Sigma^-\to\Sigma^+$ is given by \cref{eq:expmainteo}. Then $-A(B,0)=I(B)$, where $I(B)$ is the slow divergence integral associated to the $A_3$-SFS.
\end{proposition}

\begin{proof} The only relevant component is $Z$, so denote by $\Pi_Z$ the $Z$-component of $\Pi$. The factor multiplying the exponential in \cref{aux:sdi} can be taken as a constant $C>0$. Then we have that \cref{aux:sdi} for the vector field of \cref{teo:main} reads as
\eq{\frac{\partial\Pi_Z}{\partial Z}= C\exp\left( \int_{\gamma_\varepsilon}\Div_{\Omega}X  \, d\tau \right). }

Using the properties of the slow divergence integral described in \cref{sec:sdi}, and since $C\neq 0$, we have
\eq{\label{mainau0}
\frac{\partial\Pi_Z}{\partial Z} &= C\exp\left( \int_{\gamma_\varepsilon}\Div_{\Omega}X \, d\tau \right)\\
&=\exp\left( \frac{1}{\varepsilon}\left(\int_{\gamma_0}\Div X_0  \, dt+\varepsilon\ln C+o(1) \right) \right)\\
&=\exp\left( \frac{1}{\varepsilon}\left(I +O(\varepsilon) \right) \right),
}

where $I$ is the slow divergence integral of $X$ along a curve in the slow manifold $S$ from $\Sigma^-$ to $\Sigma^+$. In principle, the limit $\varepsilon\to 0$ of \cref{mainau0} is not well defined. However, according to our \cref{teo:main}, we have by differentiating \cref{eq:expmainteo} w.r.t. $Z$
\eq{\label{mainau1}
  \frac{\partial\Pi_Z}{\partial Z}=\exp\left( -\frac{A(B,\varepsilon)+\varepsilon \Psi(B,Z,\varepsilon)}{\varepsilon} \right).
}

Identifying \cref{mainau0} with \cref{mainau1} and taking the limit $\varepsilon\to 0$ we have indeed that
\eq{
  \lim_{\varepsilon\to 0} (I+O(\varepsilon))=\lim_{\varepsilon\to 0} (-A(B,\varepsilon)+\varepsilon\Psi(B,Z,\varepsilon)),
}

which shows the claim. Note that the slow divergence integral in the coordinates $(a,b,z)$ reads as
\eq{
  I(b)=\tilde I(b,\zeta^+)-\tilde I(b,\zeta^-),
}

where straightforward computations show that
\eq{\tilde I(b,\zeta)=\frac{9}{5}\zeta^5+2\zeta^3b+b^2\zeta,
}

and where $\zeta^\pm$ is a constant defined by $(a^{\pm},b,\zeta^{\pm})\in\Sigma^{\pm}\cap S$.

On the other hand, in normal coordinates and along regular parts of the slow manifold, the $A_3$-SFS can be written as (see \cref{sec:nf_reg})
\eq{
X(A,B,Z,\varepsilon)=\varepsilon\parc{A}+0\parc{B}-Z\parc{Z}+0\parc{\varepsilon}.
}%

In these coordinates the slow divergence integral reads as
\eq{
  I=A^+-A^-,
}%
where $A^+$ and $A^-$ are the corresponding parametrizations of $\Sigma^+$ and $\Sigma^-$ (respectively) in the coordinates $(A,B,Z,\varepsilon)$.

\end{proof}

}

\subsection{Blow-up and charts}\label{sec:blowup}

Let us briefly recall the blow up technique, for more details see e.g. \cite{DumRou1,DumRou2,Krupa1}. The vector field $X$ \cref{eq:cusp1} is quasihomogeneous \cite{Arnold_singularities,Jardon2}. Therefore, it is convenient to use the \emph{quasihomogeneous blow up}. This technique consists on performing a coordinate transformation defined by
\eq{\label{blowup}
\a =r^{3}\bar \a , \, \b  = r^{2}\bar \b, \,   z=r\bar z, \, \varepsilon=r^{5}\bar\varepsilon,
}

which is called the blow up map, and where $\bar \a^2+ \bar \b^2+\bar z^2+\bar\varepsilon^2=1$ and $r\in[0,+\infty)$.   That is $(\bar a, \bar b, \bar z,\bar\varepsilon,r)\in S^{3}\times\mathbb{R}^+$. Since $\varepsilon\geq 0$, we can restrict the coordinates to $\bar\varepsilon\geq 0$. Note that $S^{3}\times\left\{ 0 \right\}$ is mapped, via the blow up map \cref{blowup}, to the origin of $\mathbb{R}^{4}$. The powers or weights of the blow up map \cref{blowup} are obtained from the type of quasihomogeneity of $X$.

Let us denote by $\Phi(\bar \a,\bar\b, \bar z,\bar \varepsilon)$ the blow up map \cref{blowup}. This map induces a smooth vector field $\tilde X$ on $S^{3}\times\mathbb{R}^+$  defined by $\Phi_*\tilde X=X$. It is often the case in which the vector field $\tilde X$ is degenerate along $S^{3}\times\left\{ 0 \right\}$. Then one defines another vector field $\bar X$ by $\bar X=\frac{1}{r^m}\tilde X$ for a well chosen positive integer $m$ so that $\bar X$ is non-degenerate along $S^{3}\times\left\{ 0 \right\}$. Since $r\in\mathbb{R}^+$, the phase portraits of $\tilde X$ and $\bar X$ are equivalent outside $S^{3}\times\left\{ 0 \right\}$, and therefore it is equally useful to study $\bar X$ instead of $\tilde X$. One obtains a complete description of the local flow of $X$ near the the cusp point by studying the flow of $\bar X$ for $(\bar \a,\bar\b, \bar z,\bar\varepsilon,r)\in S^{3}\times [0,r_0)$ with $r_0>0$ sufficiently small.

For problems of dimension greater than $2$, performing computations in spherical coordinates becomes tedious. Therefore, it is more convenient to consider charts which parametrize hemispheres of the ball $S^{3}\times[0,r_0)$. In the present context, the useful charts are
\eq{\label{charts}
K_{\en}=\left\{ \bar \a=-1 \right\}, \; K_{\ex}=\left\{ \bar\a=1 \right\}, \; K_{\bar\varepsilon}=\left\{ \bar\varepsilon=1 \right\},\; K_{\pm}=\left\{ \bar\b=\pm 1 \right\}
}

and we always keep $r\in[0,r_0)$. The previous setting is also known as directional blow up. A qualitative picture of the charts is given in \cref{fig:charts}.

Briefly speaking, our analysis goes as follows: first, we perform a local analysis on each chart given in \cref{charts}. Next, we compose (``glue'') the local results to provide a full description of the flow of $X$ \cref{eq:cusp1} in a small neighborhood of the cusp point. In this way, we construct an invariant manifold from $\Sigma^{\en}$ to $\Sigma^{\ex}$. Later we ``push away'' this invariant manifold all the way up to the sections $\Sigma^-$ and $\Sigma^+$ along regular parts of the slow manifold $S$.

\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics[]{blowup.pdf}}
\node at (2.5,1.8) {$S^3\times [0,r_0)$};
\node at (1.5,2.5) {$S^3$};
\node at (-.2,2.5) {$\bar z$};
\node at (-2,-0.8) {$\bar \varepsilon$};
\node at (1.8,-0.5) {$\bar a$};
\node at (-2,-2.6) {$K_{\bar \varepsilon}$};
\node at (1.8,-2.3) {$K_{\bar a}$};
\end{tikzpicture}
\caption{The blow up space and the charts. Each chart $K_\ell$ parametrizes a region of the ball $S^3\times [0,r_0)$. A local analysis in the charts provides a full picture of the dynamics of the vector field $\bar X$.}
\label{fig:charts}
\end{figure}

 To avoid confusion of the coordinates we adopt the following notation. Any object $O$ defined in the chart $K_{\en}$ is denoted by $O_1$. Similarly any object defined in the chart $K_{\ex}$ is denoted by $O_3$. Finally, an object $O$ defined in either of the charts $K_{\bar\varepsilon}$ or $K_{\pm}$ is denoted by $O_{2}$.

\subsection{Analysis in the chart $K_{\en}$}\label{Ken}
  Taking into account our notation convention, the blow-up map in this chart is given by
\eq{\label{blowupKen}
a=-r_1^3, \; b=r_1^2b_1, \; z=r_1^3z_1, \; \varepsilon=r_1^5\varepsilon_1.
}
\renewcommand{\r}{r_1}
\newcommand{a_1}{a_1}
\newcommand{b_1}{b_1}
\renewcommand{z}{z_1}
\newcommand{\ve_1}{\varepsilon_1}
\renewcommand{\mathcal{W}^{^C}}{\mathcal{W}_1^{^C}}

The corresponding vector field in this chart (after multiplication by $3$) has the form
\eq{X_{\en}: \begin{cases}
\r' &= - \ve_1\r\left( 1+ \tilde f_1\right)\\
b_1' &= 2 \eb_1\left( 1+ \tilde f_1\right)+\r^6\ve_1^2\tilde f_2\\
z' &= -3\left( z^3+\yz-1-\frac{1}{3}\ez \right)+\r^2\ve_1\tilde f_3\\
\ve_1' &= 5\ve_1^2\left( 1+ \tilde f_1\right)
\end{cases}
}
where the functions $\tilde f_i=f_i(\r,b_1,z,\ve_1)$ are flat along $\r=0$, recall that $S^3\times\left\{ r=0 \right\}\mapsto 0\in\mathbb{R}^4$ via the blow up map. We study a transition $\Pi_1:\Delta_{1}^{\en}\to\Delta_1^{\ex}$ where
\eq{
\Delta_{1}^{\en} &= \left\{ (\r,b_1,z,\ve_1)\in\mathbb{R}^4 \, | \, \r=r_0, \ve_1<\delta, \, z>0 \right\}\\
\Delta_1^{\ex} &= \left\{ (\r,b_1,z,\ve_1)\in\mathbb{R}^4 \, | \, \ve_1=\delta, \r<r_0 \right\},
}
where $r_0$ and $\delta$ are sufficiently small positive constants.

\begin{remark}
  The section $\Delta_1^{\en}$ corresponds to $\Sigma^{\en}$ in the blow-up space, that is $\Sigma^{\en}=\Phi(\Delta_1^{\en})$, where $\Phi$ is the blow-up map \cref{blowupKen}. This implies that trajectories of $X$ crossing $\Sigma^{\en}$ correspond to trajectories of $X_{\en}$ crossing $\Delta_1^{\en}$.
\end{remark}

Before going any further, let us provide a qualitative description of $X_{\en}$ as in \cite{BKK}. This process can be repeated, following similar arguments, in all the local charts; however, for brevity we only detail it for the current one.

\paragraph{Qualitative description of the flow of $X_{\en}$}{

The subspaces $\left\{ \r=0 \right\}$, $\left\{ \ve_1=0 \right\}$ and $\left\{ \r=0 \right\}\cap\left\{ \ve_1=0 \right\}$ are invariant. Therefore, it is useful to study the flow of $X_{\en}$ restricted to the aforementioned subspaces.

\begin{description}[leftmargin=0cm]
\item[Restriction to $\left\{ \r=0 \right\}\cap\left\{ \ve_1=0 \right\}$.] In this space $X_{\en}$ is reduced to
\eq{\label{eq:K_en1}
b_1' &   = 0\\
z' &   = -3\left( z^3+\yz-1 \right).\\
}

The set
\eq{\label{gamma1}
\gamma_1 = \left\{ (b_1,z)\, | \,  z^3+\yz-1=0  \right\}
}

is a curve of equilibrium points. The phase portrait of \eqref{eq:K_en1} is shown in figure \ref{fig:Ken1}.

\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics[scale=1]{cusp2_2.pdf}}
\node at (2.2,0.2) {$b_1$};
\node at (0.1,2.3) {$z_1$};
\end{tikzpicture}
\caption{The phase portrait of $X_{\en}$ restricted to the invariant space $\left\{ \r=0 \right\}\cap\left\{ \ve_1=0 \right\}$. The shown curve is $\gamma_1$ and it comprises a set of equilibrium points. Note that locally, all trajectories with initial condition $z_1(0)>0$ are attracted to $\gamma_1|_{\left\{z>0\right\}}$.}
\label{fig:Ken1}
\end{figure}

\begin{remark} All the trajectories of \cref{eq:K_en1} restricted to an initial condition $z_0>0$ are attracted to the curve $\gamma_1|_{z_1>0}$. Furthermore, due to our definition of $\Delta_1^{\en}$, we are interested \emph{only} in trajectories satisfying this initial condition. Thus, from now on, we restrict our analysis to the subspace $\left\{ z>0 \right\}$.

\end{remark}

\item[Restriction to $\left\{ \ve_1=0 \right\}$.] In this space $X_{\en}$ is reduced to
\eq{\label{eq:K_en2}
\r' &   = 0\\
b_1' &   = 0\\
z' &   = -3\left( z^3+\yz-1\right).\\
}

The set $\Gamma_1 = \left\{ (\r,b_1,z)\, | \,  z^3+\yz-1=0  \right\}$ is a surface of equilibrium points given by $\Gamma_1=(r_1,\gamma_1)$. Since $\r'=0$, the phase space of \cref{eq:K_en2} is foliated by two dimensional leaves in which the flow looks like \cref{fig:Ken1}.

\item[Restriction to $\left\{ \r=0 \right\}$.] In this space $X_{\en}$ is reduced to
\eq{\label{eq:K_en3}
b_1' &   = 2\eb_1  \\
z' &   = -3\left( z^3+\yz-1-\frac{1}{3}\ez \right) \\
\ve_1' & = 5\ve_1^2,
}

Once again, the set $\gamma_1 = \left\{ (b_1,z,\ve_1)\, | \, \ve_1=0, \,  z>0,\,  z^3+\yz-1=0  \right\}$ is a curve of equilibrium points. The Jacobian of \cref{eq:K_en3} evaluated along $\gamma_1$ shows that, for small enough $\ve_1$, there exists an invariant center manifold that passes through $\gamma_1$. Furthermore, the non-zero eigenvalue corresponding to the $z$-direction is negative along $\gamma_1$. The phase portrait of \eqref{eq:K_en3} is shown in figure \ref{fig:Ken3}.\\

\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics[scale=1]{cusp3_2.pdf}}
\node at (2.5,0) {$b_1$};
\node at (0.2,2.1) {$z_1$};
\node at (-1.5,-1.6) {$\varepsilon_1$};
\end{tikzpicture}

\caption{Phase portrait of \eqref{eq:K_en3} restricted to $z>0$. The shown surface is an invariant center manifold, which is attracting in the $z_1$-direction. }
\label{fig:Ken3}
\end{figure}

Observe that the $b_1$ and the $\ve_1$ directions are expanding. It is important to know the relation between such two expanding variables. We have
\eq{
\frac{db_1}{d\ve_1}=\frac{2}{5}\frac{b_1}{\ve_1},
}

which has the solution
\eq{
b_1=b_1^*\left( \frac{\ve_1}{\ve_1^*} \right)^{2/5},
}

where $b_1^*\leqb_1$ and $\ve_1^*\leq\ve_1$ are the initial conditions, that is $(b_1^*,\ve_1^*)=(b_1,\ve_1)|_{\Delta_1^{\en}}$. It is important to look at the ratio of initial conditions $\tfrac{b_1^*}{\left(\ve_1^*\right)^{2/5}}$. This ratio tells us that $b_1$ is bounded as $\ve_1\to 0$ (and therefore as $\ve_1^*\to 0$) if and only if $b_1^*\in O\left( \left(\ve_1^*\right)^{2/5} \right)$. In other words, if the initial condition $b_1^*$ is not of order $O((\ve_1^*)^{2/5})$ then the value of $b_1$ at $\Delta_1^{\ex}$ blows up as $\ve_1^*\to 0$. This leads us to partition the section $\Delta_1^{\en}$ into three open regions as follows.

\eq{
\Delta_1^{\en,\inn} &= \Delta_1^{\en}|_{|b_1|<M\ve_1^{2/5}}\\
\Delta_1^{\en,b_1} &= \Delta_1^{\en}|_{b_1>K\ve_1^{2/5}}\\
\Delta_1^{\en,-b_1} &= \Delta_1^{\en}|_{-b_1>K\ve_1^{2/5}},
}

where $0<K<M<\infty$. Observe that the open sets $\Delta_1^{\en,\inn}$, $\Delta_1^{\en,b_1}$ and $\Delta_1^{\en,-b_1}$ form an open cover of $\Delta_1^{\en}$. Accordingly, these sets induce an open cover of the entry section $\Sigma^{\en}$ via the blow up map \cref{blowupKen}. See \cref{fig:region_partition} for a representation of the aforementioned partition.\bigskip

\begin{figure}[htbp]\centering
\begin{tikzpicture}\node at (-4,0){
\pgftext{\includegraphics[scale=1]{region_partition.pdf}}};
\node at (-4,1) {$\Delta_1^{\en,\inn}$};
\node at (-5,-.5) {$\Delta_1^{\en,-b_1}$};
\node at (-3,-.5) {$\Delta_1^{\en,b_1}$};
\node at (0,-1.5) {$ $};
\end{tikzpicture}

\caption{Partition of $\Delta_1^{\en}$. Trajectories crossing through $\Delta_1^{\en,\ve_1}$ corresponding to the inner wedge area, have a continuation on the chart $K_{\bar\varepsilon}$. On the other hand, outside $\Delta_1^{\en,\ve_1}$ we must consider the lateral regions $\Delta_1^{\en,b_1}$ and $\Delta_1^{\en,-b_1}$.  }
\label{fig:region_partition}
\end{figure}

Based on the partition of the entry section $\Delta_1^{\en}$, we define three transitions as follows
\eq{\label{trKen}
\Pi_1^{\inn}&:\Delta_1^{\en,\inn} \to \Delta_1^{\ex}\\
\Pi_1^{+b_1}&:\Delta_1^{\en,+b_1} \to \Delta_1^{\ex,+b_1}\\
\Pi_1^{-b_1}&:\Delta_1^{\en,-b_1} \to \Delta_1^{\ex,-b_1},
}

where

\eq{
\Delta_1^{\ex} &= \left\{ (\r,b_1,z,\ve_1)\in\mathbb{R}^4 \, | \, \ve_1=\delta, \r<r_0 \right\},\\
\Delta_1^{\ex,\pmb_1} &=\left\{ (\r,b_1,z,\ve_1)\in\mathbb{R}^4 \, | \, b_1=\pm\eta, \r<r_0 \right\}.
}

To finish with the qualitative description, note that there exists a (non-unique) $3$-dimensional center manifold $\mathcal{W}^{^C}$, which is shown to exist by evaluating the Jacobian of $X_{\en}$ all along the surface

\eq{
\Gamma_1=\left\{ (\r,b_1,z,\ve_1)\, | \, \ve_1=0, \, z>0 \,z^3+\yz-1=0  \right\}.
}

Moreover, by the analysis provided above, the center manifold $\mathcal{W}^{^C}|_{z>0}$ is attracting for $\ve_1$ small enough. Note that $\mathcal{W}^{^C}|_{\ve_1=0}=\Gamma_1$. This means that $\mathcal{W}^{^C}$ can be interpreted as a perturbation of the slow manifold $S$, written in the coordinates of the current chart. See \cref{fig:Ken2} for a representation of the previous exposition.

\begin{figure}[htbp]\centering
\begin{tikzpicture}
  \pgftext{\includegraphics{foldKen2.pdf}}
  \node at (3,-.5) {$b_1$};
  \node at (-2,-2.8) {$\r$};
  \node at (0.25,2.8) {$\ve_1$};

  \node at (-1,-2.4) {$\Delta_1^{\en}$};
  \node at (3.25,1) {$\Delta_1^{\ex,+b_1}$};
  \node at (-3.,1) {$\Delta_1^{\ex,-b_1}$};
  \node at (1,2.15) {$\Delta_1^{\ex,\ve_1}$};
\end{tikzpicture}
\caption{Phase portrait of the trajectories of $X_{\en}$ depending on their initial condition. If the trajectories satisfy the estimate $y\in O(\varepsilon^{2/5})$, then they arrive to $\Delta_1^{\ex,\varepsilon_1}$ in finite time. If the estimate $y\in O(\varepsilon^{2/5})$ is not satisfied, then we must choose one of the outgoing sections $\Delta_1^{\ex,\pm b}$ in order to have a well defined transition map. }
\label{fig:Ken2}
\end{figure}
\end{description}

Let us recall that the vector field $X_{\en}$ is of the form

\eq{\label{eq:Ken_original} X_{\en}: \begin{cases}
\r' &= - \ve_1\r\left( 1+ \tilde f_1\right)\\
b_1' &= 2 \eb_1\left( 1+ \tilde f_1\right)+\r^6\ve_1^2\tilde f_2\\
z' &= -3\left( z^3+\yz-1-\frac{1}{3}\ez \right)+\r^2\ve_1\tilde f_3\\
\ve_1' &= 5\ve_1^2\left( 1+ \tilde f_1\right)
\end{cases}
}

We now proceed to describe the transitions $\Pi_1$ given by \cref{trKen}. For this, first we write \cref{eq:Ken_original} in a suitable normal form. Next, based on this normal form, we compute the corresponding transition. \bigskip

First of all, let us move the origin to the point $(\r,\b,z,\ve_1)=(0,0,1,0)$. This is done by defining a new variable $\zeta_1$ by $\zeta_1=z_1-1$. With this variable we have a new local vector field $Y_{\en}$ which is defined by

\eq{\label{eq:Ken_normal0}
  Y_{\en}:\begin{cases}
    \r' &= - \ve_1\r\left( 1+ \tilde f_1\right)\\
    b_1' &= 2 \eb_1\left( 1+ \tilde f_1\right)+\r^6\ve_1^2\tilde f_2\\
    \ve_1' &= 5\ve_1^2\left( 1+ \tilde f_1\right)\\
    \zeta_1' &= -3 G(b_1,\ve_1,\zeta_1)+\ve_1\tilde h,
  \end{cases}
}
where $G(0,0,0)=0$ and $\tparcs{G}{\zeta_1}(0,0,0)=3$. Now, we want to write $Y_{\en}$ in a suitable normal form. From \cref{prop:nf_semihyp}, we know that $Y_{\en}$ is $\mathcal{C}^{\ell}$ equivalent to

\renewcommand{b_1}{B_1}

\eq{\label{eq:Ken_normal1}X_{\en}^N: \begin{cases}
\r' &= - \ve_1\r\\
b_1' &= 2 \eb_1 \\
\ve_1' &= 5\ve_1^2\\
Z_1' &= -9(1+H_1(\r,b_1,\ve_1))Z_1,
\end{cases}
}

where $H_1$ is a $C^\ell$-function vanishing at the origin. This normal form $X_{\en}^{N}$ is convenient since the chosen center manifold $\mathcal{W}^{^C}$ is now simply given by $\mathcal{W}^{^C}=\left\{  Z_1=0 \right\}$. Furthermore, from the format of $X_{\en}^{N}$, it is evident the ``hyperbolic nature'' of the flow restricted to the center manifold: the restriction of $X_{\en}^N$ to the center manifold $\mathcal{W}^{^C}$ has a simple structure, namely

\eq{
X_{\en}^N|_{\mathcal{W}^{^C}}: \begin{cases}
\r' &= - \ve_1\r\\
b_1' &= 2 \eb_1 \\
\ve_1' &= 5\ve_1^2.
\end{cases}
}

Note that for $\ve_1\neq 0$, the vector field $\tfrac{1}{\ve_1}X_{\en}^N|_{\mathcal{W}^{^C}}$ is hyperbolic.\bigskip

\newcommand{\tilde{r}_1}{\tilde{r}_1}
\newcommand{\tilde{B}_1}{\tilde{B}_1}
\newcommand{\tilde{\ve}_1}{\tilde{\varepsilon}_1}
\newcommand{\tilde{Z}_1}{\tilde{Z}_1}

The vector field $X_{\en}^N$ is of the form studied in \cref{prop:tr_semihyp}, therefore we have that the transition
\eq{
\Pi_1^{\inn}:(b_1,\ve_1,z)\mapsto(\tilde{r}_1,\tilde{B}_1,\tilde{Z}_1)
}

is of the form
\eq{
\tilde{r}_1 &=r_0\left( \frac{\ve_1}{\delta} \right)^{1/5}\\
\tilde{B}_1 &=b_1\left( \frac{\delta}{\ve_1} \right)^{2/5}\\
\tilde{Z}_1&=Z_1\exp\left( -\frac{9}{5\ve_1}(1+\alpha_1\ve_1\ln\ve_1+\ve_1 G_1) \right),
}

where $\alpha_1=\alpha_1(r_0|b_1|^{1/2},r_0\ve_1^{1/5})$ and $ G_1= G_1(r_0|b_1|^{1/2}, r_0\ve_1^{1/5},\mu)$ where $\mu=b_1\ve_1^{-2/5}$. Recall that for this transition we have the condition $b_1\in O(\ve_1^{2/5})$ so $\mu$ is well defined.\bigskip

On the other hand, the transition
\eq{
\Pi_1^{\pmb_1}:(b_1,\ve_1,Z_1)\mapsto(\tilde{r}_1,\tilde{\ve}_1,\tilde Z_1)
}

is (see \cref{prop:tr_semihyp}) of the form
\eq{
\tilde{r}_1 &=r_0\left( \frac{b_1}{\eta} \right)^{1/2}\\
\tilde{\ve}_1 &=\ve_1\left( \frac{\eta}{b_1} \right)^{5/2}\\
\tilde{Z}_1&=Z_1\exp\left( -\frac{9}{5\ve_1}(1+\beta_1\ve_1\ln(|b_1|)+\ve_1 H_1) \right),
}

where $\beta_1=\beta_1(r_0|b_1|^{1/2},r_0\ve_1^{1/5})$ and $H_1=H_1(r_0|b_1|^{1/2},r_0\ve_1^{1/5},\sigma)$, where $\sigma=\ve_1|b_1|^{-5/2}$. Note that since $b_1\notin O(\ve_1^{2/5})$, $\sigma$ is well defined. We observe that the transitions $\Pi_1^{\ve_1}$ and $\Pi_1^{\pmb_1}$ are exponential type maps.

\subsection{Analysis in the chart $K_{\bar\varepsilon}$}\label{Ke}
Taking into account our notation convention, the blow-up map in this chart is given by

\eq{
a=r_2^3a_2, \; b=r_2^2b_2, \; z=r_2^3z_2, \; \varepsilon=r_2^5.
}
\newcommand{\bar\ve}{\bar\varepsilon}
\renewcommand{\r}{r_2}
\renewcommand{a_1}{a_2}
\renewcommand{b_1}{b_2}
\renewcommand{z}{z_2}
\renewcommand{\ve_1}{\varepsilon_2}

Then, the blown up vector field reads as

\eq{ \label{eq:X2}X_{\bar\ve}:\begin{cases}
\r' &=0\\
a_1' &=1+\tilde g_1\\
b_1' &=r^6\tilde g_2\\
z' &=-\left(z^3+\yz+a_1\right)+\tilde g_3,
\end{cases}
}

where the function $\tilde g_i=\tilde g_i(\r,a_1,b_1,z)$ are flat along $\r=0$. Note that in this chart $\r$ acts as a parameter and that the flow is regular. Furthermore, note that $X_{\bar\ve}$ is not a slow-fast system, but a regular vector field. From the equation $a_1'=1+\tilde g_1$, we define the following ``entry'' and ``exit'' sections.
\eq{
\Delta_2^{\en,\bar\ve} &=\left\{ (\r,a_1,b_1,z)\, | \, a_1=-A_0, \, z\geq 0 \right\},\\
\Delta_2^{\ex,\bar\ve} &=\left\{ (\r,a_1,b_1,z)\, | \, a_1=A_0, \, z\leq 0 \right\}.\\
}
\renewcommand{\tilde{r}_1}{\tilde{r}_2}
\newcommand{\tilde{a}_2}{\tilde{a}_2}
\renewcommand{\tilde{B}_1}{\tilde{b}_2}
\renewcommand{\tilde{Z}_1}{\tilde{z}_2}
\renewcommand{\tilde{\ve}_1}{\tilde{\varepsilon}_2}

Therefore, we define a transition $\Pi_2^{\bar\ve}$ as
\eq{
\Pi_2^{\bar\ve} : &\Delta_2^{\en,\bar\ve}\to\Delta_2^{\en,\bar\ve}\\
&(\r,b_1,z)\mapsto (\tilde{r}_1,\tilde{B}_1,\tilde{Z}_1).
}

Since \cref{eq:X2} is regular, by the flow box theorem all trajectories starting at $\Delta_2^{\en,\bar\ve}$ arrive at $\Delta_2^{\ex,\bar\ve}$ in finite time. Moreover, the transition $\Pi_2^{\bar\ve}$ is a diffeomorphism and then, from \cref{eq:X2} we have that $\Pi_2^{\bar\ve}$ reads as
\eq{
  \Pi_2{\bar\ve}(\r,b_1,z) &= (\tilde{r}_1,\tilde{B}_1,\tilde{Z}_1) \\
  &=(\r,b_1+h_{b_1},\phi_1(\r,b_1)+ \phi_2(\r,b_1)(1+\phi_3(\r,b_1,z))z),
}

where the $\phi_i$'s are smooth functions. Observe that in this chart, the transition is not an exponential type map.

\subsection{Analysis in the chart $K_{\ex}$}\label{Kex}

Taking into account our notation convention, the blow-up map in this chart is given by

\eq{
a=r_3^3, \; b=r_3^2b_3, \; z=r_3^3z_3, \; \varepsilon=r_3^5\varepsilon_3.
}
  \renewcommand{\r}{r_3}
  \renewcommand{a_1}{a_3}
  \renewcommand{b_1}{b_3}
  \renewcommand{z}{z_3}
  \renewcommand{\ve_1}{\varepsilon_3}
  \renewcommand{\mathcal{W}^{^C}}{\mathcal{W}_3^{^C}}

  \renewcommand{\tilde{r}_1}{\tilde{r}_3}
  \renewcommand{\tilde{a}_2}{\tilde{a}_3}
  \renewcommand{\tilde{B}_1}{\tilde{b}_3}
  \renewcommand{\tilde{Z}_1}{\tilde{z}_3}
  \renewcommand{\tilde{\ve}_1}{\tilde{\varepsilon}_3}

  \renewcommand{\bar\ve}{\bar\varepsilon}
Then, the blown up vector field reads as

\eq{X_{\ex}: \begin{cases}
  \r' &=  \ve_1\r\left( 1+ \tilde f_1\right)\\
  b_1' &= -2 \eb_1\left( 1+ \tilde f_1\right)+\r^6\ve_1^2\tilde f_2\\
  z' &= -3\left( z^3+\yz+1+\frac{1}{3}\ez \right)+\r^2\ve_1\tilde f_3\\
  \ve_1' &= -5\ve_1^2\left( 1+ \tilde f_1\right)
\end{cases}
}

where the function $\tilde f_i=\tilde f_i(\r,b_1,\ve_1,z)$ are flat  along $\r=0$. Observe that the vector field $X_{\ex}$ resembles the vector field $X_{\en}$. Therefore, we have a similar behavior of the trajectories, the main difference is that in the case of $X_{\ex}$, there is one expanding ($\r$) and three contracting ($b_1$, $\ve_1$ and $z$) directions. The flow of $X_{\ex}$ is obtained following similar arguments as for the flow of $X_{\en}$.\bigskip

From the fact that $X_{\ex}$ has three contracting and one expanding direction, we define the entry sections

\eq{
\Delta_{3}^{\en,\bar\ve} &=\left\{ (\r,b_1,\ve_1,z)\, : \, \ve_1=\delta,\, z<0, \, \r<r_0\right\}\\
\Delta_{3}^{\en,+b_1} &=\left\{(\r,b_1,\ve_1,z)\, : \,b_1=\eta ,\, z<0, \, \r<r_0\right\}\\
\Delta_{3}^{\en,-b_1} &=\left\{(\r,b_1,\ve_1,z)\, : \,b_1=-\eta,\, z<0, \, \r<r_0 \right\},
}

where all the constants are positive and sufficiently small, and the exit section

\eq{
\Delta_{3}^{\ex} &=\left\{ (\r,b_1,\ve_1,z)\, : \, \r=r_0, \, z<0,\, \ve_1<\delta, \, |b_1|<\eta \right\}.
}

Then, accordingly, we define three transition maps as follows

\eq{\label{Kextrs}
\Pi_3^{\ve_1} &:\Delta_{3}^{\en,\bar\ve} \to \Delta_{3}^{\ex}\\
&:(\r,b_1,z)\mapsto(\tilde{B}_1,\tilde{\ve}_1,\tilde{Z}_1)\\[2ex]
\Pi_3^{+b_1} &:\Delta_{3}^{\en,+b_1}\to \Delta_{3}^{\ex}\\
&:(\r,\ve_1,z)\mapsto(\tilde{B}_1,\tilde{\ve}_1,\tilde{Z}_1)\\[2ex]
\Pi_3^{-b_1} &:\Delta_{3}^{\en,-b_1}\to \Delta_{3}^{\ex}\\
&:(\r,\ve_1,z)\mapsto(\tilde{B}_1,\tilde{\ve}_1,\tilde{Z}_1).
}

\renewcommand{b_1}{B_3}
\renewcommand{\tilde{B}_1}{\tilde{B}_3}
Now we proceed to write $X_{\ex}$ in a normal form just as we did with $X_{\en}$ in \cref{Ken}. Following \cref{prop:nf_semihyp} we have that $X_{\ex}$ is $\mathcal{C}^{\ell}$ equivalent to

\eq{X_{\ex}^N:\begin{cases}
  \r' &=  \ve_1\r\\
  b_1' &= -2 \eb_1\\
  \ve_1' &= -5\ve_1^2\\
  Z_3' &= -9(1+H_3)Z_3,
\end{cases}
}

where $H_3=H_3(\r,b_1,\ve_1)$ is a $C^\ell$ function vanishing at the origin. Just as in the chart $K_{\en}$, there exists a three dimensional center manifold $\mathcal{W}^{^C}$ associated to $X_{\ex}^N$ and which has been chosen such that $\mathcal{W}^{^C}=\left\{ Z_3=0 \right\}$. Since $\r$ is the only expanding direction, we take as transition time $T_3=\ln\left( \tfrac{r_0}{\r} \right)$. This transition time is computed from the dynamics restricted to $\mathcal{W}^{^C}$, that is, from the equation $\r'=\r$. In contrast to what happened in the chart $K_{\en}$, the time $T_3$ is well defined for all the three transitions $\Pi_3^{\ve_1}$, $\Pi_3^{+b_1}$ and $\Pi_3^{-b_1}$. Following \cref{prop:tr_semihyp} we have

\eq{
  \tilde{B}_1 &= b_1 \left( \frac{\r}{r_0} \right)^{2}\\
  \tilde{\ve}_1 &=\ve_1 \left( \frac{\r}{r_0} \right)^{5}\\
  \tilde Z_3 &= Z_3\exp\left( -\frac{9}{5\ve_1}\left( \left( \frac{r_0}{\r} \right)^{5}-1+\alpha_3\ve_1\ln\r+\ve_1 H_3 \right) \right),
}
where $\alpha_3=\alpha_3(\r|b_1|^{1/2}, \r\ve_1^{1/5})$ and $H_3=H_3(\r|b_1|^{1/2}, \r\ve_1^{1/5},\r)$. Therefore, by taking the definitions of the entry sections we have

\eq{
  \Pi_3^{\varepsilon_3}(r_3,B_3,Z_3) &= \left( b_1 \left( \frac{\r}{r_0} \right)^{2}, \,  \delta \left( \frac{\r}{r_0} \right)^{5},\, Z_3\exp\left( -\frac{9}{5\delta}\left( \left( \frac{r_0}{\r} \right)^{5}-1+\alpha_3\delta\ln\r+\delta H_3 \right) \right) \right)\\
  \Pi_3^{\pm b_3}(r_3,\varepsilon_3,Z_3) &= \left( \pm\eta \left( \frac{\r}{r_0} \right)^{2}, \,  \varepsilon_3 \left( \frac{\r}{r_0} \right)^{5},\, Z_3\exp\left( -\frac{9}{5\varepsilon_3}\left( \left( \frac{r_0}{\r} \right)^{5}-1+\alpha_3\varepsilon_3\ln\r+\varepsilon_3 H_3 \right) \right) \right).
}

Observe that these transitions are of exponential type.

\subsection{Analysis in the charts $K_{\pm\bar b}$}\label{Kpm}
In this section we study the local flow at the charts $K_{+\bar b}$ and $K_{-\bar b}$. In a qualitative sense, these charts come into play when the initial condition $b_0=b|_{\Sigma^{\en}}$ does not satisfy the estimate $b_0\in O(\varepsilon^{2/5})$. This implies that the corresponding trajectory passes away from the cusp point. The chart $K_{+\bar b}$ ``sees'' trajectories with initial condition $b|_{\Sigma^{\en}}>0$ while $K_{-\bar b}$ ``sees'' trajectories with initial condition $b|_{\Sigma^{\en}}<0$.

\subsection*{Analysis in the chart $K_{+\bar{b}}$}
In this chart the blow-up maps reads
\eq{
  a=r_2^3a_2, \, b=r_2^2, \, z=r_2z_2, \, \varepsilon = r_2^5\varepsilon_2.
}

\renewcommand{\r}{r_2}
  \renewcommand{a_1}{a_2}
  \renewcommand{b_1}{b_2}
  \renewcommand{z}{z_2}
  \renewcommand{\ve_1}{\varepsilon_2}
  \renewcommand{\mathcal{W}^{^C}}{\mathcal{W}_2^{^C}}

\renewcommand{\tilde{r}_1}{\tilde{r}_2}
  \renewcommand{\tilde{a}_2}{\tilde{a}_2}
  \renewcommand{\tilde{B}_1}{\tilde{b}_2}
  \renewcommand{\tilde{Z}_1}{\tilde{z}_2}
  \renewcommand{\tilde{\ve}_1}{\tilde{\varepsilon}_2}

Then we have that the blow-up vector field is given by
\eq{
  X_{+\bar{b}}:\begin{cases}
    \r' &= \ve_1\bar f_r\\
    a_1' &= \ve_1(1+\bar f_{a_1})+\ve_1\bar g_{a_1}\\
    \ve_1' &= -\ve_1\bar f_{\ve_1}\\
    z' &= -(z^3+z+a_1)+\ve_1\bar f_{z}
  \end{cases}
}

where all the functions $\bar f_\ell$ are flat along $\left\{ \r=0 \right\}$. Observe that the set

\eq{\label{gamma2}
  \Gamma_2=\left\{ (\r,a_1,\ve_1,z) \, |\, \ve_1=0,\, z^3+z+a_1=0 \right\}
}

is a NHIM of $X_{+\bar{b}}$. However, $X_{+\bar{b}}$ is not exactly a slow-fast system since $\ve_1'\neq 0$, but the restriction of $X_{+\bar{b}}$ to $\left\{\r=0\right\}$ is indeed a slow-fast system. This restriction reads as
\eq{\label{Kb1}
  X_{+\bar{b}}|_{\left\{ \r=0 \right\}}:\begin{cases}
    a_1' &= \ve_1\\
    \ve_1' &= 0\\
    z' &= -(z^3+z+a_1).
  \end{cases}
}

\begin{remark} The subspace $\left\{\r=0\right\}$ is invariant. Moreover, since $X_{+\bar{b}}$ is a flat perturbation of $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$, it is equally useful to study the restriction $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$. After all, by regular perturbation theory, their flows are equivalent.
\end{remark}

The slow manifold  of $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$ is defined by $\Gamma_2|_{\r=0}$ and is normally hyperbolic. Let us define the sections
\eq{\label{Kbsections}
  \Delta_2^{\en,+b_1} &= \left\{ (\r,a_1,\ve_1,z)\in\mathbb{R}^4\, | \, a_1=-A_0 \right\}\\
  \Delta_2^{\ex,+b_1} &= \left\{ (\r,a_1,\ve_1,z)\in\mathbb{R}^4\, | \, a_1=A_0 \right\}.
}

Accordingly, we study the transition
\eq{
  \Pi_2^{+b_1}: & \Delta_2^{\en,+b_1}\to\Delta_2^{\ex,+b_1}\\
  &(\r,\ve_1,z)\mapsto (\tilde{r}_1,\tilde{\ve}_1,\tilde{Z}_1).
}

For a qualitative description of $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$ and the objects defined above see \cref{fig:Ky1}.

\begin{figure}[htbp]\centering
  \begin{tikzpicture}
    \pgftext{\includegraphics[scale=1]{ky1.pdf}}
    \node at (1.75,0) {$a_1$};
    \node at (0,1.6) {$z$};

    \node at (1.75+4,0) {$a_1$};
    \node at (0+4,1.6) {$z$};

    \node at (1.75-3.9,0) {$a_1$};
    \node at (0-3.9,1.6) {$z$};

    \node at (3,1.5) {$\Delta_2^{\en,+b_1}$};
    \node at (5.5,-1.5) {$\Delta_2^{\ex,+b_1}$};
    \end{tikzpicture}
  \caption{Left: phase portrait of the corresponding layer equation of $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$. Center: phase portrait of the corresponding CDE of $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$. Right: Since the critical manifold is regular, by Fenichel theory we know that the manifold $\Gamma_2$ is perturbed to an invariant manifold $\Gamma_{2,\varepsilon_2}$ which is at distance of order $O(\varepsilon_2)$ from $\Gamma_2$. }
  \label{fig:Ky1}
\end{figure}

We know from \cref{sec:nf_reg} that for sufficiently small $\ve_1$, there exists a $C^\ell$ change of coordinates that transforms $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$ into the vector field

\eq{Y^N:\begin{cases}
  a_1' &= \ve_1\\
  \ve_1' &= 0\\
  Z_2' &= -Z_2,
\end{cases}
}

From the definition of the entry and exit sections \cref{Kbsections}, the time of integration is $T=2A_0$. To obtain the component $Z_2$ of the transition $\Pi_2^{+b_1}|_{\left\{ \r=0 \right\}}$ we need to integrate

\eq{
  Z_2' = -\frac{1}{\ve_1}Z_2,
}

and then $\tilde Z_2=Z_2(T)$. Therefore we have that after choosing a center manifold $\mathcal{W}^{^C}$, the transition $\Pi_2^{+b_1}$ reads as

\eq{
  \Pi_2^{+b_1}(0,\ve_1,Z_2) &=\left( 0,\ve_1, Z_2\exp\left( -\frac{2A_0}{\ve_1}\right)\right).
}

Note that $\Pi_2^{+b_1}$ is an exponential type map.

\subsection*{Analysis in the chart $K_{-\bar b}$}

In this chart the blow-up maps reads
\eq{
  a=r_2^3a_2, \, b=-r_2^2, \, z=r_2z_2, \, \varepsilon = r^5\varepsilon_2.
}

\renewcommand{\r}{r_2}
\renewcommand{a_1}{a_2}
\renewcommand{b_1}{b_2}
\renewcommand{z}{z_2}
\renewcommand{\ve_1}{\varepsilon_2}
\renewcommand{\mathcal{W}^{^C}}{\mathcal{W}_2^{^C}}

\renewcommand{\tilde{r}_1}{\tilde{r}_2}
\renewcommand{\tilde{a}_2}{\tilde{a}_2}
\renewcommand{\tilde{B}_1}{\tilde{b}_2}
\renewcommand{\tilde{Z}_1}{\tilde{z}_2}
\renewcommand{\tilde{\ve}_1}{\tilde{\varepsilon}_2}

Then we have that the blow-up vector field is given by
\eq{
  X_{-\bar{b}}:\begin{cases}
    \r' &= -\ve_1\bar f_r\\
    a_1' &= \ve_1(1+\bar f_{a_1})+\ve_1\bar g_{a_1}\\
    \ve_1' &= \ve_1\bar f_{\ve_1}\\
    z' &= -(z^3-z+a_1)+\ve_1\bar f_{z}
  \end{cases}
}

where all the functions $\bar f_\ell$ and $\bar g_{a_1}$ are flat along $\left\{ \r=0 \right\}$. Observe that, as in the previous section, the subspace $\left\{ \r=0 \right\}$ is invariant. The restriction of $X_{-\bar{b}}$ to this subspace reads as
\eq{
  X_{-\bar{b}}|_{\left\{ \r=0 \right\}}:\begin{cases}
    a_1' &= \ve_1\\
    \ve_1' &= 0\\
    z' &= -(z^3-z+a_1).
  \end{cases}
}

The flow of $X_{-\bar{b}}$ is a flat perturbation of the flow of $X_{-\bar{b}}|_{\left\{ \r=0 \right\}}$. Therefore, let us continue our analysis restricted to the invariant space $\left\{ \r=0 \right\}$.\bigskip

The manifold $\Gamma_2$, which is defined by
\eq{
  \Gamma_2=\left\{ (\r,a_1,\ve_1,z) \, |\, \r=0,\, \ve_1=0, \, z^3-z+a_1=0 \right\}
}

is normally hyperbolic except at the two points $p_{\pm}=\pm\left( \frac{2}{3\sqrt{3}},\frac{1}{\sqrt{3}} \right)$. Let us define the sections
\eq{
  \Delta_2^{\en,-b_1} &= \left\{ (\r,a_1,\ve_1,z)\in\mathbb{R}^4\, | \, a_1=-A_0 \right\}\\
  \Delta_2^{\ex,-b_1} &= \left\{ (\r,a_1,\ve_1,z)\in\mathbb{R}^4\, | \, a_1=A_0 \right\},
}

where $A_0>0$ is a sufficiently large constant. We are interested in the transition
\eq{
  \Pi_2^{-b_1}: & \Delta_2^{\en,-b_1}\to\Delta_2^{\ex,-b_1}\\
  &(\r,\ve_1,z)\mapsto (\tilde{r}_1,\tilde{\ve}_1,\tilde{Z}_1).
}

For a qualitative description of $X_{-\bar{b}}|_{\left\{ \r=0 \right\}}$ and the objects defined above see \cref{fig:Ky2}.

\begin{figure}[htbp]\centering
  \begin{tikzpicture}
    \pgftext{\includegraphics[scale=1]{ky2.pdf}}
    \node at (1.75,0) {$a_1$};
    \node at (0,1.6) {$z$};

    \node at (1.75+4,0) {$a_1$};
    \node at (0+4,1.6) {$z$};

    \node at (1.75-3.9,0) {$a_1$};
    \node at (0-3.9,1.6) {$z$};

    \node at (3,1.5) {$\Delta_2^{\en,+b_1}$};
    \node at (5.5,-1.5) {$\Delta_2^{\ex,+b_1}$};
    \end{tikzpicture}
  \caption{Left: phase portrait of the corresponding layer equation of $X_{-\bar{b}}|_{\left\{ \r=0 \right\}}$. Center: phase portrait of the corresponding CDE of $X_{-\bar{b}}|_{\left\{ \r=0 \right\}}$. Right: The expected perturbed invariant manifold obtained from the flow of the corresponding CDE and layer equation.}
  \label{fig:Ky2}
\end{figure}

Away from the fold points $p_\pm$, the manifold $\Gamma_2$ is regular and thus, Fenichel's theory applies. However, we need to take care of the transition near the fold point $p_+$. The local transition of a slow-fast system near a fold point is investigated in e.g. \cite{Krupa3}. However, in our current problem this transition is not essential. By this we mean that the passage through the fold point is seen as a flat perturbation of the trajectory along the stable branch of $\Gamma_2$. In a qualitative sense, this is due to the fact that the transition $\Pi_2^{-b_1}$ goes along a large NHIM, which fails to be normally hyperbolic only at one point.

\begin{proposition} We can choose appropriate coordinates $(Z_2,\ve_1)$ in $\Delta_2^{\en,-b_1}$ such that the transition $\Pi_2^{-b_1}: \Delta_2^{\en,-b_1}\to\Delta_2^{\ex,-b_1}$, restricted to $\r=0$, is an exponential type map of the form

\eq{
\Pi_2^{-b_1}(0,\varepsilon_2,Z_2)=\left( 0,\ve_1,\phi_2(\ve_1)+Z_2\exp\left( -\frac{1}{\ve_1}(A_0+\ve_1\psi_2(Z_2,\ve_1)) \right)\right),
}

where $\phi_2$ are flat at $\ve_1=0$, $\psi_2$ is $\mathcal{C}^{\ell}$-admissible, and where $A_0$ is given by the slow divergence integral of $X_{-\bar{b}}|_{\left\{ \r=0 \right\}}$.

\end{proposition}

\begin{proof}

To prove that $A_0$ is given by the slow divergence integral we proceed along the same reasoning as in \cref{prop:slowdiv}, so we do not repeat it here. In figure \cref{fig:ky3} we see the three transitions that we must consider.

\begin{figure}[htbp]\centering
  \begin{tikzpicture}
    \pgftext{\includegraphics[scale=1.5]{ky3.pdf}}
    \node at (0,2.5) {$z_2$};
    \node at (2.5,0) {$a_2$};
    \node at (-2.25,2.1) {$\Delta_2^{\en,-b_2}$};
    \node at (2.25,-2.1) {$\Delta_2^{\ex,-b_2}$};
    \node at (-.45,.35) {$\Omega^{\en}$};
    \node at (1.5,0.3) {$\Omega^{\ex}$};
    \end{tikzpicture}
  \caption{The three different transitions in which $\Pi_2^{-b_1}$ is decomposed. The central transitions is locally an $A_2$ problem. The other two transitions at the sides are regular.}
  \label{fig:ky3}
\end{figure}

The three transitions are defined as

\eq{
\Pi_2^{reg_1} &:\Delta_2^{\en,-b_2}\to \Omega^{\en}\\
\Pi_2^{fold} &: \Omega^{\en}\to \Omega^{\ex}\\
\Pi_2^{reg_2} &:\Omega^{\ex}\to \Delta_2^{\ex,-b_2},
}

where we define $\Omega^{\en}$ and $\Omega^{\en}$ as

\eq{
  \Omega^{\en} &=\left\{ (a_1,\ve_1,Z_2)\in\mathbb{R}^3 \, | \, a_1=-a_{2,\en}\right\}\\
  \Omega^{\ex} &=\left\{ (a_1,\ve_1,Z_2)\in\mathbb{R}^3 \, | \, Z_2=-Z_{2,\ex}\right\},
}

where $a_{2,\en}$ and $Z_{2,\ex}$ are sufficiently small positive constants. The total transition $\Pi_2^{+b_1}$ is given by $\Pi_2^{b_1}=\Pi_2^{reg_2}\circ\Pi_2^{fold}\circ\Pi_2^{reg_1}$. Recall from \cref{sec:Exp_trans} that if we want to write the transition  $\Pi_2^{+b_1}$ as an exponential type map, we require that $\Pi_2^{reg_1}$ is expressed as an exponential type map with no shift. The transition $\Pi_2^{fold}$  is studied in e.g. \cite{JardonThesis,Krupa3}. In \cite{JardonThesis} is proved that there are local coordinates $(\bar Z_2,\varepsilon)$ in $\Omega^{\en}$,  and $(\tilde a_2,\tilde\varepsilon)$ in $\Omega^{\ex}$, such that the transition $\Pi_2^{fold}$ is given by

\eq{
\Pi_2^{fold}(\bar Z_2,\ve_1)&=(\tilde a_2,\tilde\ve_1)\\
&=\left(\ve_1^{2/3}+O(\ve_1), \ve_1\right).
}

Assume now that we have characterized an invariant manifold $\mathcal{M}_{\ve_1}^{fold}$ from $\Omega^{\en}$ to $\Omega^{\ex}$ via the map $\Pi_2^{fold}$. Now we want to ``extend'' $\mathcal{M}_{\ve_1}^{fold}$ all the way up to the sections $\Delta_2^{\en,-b_2}$ and $\Delta_2^{\ex,-b_2}$ via transitions along normally hyperbolic regions of $\Gamma_2$. For this, it is more convenient to regard $\mathcal{M}_{\ve_1}^{fold}$ as a graph $\zeta_2=\phi_{\ve_1}(A_2)$ where $(\zeta_2,A_2)$ are local coordinates around the fold point $p_+$ and where $\phi_{\ve_1}$ is a diffeomorphism for $\ve_1>0$. In this way we can equivalently express the map $\Pi_2^{fold}$ as
\eq{
  \Pi_2^{fold}(\zeta,\ve_1) &=(\tilde\zeta_2,\tilde\ve_1)\\
  &=(\psi_{\ve_1}(\zeta),\ve_1)
}

where $\psi_{\ve_1}$ is a diffeomorphism for $\ve_1>0$ and only a homeomorphism for $\ve_1=0$. Next, following \cref{sec:nf_reg} we can find coordinates $(Z_2,\ve_1)$ in $\Delta_2^{\en,-b_2}$, and coordinates $(\tilde Z_2,\ve_1)$ in $\Delta_2^{\ex,-b_2}$ in such a way that the transitions $\Pi_2^{reg_1}$ and $\Pi_2^{reg_2}$ are given as

\eq{
\Pi_2^{reg_1}(Z_2,\ve_1) &=\left( Z_2\exp\left( -\frac{1}{\ve_1}(A_0-a_{2,\en}) \right) \right) = (\bar Z_2,\ve_1)\\
\Pi_2^{reg_2}(-Z_{2,\ex},\ve_1) &=\left( -Z_{2,\ex}\exp\left( -\frac{1}{\ve_1}(A_0-\tilde a_2) \right) \right) = (\tilde Z_2,\ve_1).
}

\begin{remark}
  Recall that along normally hyperbolic slow manifolds, it is possible to make a normal form transformation in such a way that this transformation respects certain constraint or structure of the vector field, \cite{Bonckaert1,Bonckaert2}. In this particular case, we respect the choice of the invariant manifold $\mathcal{M}_{\ve_1}^{fold}$.
\end{remark}

Next, we can compute the composition $\Pi_2^{-b_1}=\Pi_2^{reg_2}\circ\Pi_2^{fold}\circ\Pi_2^{reg_1}$ by following \cref{sec:Exp_trans} and it thus follows that

\eq{
  \Pi_2^{-b_1}(0,Z_2,\ve_1)=\left(0,\bar\psi_{\ve_1}+Z_2\exp\left( -\frac{1}{\ve_1}(A_1+A_3+\ve_1\psi_2) \right),\ve_1 \right),
}%
where $\bar\psi_{\ve_1}=\psi_{\ve_1}(0)\exp\left( -\frac{A_3}{\ve_1} \right)$ and where $\psi_2=\psi_2(Z,\ve_1)$ is a $\mathcal{C}^{\ell}$-admissible function. Note that $\bar\psi_{\ve_1}$ is flat at $\ve_1=0$.

\end{proof}

\subsection{Proof of \cref{teo:main}}\label{sec:proofofmain}
\renewcommand{\bar\ve}{\bar{\varepsilon}}
\newcommand{\bar{b}}{\bar{b}}
\newcommand{\mathcal{A}}{\mathcal{A}}

Let us first recall that, within the blow up space, we have three types of transitions according to the initial condition $b_1|_{\Delta_1^{\en}}$, namely
\begin{itemize}
  \item If $b_1|_{\Delta_1^{\en}}\in O(\varepsilon_1^{2/5})$ then we construct a transition passing through the charts $K_{\en}\to K_{\bar\ve}\to K_{\ex}$.
  \item If $b_1|_{\Delta_1^{\en}}\notin O(\varepsilon_1^{2/5})$ and $b_1|_{\Delta_1^{\en}}>0$ then we construct a transition passing through the charts $K_{\en}\to K_{+\bar{b}}\to K_{\ex}$.
  \item If $b_1|_{\Delta_1^{\en}}\notin O(\varepsilon_1^{2/5})$ and $b_1|_{\Delta_1^{\en}}<0$ then we construct a transition passing through the charts $K_{\en}\to K_{-\bar{b}}\to K_{\ex}$.

\end{itemize}
In \cref{fig:cusp_comp} we give a qualitative diagram of the local transitions obtained and their relationship.

\tikzstyle{stuff_fill}=[rectangle,draw,preaction={fill=white}]

\begin{figure}[htbp]\centering
  \begin{tikzpicture}
    \pgftext{\includegraphics[scale=0.75]{cusp_comp}}
    \node at (-5,-2.3) {$K_{\en}$};
    \node at (0,-2.3) {$K_{\bar\ve}$};
    \node at (5,-2.3) {$K_{\ex}$};
    \node at (0,2.7) {$K_{+\bar{b}}$};
    \node at (0,-7.6) {$K_{-\bar{b}}$};

    \node at (-6.05,-.8) {\scriptsize{$r_1$}};
    \node at (-4.3,1.75) {\scriptsize{$b_1$}};
    \node at (-2.4,0) {\scriptsize{$\varepsilon_1$}};

    \node at (-6.05+8.475,.05) {\scriptsize{$\varepsilon_3$}};
    \node at (-4.3+8.65,1.75) {\scriptsize{$b_3$}};
    \node at (-2.4+8.45,-.9) {\scriptsize{$r_3$}};

    \node at (-5.6,-1.45) {\scriptsize{$\Delta_1^{\en}$}};
    \node at (-3.7,-1.65) {\scriptsize{$\Delta_1^{\ex,-b_1}$}};
    \node at (-3.65, 0.95) {\scriptsize{$\Delta_1^{\ex,+b_1}$}};
    \node at (-2.75, -.8) {\scriptsize{$\Delta_1^{\ex,\varepsilon_1}$}};

    \node at (3,-.85) {\scriptsize{$\Delta_3^{\en,\varepsilon_3}$}};
    \node at (5.6,-1.65) {\scriptsize{$\Delta_3^{\en,-b_3}$}};
    \node at (5.6, 0.95) {\scriptsize{$\Delta_3^{\en,+b_3}$}};
    \node at (5.8, -.1) {\scriptsize{$\Delta_3^{\ex}$}};

    \node at (-2.25,.85) [stuff_fill] {\scriptsize{$M_{\en}^{\bar\ve}$}};
    \node at (-3.65,3.85) [stuff_fill] {\scriptsize{$M_{\en}^{+\bar{b}}$}};
    \node at (3.45,2.95) [stuff_fill] {\scriptsize{$M_{+\bar{b}}^{\ex}$}};
    \node at (1.5,0.25) [stuff_fill] {\scriptsize{$M_{\bar\varepsilon}^{\ex}$}};
    \node at (2.5,-3.95) [stuff_fill] {\scriptsize{$M_{-\bar{b}}^{\ex}$}};
    \node at (-3.5,-2.95) [stuff_fill] {\scriptsize{$M_{\en}^{-\bar{b}}$}};

    \node at (-.6,1.3) {\scriptsize{$\Delta_2^{\en,\varepsilon_2}$}};
    \node at (1,-1.3) {\scriptsize{$\Delta_2^{\ex,\varepsilon_2}$}};

    \end{tikzpicture}
  \caption{All the transitions obtained in the charts. We have to compose all such transitions through the matching maps $M_i^j$. A matching map $M_i^j$ relates the coordinates between the charts $K_i$ and $K_j$. }
  \label{fig:cusp_comp}
\end{figure}

Let us only detail the transition through the inner layer $\Delta^{\inn}$ corresponding to $b_1|_{\Delta_1^{\en}}\in O(\varepsilon_1^{2/5})$, the other cases follow the same lines.

The transition $\Pi^{\inn}:\Delta_1^{\inn}\to\Delta_2^{\ex}$ is given as

\renewcommand{a_1}{a}
\renewcommand{b_1}{b}

\eq{
  \Pi^{\inn} &= \Pi_{3}^{\varepsilon_3}\circ M_{\bar\varepsilon}^{\ex}\circ\Pi_{2}^{\varepsilon_2}\circ M_{\en}^{\bar\varepsilon}\circ\Pi_1^{\inn}
}

where the matching maps are obtained from the blow-up map. For example, to obtain the matching map from the chart $K_{\en}$ to the chart $K_{\bar\ve}$ we relate the two directional blow-up maps

\eq{
  a_1=-r_1^3, \; b_1 = r_1^2 b_1, \; z= r_1z_1, \; \varepsilon=r_1^5\varepsilon_1
}

and

\eq{
  a_1=r_2^3 a_2, \; b_1 = r_2^2 b_2, \; z= r_2z_2, \; \varepsilon=r_2^5.
}

Let us work out only with the $z$-component of the transitions as it is the only relevant one. Recall from \cref{Ken} that $\Pi_1^{\inn}$ is an exponential type map with no shift. Next, the composition $\Pi^{central}=M_{\bar\varepsilon}^{\ex}\circ\Pi_{2}^{\varepsilon_2}\circ M_{\en}^{\bar\varepsilon}$ yields a diffeomorphism as $\Pi_2^{\varepsilon_2}$ is a diffeomorphism, and the matching maps are also diffeomorphisms on their domain of definition. Next, the last transition $\Pi_3^{\varepsilon_3}$ is an exponential type map with no shift, see \cref{Kex}. Therefore, following \cref{sec:Exp_trans} we have that $\Pi_3^{\varepsilon_3}\circ\Pi^{central}\circ\Pi_1^{\inn}$ is an exponential type map of the form

\eq{
  \Pi^{\inn}_{Z_1}=\bar\phi(B_1,\varepsilon_1)+Z_1\exp\left( -\frac{1}{\varepsilon_1}\left( \bar \mathcal{A}(B_1,\varepsilon_1)+\varepsilon_1\bar\Psi(B_1,\varepsilon_1,Z_1) \right) \right),
}

where $\bar \mathcal{A}>0$ and $\phi$ and $\Psi$ are $\mathcal{C}^{\ell}$ admissible functions. The differentiability of $\phi$ and $\Psi$ with respect to monomials is evident from the results of \cref{Ken}. By blowing down we obtain that the transition $\Pi^{\inn}:\Sigma^{\en}\to\Sigma^{\ex}$ (in a small neighborhood of the cusp point and within the inner layer as domain) reads as

\eq{
  \Pi^{inner}_{Z}=\phi(B,\varepsilon)+Z\exp\left( -\frac{1}{\varepsilon}\left(  \mathcal{A}(B,\varepsilon)+\varepsilon\bar\Psi(B,\varepsilon,Z) \right) \right).
}

To obtain the transition $\Pi:\Sigma^-\to\Sigma^+$ we now need to compose $\Pi^{\inn}_{Z}$ with exponential type maps on the left and on the right corresponding to

\eq{
  \Pi^-:\Sigma^-\to\Sigma^{\en}\\
  \Pi^+:\Sigma^{\ex}\to\Sigma^+.
}

However, we must proceed with care. In order to express the transition $\Pi$ as an exponential type map, we need to choose appropriate coordinates on $\Sigma^{-}$ and on $\Sigma^+$ that respect the already chosen coordinates in $\Sigma^{\en}$ and in $\Sigma^{\ex}$. Fortunately, this is possible with the extensions of Bonckaert \cite{Bonckaert1,Bonckaert2} to the normalization results of Takens \cite{Takens_partially}.

For sake of clarity, let $(B_{\en},Z_{\en})$ be coordinates in $\Sigma^{\en}$ and $(B_{\ex}, Z_{\ex})$ be coordinates in $\Sigma^{\ex}$. We have shown that these coordinates can be chosen in such a way that the ``vertical'' component of the transition map $\Pi^{\inn}:\Sigma^{\en}\to\Sigma^{\ex}$ reads as

\eq{
  \Pi_{Z_{\en}}(B_{\en},Z_{\en},\varepsilon)&=Z_{\ex}\\
                            &=\phi(B_{\en},\varepsilon)+Z_{\en}\exp\left( -\frac{1}{\varepsilon}\left(  \mathcal{A}(B_{\en},\varepsilon)+\varepsilon\bar\Psi(B_{\en},\varepsilon,Z_{\en}) \right) \right).
}

In this case the invariant manifold, say $\mathcal{M}_{\varepsilon}$, is given by $Z_{\en}=0$. Using \cite{Bonckaert1,Bonckaert2} we can find suitable coordinates $(B_-,Z_-)$ in $\Sigma^-$ in such a way that

\eq{
  \Pi^-_{Z_-}(B_-,Z_-,\varepsilon)=Z_-\exp\left( -\frac{1}{\varepsilon}(A_0) \right)=Z_{\en}.
}

In other words, there is a change of coordinates respecting the invariant manifold $\mathcal{M}_{\varepsilon}$ under which the transition $\Pi^-$ is an exponential type map with no shift and linear. Similar arguments hold for the choice of coordinates in $\Sigma^+$. Finally, following \cref{sec:Exp_trans}, the composition $\Pi^+_{Z_+}\circ\Pi_{Z_{\en}}\circ\Pi^-_{Z_-}$ leads to the result.

redpajama_set_name: RedPajamaArXiv


text:
Four Horsemen (DVD)
April 15, 2012 By Tim Isaac Leave a Comment
Starring: Noam Chomsky, Gillian Tett, Max Keiser
Director: Ross Ashcroft
Running Time: 98 mins
Certificate: E
Release Date: April 2nd, 2012
It's very rare that something actually makes me angry, but Four Horsemen did. The documentary tries to deflect criticism by saying during the voiceover that some will dismiss it as socialist when it isn't, but my problem with it wasn't with its viewpoint, but that as an argument it's absolutely hideously constructed and an altogether frustrating experience.
Made in the wake of the banking collapse and economic crisis, Four Horsemen attempts to explain what happened, why we haven't learned any lessons from it, and why western society is on the brink of collapse and unable to come to terms with the fact its version of civilisation may have peaked. Much of it is stuff many will have heard before, especially as the economic collapse has been picked apart in detail by numerous different voices. It doesn't help that Four Horsemen has little new to add, and most of what it does have to say has been said better by others – the wonderful The Shock Doctrine and Inside Job spring to mind.
However it wouldn't necessarily be a bad thing to say it again, even with a rather 'we're all doomed and isn't western civilisation an ass?' mantra added on top. The problem I had with it though is the way its puts its point across, which is to constantly use conclusions as evidence. For example, the film has decided western civilisation as we know it is over, so rather than arguing that point from the ground up, it extrapolates backward, so that everything about the modern day becomes evidence for its imminent end. For example, perhaps idolising chefs is something that happens just before civilisations end, but rather than providing evidence, the film presents it as an odd but indisputable fact.
It's not even that what Four Horsemen say is necessarily untrue, it's just horribly argued. It results in the film becoming a complete house of cards that can be blown over ridiculously easily, not because it's wrong but because it's not structured in a way that builds towards a conclusion, but instead decides the end result and then finds the evidence for it. It means that if you generally agree with its 'Occupy Wall Street'-style philosophy, you'll lap up, but if you're on the other side of the argument, Four Horsemen makes itself way too easy to dismiss.
It's a shame as the film has a lot of very knowledgeable interview subjects who have a lot of interesting stuff to say, but then the voiceover chimes in and presents something as an absolute which is nothing of the sort, and I for one just wanted to scream at the screen, 'Why is that necessarily true?' But rather than bothering to explain, it moves onto the next reason we're all doomed. It's a rather condescending way to make a documentary and very frustrating.
It might have helped if the documentary was more focussed, but it's so keen to cover every ill of the West it can find, even if it's only tangentially linked to its core subject – from what it sees as the self-interest of foreign aid to the flaws of democracy – that it doesn't present a truly convincing argument for any of it.
Four Horsemen has a lot of important questions to ask. Why have we allowed the financial system – something that doesn't actually 'produce' anything – hold so much power? How can so much of the world's wealth be based on debt? Why has the enormous growth in sheer amount of money in the past 60 years not resulted in a fairer distribution of wealth? With its interview subjects it has the potential to say a lot of fascinating things about it, but the structure and voiceover is so busy putting the cart before the horse that it will ultimately only be of interest to those who already believe everything it's got to say and want something that feels like it's backing them up.
Overall Verdict: A bit of a missed opportunity. There's nothing intrinsically wrong with polemic, but Four Horsemen could have done with studying the art of rhetoric in order to make its arguments stronger. Actually, that's the one thing about the West it doesn't criticise but perhaps should– that generally the quality of argument has gone completely down the tubes.
Special Features: Trailers, 'The Making Of Four Horsemen' Featurette, Billion Dollar Bill Animation, Scott Carver – The World's Best Banker, The Washed Up World Bank Banker
Reviewer: Tim Isaac
Filed Under: DVD and BLU-RAY REVIEWS

redpajama_set_name: RedPajamaCommonCrawl


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Latte lovers rejoice! Study reveals drinking coffee could lower the risk of Alzheimer's disease
Articles By Vivek G (Outcomes 1
Max Q: Elon Musk talks Starship, envisions first orbital launch in January 2022
Tech News / Technology Information
PD November 23, 2021 by Balmond
Hello and welcome back to Max Q, your very favorite space news digest. A ton happened this week — launches, financing deals, contract awards, etc., etc. — so let's dive in.
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Source: SpaceX
Musk: SpaceX to conduct first orbital flight of Starship in January
SpaceX CEO Elon Musk gave a talk at the National Academies' first virtual joint fall meeting of the Space Studies Board and the Board on Physics and Astronomy on Wednesday night, followed by a Q&A session with Academy members on the technical and operational aspects of Starship, SpaceX's ultra-super-heavy launch system.
First, the logistical details: Musk said he anticipates conducting the first orbital flight of Starship in January, followed by dozens or possibly more launches through the rest of the year. He added candidly that while the first orbital launch is not likely to be successful, he's "comfortable" Starship will reach orbit at some point next year.
This is significant and not totally unsurprising. While Musk has made a handful of predictions before about when Starship might attempt an orbital launch, all of that has been dependent on the Federal Aviation Administration completing its regulatory review of SpaceX's Boca Chica launch site, Starbase. Well now, the FAA has officially confirmed that it anticipates the review will be complete by December 31, so a January launch is looking more likely than ever.
Musk also reaffirmed his vision of Starship as a tool that can make humans truly multi-planetary for the first time in history. He described the vehicle as "a generalized transport mechanism for the greater solar system" — that means the moon, Mars and even beyond.
Rocket Lab acquires PSC, recovers booster after sending BlackSky satellites to orbit
Rocket Lab was busy this week. First, the launch company announced it was acquiring spacecraft separation systems company Planetary Systems Corporation, Rocket Lab's third acquisition in around 18 months or so. Then there was the company's third-quarter earnings call. On Wednesday evening, the company — drumroll, please — successfully recovered the first stage of its Electron launcher for the third time.
Image Credits: Rocket Lab (opens in a new window)
Rocket Lab has successfully recovered the first-stage booster twice in its history. During all three flights, this booster made a splashdown via parachute. But Wednesday night's recovery included an additional element: the presence of a helicopter, which hovered near the splashdown area to track and observe the booster as it made its descent. While the helicopter didn't actually assist with recovery, its presence is significant, as it indicates that Rocket Lab is also a step closer to executing its ultimate reusability plan: using a parachute to slow the velocity of the booster and capturing it mid-air.
Finally, Rocket Lab finished out the week with even more news: It has entered into an exclusive license agreement with Johns Hopkins University to commercialize small spacecraft telemetry and control radio technology. It's calling the software-defined radio product "Frontier-S by Rocket Lab."
More news from TC and beyond
Blue Origin reportedly upped its offer to cover the costs of developing a lander for NASA's Human Landing System program from $2 billion to $3 billion in an effort to receive a contract. NASA has said that budgetary issues were partly to blame for why it chose a single bidder — SpaceX — for the HLS contract.
Hydrosat, a geospatial data startup, has closed $10 million in seed funding to accelerate the commercialization of its ground temperature analytics product. The startup plans on sending a satellite equipped with a thermal infrared sensor to orbit next year.
Fleet Space raised $26.4 million in Series B financing led by existing investors Artesian Venture Partners, Blackbird Ventures, Grok and Horizons Ventures. The Australian startup is manufacturing and launching satellites for Internet of Things devices, with the goal of operating a 140-satellite constellation.
Intuitive Machines landed a $77.5 million contract with NASA under the agency's Commercial Lunar Payload Services program. Intuitive Machines will deliver four research payloads to the moon in 2024 using its Nova-C lander.
Inversion, a startup developing a space capsule to send and return small payloads from orbit, raised a $10 million seed round led by Spark Capital.
Northrop Grumman is leading a team that includes AVL, Intuitive Machines, Lunar Outpost and Michelin to design a crewed lunar terrain vehicle. That vehicle design will very likely be submitted as a bid to NASA for a forthcoming contract.
Sierra Space — the company that has plans to build a commercial space station along with Blue Origin, Boeing and others — has raised $1.4 billion in new capital, bringing its valuation to $4.5 billion. There were a lot of questions about how the company was going to finance the space station, and its Dreamchaser space plane and well… looks like we're getting an answer.
Starlink satellite dishes have been spotted at some Supercharger locations. There's been no comment from Tesla or SpaceX, but it's a notable development.
The agenda for TC Sessions: Space is here
Last year we held our first dedicated space event, and it went so well that we decided to host it again in 2021. This year, it's happening December 14 and 15, and it's once again going to be an entirely virtual conference, so people from all over the world will be able to join — and you can, too.
Check out a sneak peek of the early agenda by clicking here. Suffice to say, you won't want to miss it.
Read more stories on TechCrunch.com


ARSENAL secretary David Miles has revealed his family told him not to come home if they were drawn against Barcelona.
Miles was the only Gunners representative for the draw in Nyon earlier today and witnessed his club get drawn with the Catalan giants for the third time in five years today.
Last season's champions swept into the knockout stages comfortably and lead La Liga by a point ahead of Real Madrid.
Despite the draw, Miles insists Arsenal have the players to trouble Luis Enrique's side.
"We've got world class players and it will be two fantastic games that we are looking forward to," he added.
"We're sitting top of the Premier League, come February-March we'll have some players back and it would be great to take them on with our full-strength side."


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