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The computer programs used in the proof are in the file ada.tgz
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Continuum mechanics, homoclinic solutions, computer assisted proofs
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\begin{document}

\title[Homoclinic solution]{A homoclinic solution for excitation waves \\
       on a contractile substratum}
\date{\today}
\author{D.~Ambrosi$^1$, G.~Arioli$^1$, H.~Koch$^2$}      

\date{\today}
\maketitle
\begin{center}
{\small $^1$MOX -- Modellistica e Calcolo Scientifico\\
        Dipartimento di Matematica ``F. Brioschi'' \\
        Politecnico di Milano \\
        via Bonardi 9, 20133 Milano, Italy\\}

\smallskip
{\small $^2$Department of Mathematics\\
        University of Texas at Austin\\
        Austin, TX 78712, USA\\}
\smallskip

{\tt \{davide.ambrosi,gianni.arioli\}@polimi.it, koch@math.utexas.edu}
\end{center}

\begin{abstract}
We analyze a model of electric signalling in biological tissues
and prove that this model admits a travelling wave solution.
Our result is based on a new technique for computing rigorous bounds
on the stable and unstable manifolds at an equilibrium point
of a dynamical system depending on a parameter.
\end{abstract}

\section{Modeling, motivations and main result}
The mathematical modelling of electric signalling in biological tissues, and in the cardiac muscle
in particular, is a longstanding problem that has attracted a number of efforts.
The mathematical structure of a typical model consists of one reaction--diffusion 
equation, linear in the diffusion and nonlinear in the reaction term, coupled 
with a one or more ordinary differential equations (and in some cases many) \cite{keenersneyd}.  
The classical model that incorporates this basic mathematical structure,
while introducing the minimum amount of algebraic complications is
given by the Fitzhugh--Nagumo equations:
\begin{align}
\label{1}
      \frac{\partial v}{\partial t} - \nabla \cdot \left(\D \nabla v \right) & = 
       - A v (v-\alpha)(v-1) -A w\,, \\
\label{2}
      \frac{\partial w}{\partial t}  & = v - \frac{w}{\tau}\,,       
\end{align}
where $v(\bX,t)$ is the action potential and $w(\bX,t)$ is the gate variable. In general $\D$ is a 
symmetric positive definite tensor and $A \simeq ||D|| \gg 1$, where
$\|\cdot\|$ denotes some suitable norm.

Recent papers \cite{teresi,whiteley} have pointed out the role of the contractility of the substratum
in real physiological conditions, where the electrical potential actually modulates the contraction 
of the muscle fibers and consequently the strain of the material. Equations (\ref{1},\ref{2}) are therefore to
be rewritten in moving coordinates, where the strain of the domain is driven by the
action potential itself. It is convenient to rewrite the equations in material coordinates, 
using a mapping between  current positions and reference ones.  
Adopting the standard terminology of continuum mechanics, we denote by $\bx=\bx(\bX,t)$  the 
position at time $t$ of the material point that was at time $t=0$ at $\bX$. The gradient of deformation is
therefore  $\F = \frac{\partial \bx}{\partial \bX}$ and the  \cref{1,2} rewritten 
on a contractile substratum in material coordinates read \cite{teresi, whiteley}, 
\begin{align}
\label{3}
   \frac{\partial}{\partial t} (Jv)  -  \Div \left(J \F^{-1} \D \F^{-T} \Grad v  \right) & = - A J v (v-\alpha)(v-1) - A J w, \\
\label{4}
      \frac{\partial}{\partial t}  (Jw)  & =   Jv - \frac{Jw}{\tau},
\end{align}
where $J=\det(\F)$ and the symbols $\Div$ and $\Grad$ denote operators
with respect to $\bX$.\\
Our goal in this paper is a rigorous, computer assisted, analysis of the one dimensional counterpart of \cref{3,4}   
\begin{align}
\label{5}
      \frac{\partial}{\partial t} (Jv) -  
      \frac{\partial}{\partial X} D \left(J^{-1} \frac{\partial v}{\partial X} \right) & = 
       - A J v (v-\alpha)(v-1) - A J w, \\
\label{6}
      \frac{\partial}{\partial t}  (Jw)  & =   Jv - \frac{Jw}{\tau}, 
\end{align}
where the (scalar) diffusion coefficient $D$ is taken constant and, simply, 
$J=\partial x/\partial X$.\\ 
To close the problem, we need to introduce a relation between the contraction of the substrate 
and the action potential. While this relation in real biological tissues is 
quite complicated, we choose the simple linear relation 
\begin{equation}
\label{7}
      \frac{\partial x}{\partial X} = 1-\beta v,
\end{equation}
where $\beta\in(0,1)$ is a constant. The end result is \cite{aanq}  
\begin{align}
\label{8}
 \frac{\partial}{\partial t} ((1- \beta v)v)
  -  D \frac{\partial}{\partial X} \left( \frac{1}{1- \beta v} 
       \frac{\partial v}{\partial X} \right)  &= - (1- \beta v) v (v-\alpha)(v-1) - (1- \beta v) w, \\ 
\label{8.1}
      \frac{\partial }{\partial t} \left((1-\beta v) w \right)   & =  
      (1-\beta v) \left( v - \frac{w}{\tau} \right).
\end{align}
where we have directly taken $A=D=\varepsilon^{-1}$. \\

In \cite{aanq}  it was found that \cref{8,8.1}, admit travelling pulse solutions
that travel faster than in the rigid case. Our aim here is to give a proof for the existence
of such a solution.
To be more specific, we are interested in solutions of \cref{8,8.1} of travelling wave type and
finite energy, that is solutions $v(t,X)=V(X-ct)$,
$w(t,X)=W(X-ct)$, where $V$ and $W$ are homoclinic to $0$. 
The system becomes
\begin{equation}
\left\{
\begin{array}{lll}
\frac{V''}{1-\beta V}=&-\varepsilon c(1-2\beta
V)V'-\frac{\beta(V')^{2}}{(1-\beta V)^{2}}\\
&+(1-\beta V)V(V-\alpha)(V-1)+(1-\beta V)W\\
W'=&\frac{W}{c\tau}-\frac{V}{c}-\frac{\beta}{c}V^2\,,
\end{array}
\right.\label{eq:ode}
\end{equation}

\begin{thm}\label{th:hs}
Let $\alpha=0.1$, $\beta=0.3$, $\varepsilon=0.01$, $\tau=0.2$.
There exists $c\in[c_0-\delta,c_0+\delta]$, with
$c_0=\frac{719578791}{12160570}\approx 59$
and $\delta=2^{-43}$, such
that the system \eqref{eq:ode} admits a solution homoclinic to $0$.
\end{thm}

The proof of Theorem \ref{th:hs} is divided into several parts.
In Section \ref{sec:Invariant-manifolds} 
we describe a general method to compute bounds on the invariant manifolds of
nonlinear dynamical systems.
This method is then applied to the system (\ref{eq:ode}) as described in
Section 3.
It reduces the problem to a number of specific estimates
that can be carried out with the aid of a computer.
These estimates are described in the first part of Section 4.
In the second part we discuss some of the details
of our computer-assisted proof.


\section{A parametrization for the invariant manifolds\label{sec:Invariant-manifolds}}

Consider the nonlinear dynamical system in $\mathbb{R}^{N}$
\begin{equation}
y'=Ay+B(y)\,,\label{eq:generic}
\end{equation}
 for a curve $y:\mathbb{R}\to D$, where $D$ is some open domain in
 $\mathbb{R}^N$ containing the origin, $A$ is an invertible linear map on $\mathbb{R}^N$,
$B:D^*\to\mathbb{C}^N$ (where $D^*$ is some open domain in
 $\mathbb{C}^N$ that includes the closure of $D$) is analytic, $B(x)\in \mathbb{R}^N$ if
$x\in D$ and $B(y)=O(|y|^2)$.
By definition, the unstable manifold at $0$ is the set of all initial
conditions $y(0)$ such that $\lim_{t\to-\infty} y(t)=0$. 
In the following, with no loss of generality, we will only
consider the unstable manifold, since the stable manifold of
(\ref{eq:generic}) is the unstable manifold of $\dot{y}=-(Ay+B(y))$.
It is well known
that, under these assumptions,
the unstable manifold is tangent in $0$ to the eigenspace of
$A$ corresponding to eigenvalues with positive real part.
We are interested in computing a parametrization of the local unstable manifold
in a neighborhood of the origin, with rigorous bounds on the error.
This problem was addressed in \cite{jt},
where an explicit algorithm was introduced in the case where
the matrix $A$ only has real eigenvalues. 
Here we extend the approach
to the generic case when $A$ can have complex conjugate eigenvalues,
and we also provide a method for carrying out the computation and controlling the errors.
There are many other ways of constructing
invariant manifolds; see e.g. \cite{cfd1,cfd2,cfd3}.
The advantage of the method presented here
is that it is simple and can easily be implemented
on a computer, see also \cite{wittig}.
The following theorem addresses the problem of computing the unstable
manifold tangent to a simple real eigenvector.

\begin{thm}\label{th:real}
If $A$ admits a real, simple, strictly positive eigenvalue $\lambda$ with
corresponding eigenvector $v$ and $z:(-1,1)\to\mathbb{R}^N$ satisfies
\begin{equation}
\lambda s  z'(s)=Az(s)+B(sv+z(s))\label{eq:z}
\end{equation}
and
\begin{equation}
y(t)=e^{\lambda t}v+z(e^{\lambda t})\label{eq:def-y}
\end{equation}
satisfies $y(t)\in D$ for all $t<0$, then $y(t)$
is a solution of (\ref{eq:generic}) for $t<0$ and
$y(t)=e^{\lambda t}v+O(e^{2\lambda t})$ as $t\to-\infty$.
\end{thm}

\begin{proof}
Consider the change of variable
\begin{equation}
s=e^{\lambda t}\,,\label{eq:changeV}
\end{equation}
and write $y(t)=\tilde{y}(s)=z(s)+sv$.
If $z$ satisfies (\ref{eq:z}), then
$$
\lambda s\left(v+z'(s)\right)=s\lambda v+Az(s)+B(sv+z(s))\,,
$$
and
\begin{equation}
\lambda
s\tilde{y}'(s)=A\tilde{y}(s)+B(\tilde{y}(s))\,.\label{eq:system2}
\end{equation}
Since $y'(t)=\lambda e^{\lambda t}\tilde{y}(e^{\lambda t})=\lambda s\tilde{y}'(s)$,
then $y(t)=\tilde{y}(e^{\lambda t})$ satifies (\ref{eq:generic}).
Furthermore, 
$y(t)=e^{\lambda t}v+O(e^{2\lambda t})$ as $t\to-\infty$.
\end{proof}

Next, we extend the method to the unstable manifold corresponding to
a pair of complex conjugate eigenvalues.
Assume that $v$ and $\bar v$ are eigenvectors of $A$,
with eigenvalues $\lambda$ and $\bar\lambda$  and ${\rm  Im}\lambda\ne0$.
If ${\rm Re}\lambda>0$, then equation (\ref{eq:generic})
admits an unstable manifold at $0$ tangent to the span of $({\rm Re}v,{\rm Im}v)$.
In order to compute an explicit expression for such a manifold we use the following theorem:

\begin{thm}\label{th:complex}
Let $B_1(0)$ be the unit disk in $\mathbb{C}$ centered at $0$, assume that
$$Z:[B_1(0)]^2\to \mathbb{C}^N$$
satisfies
\begin{equation}
\lambda s_1Z_1(s_1,s_2)+\bar\lambda s_2Z_2(s_1,s_2)=
AZ(s_1,s_2)+B\bigl(s_1v+s_2\bar v+Z(s_1,s_2)\bigr)\,,\label{eq:z2}
\end{equation}
where $Z_k=\frac{\partial Z}{\partial s_k}$.
Then, for all $(r_1,r_2)\in[B_1(0)]^2$ and all $t\le0$
the function
\begin{equation}
y(t)=r_1 e^{\lambda t}v+r_2 e^{\bar\lambda t}\bar v+
Z\left(r_1 e^{\lambda t},r_2 e^{\bar\lambda t}\right)\label{eq:y2}
\end{equation}
is a solution of equation (\ref{eq:generic}). If additionally
$r_2=\bar r_1$, then $y(t)\in\mathbb{R}^N$ for all $t\le0$.
\end{thm}
\begin{proof}
Differentiating (\ref{eq:y2}) with respect to $t$ we have
$$
y'(t)=r_1 \lambda e^{\lambda t}v+r_2 \bar\lambda e^{\bar\lambda t}\bar v+
Z_1\left(r_1 e^{\lambda t},r_2 e^{\bar\lambda t}\right)r_1 \lambda e^{\lambda t}+
Z_2\left(r_1 e^{\lambda t},r_2 e^{\bar\lambda t}\right)r_2 \bar\lambda e^{\bar\lambda t}\,.
$$
We also have
$$
Ay(t)=r_1 \lambda e^{\lambda t}v+\bar \lambda r_2 e^{\bar\lambda t}\bar v+
AZ\left(r_1 e^{\lambda t},r_2 e^{\bar\lambda t}\right)\,,
$$
so $y(t)$ is a solution of (\ref{eq:generic}) if and only if
\begin{align}\begin{split}
Z_1\left(r_1 e^{\lambda t},r_2 e^{\bar\lambda t}\right)r_1 \lambda e^{\lambda t}+
Z_2\left(r_1 e^{\lambda t},r_2 e^{\bar\lambda t}\right)r_2 \bar\lambda e^{\bar\lambda t}=\\
AZ\left(r_1 e^{\lambda t},r_2 e^{\bar\lambda t}\right)+
B\left(r_1e^{\lambda t}v+r_2e^{\bar\lambda t}\bar v+
Z\left(r_1 e^{\lambda t},r_2 e^{\bar\lambda t}\right)\right)\,,
\end{split}\end{align}
which follows from equation (\ref{eq:z2}) by choosing $s_1=r_1 e^{\lambda t}$ and
$s_2=r_2 e^{\bar\lambda t}$.
Now set $W(s_1,s_2)=Z(s_2,s_1)$. A direct computation shows that
$W$ satisfies
\begin{equation}
\lambda s_1 W_2(s_2,s_1)+\bar\lambda s_2W_1(s_2,s_1)=
AW(s_2,s_1)+B\bigl(s_1\bar v+s_2 v+W(s_2,s_1)\bigr)\,;\label{eq:W}
\end{equation}
if we take the complex conjugate of (\ref{eq:W}) and set
$t_1=\bar s_2$ and $t_2=\bar s_1$ we get
\begin{equation}
\lambda t_1 \bar W_1(\bar t_1,\bar t_2)+\bar\lambda t_2 \bar W_2(\bar t_1,\bar t_2)=
A\bar W(\bar t_1,\bar t_2)+
\bar B\bigl(\bar t_2\bar v+\bar t_1 v+W(\bar t_1,\bar t_2)\bigr)\,.\label{eq:W2}
\end{equation}
Recalling that $B$ is an analytic function that takes real values for
real arguments, we have
$$
\bar B\bigl(\bar t_2\bar v+\bar t_1 v+W(\bar t_1,\bar t_2)\bigr)
=B\bigl(t_2 v+t_1 \bar v+\bar W(\bar t_1,\bar t_2)\bigr)\,,
$$
therefore $\bar W(\bar t_1,\bar t_2)$ satisfies equation (\ref{eq:z2}). Then
$\bar W(\bar t_1,\bar t_2)=Z(t_1,t_2)=\bar Z(\bar t_2,\bar t_1)$, so
$Z(s,\bar s)=\bar Z(s,\bar s)$ and $y(t)$ takes real values when $r_2=\bar r_1$.
\end{proof}

It is straightforward to extend Theorem \ref{th:complex} to the case of a manifold
of higher dimension, corresponding to real or complex conjugate eigenvalues.

\section{Analytic framework\label{sec:Analytic-framework}}

 We represent the functions $z$ and $Z$ that are needed for
 Theorems \ref{th:real} and \ref{th:complex} as power series.
 Using the equations \eqref{eq:z} and \eqref{eq:z2},
 it is straightforward to determine the coefficients
 of $z$ and $Z$ recursively.
 Combining the result with an estimate on the radius of convergence
yields the desired parametrization of the invariant manifolds.
 However, an exact computation of all these coefficients
 is impossible, except in some very special cases.
 Besides, the equations involve parameters
 (eigenvalues and eigenvectors) that may not be
 computable exactly, depending on the matrix $A$.
 The goal therefore is to determine approximate values
 and rigorous error bounds,
 first for the eigenvalues and eigenvectors,
 and then for the Taylor coefficients of $z$ and $Z$.
 As can easily be guessed,
 this goes far beyond what could be done by hand.
 Fortunately, the most tedious part
 is trivial enough that it can be done with a computer.
 But the computer has only a limited number of states
 and works at a finite speed,
 so we need a finite (and not too large)
 dimensional approximation of the problem, with a
 precise control on the error involved.
 In Section \ref{sec:cap} we describe how to handle this task.

We start by introducing a suitable analytic framework. We
discuss here the case of the complex invariant manifold, the other
case being simpler.
Let $\calXX$ be the space of functions of two
complex variables with domain $[B_1(0)]^2$,
which can be written as a power series
\begin{equation}
u(s_1,s_2)=\sum_{j,k=0}^{\infty}u_{jk}s_1^js_2^{k}\label{eq:powerseries}
\end{equation}
with $u_{jk}\in\mathbb{C}$ and such that
$$
\|u\|:=\sum_{j,k=0}^{\infty}|u_{jk}|<+\infty\,.
$$
The following lemma is straighforward:
\begin{lem}
\label{lem:Banach}The space $\calXX$ is a Banach algebra;
in particular, for all $u,v\in\calXX$ we have $uv\in\calXX$
and $||uv||\leq||u||||v||$.
\end{lem}

In order to compute a function in $\calXX$ which satisfies the
assumptions of Theorem \ref{th:complex}, 
we define $\calZ$ as the subalgebra of the functions
in $\calXX$ with the coefficients of order 0 and 1 (that
is the constant and first order terms) equal to $0$, then we 
define $\calZ^N$ as the Banach space of $N$-tuples of functions in
$\calZ$ with norm
$$
\|Z\|=\sum_{k=1}^N\|Z_k\|
$$
and we write equation (\ref{eq:z2}) as
\begin{equation}\label{eq:Zss}
Z(s_1,s_2)=D_{\lambda}^{-1}(AZ(s_1,s_2)+B(s_1v+s_2\bar v+Z(s_1,s_2)))\,,
\end{equation}
where $Z\in\calZ^3$ and $D_\lambda^{-1}$ is defined by
$$
D_\lambda^{-1}(s_1^{j}s_2^{k})=\frac{s_1^{j}s_2^{k}}{j\lambda+k\bar\lambda}\,,
$$
(we abuse the notation by calling $D_\lambda^{-1}$ both the operator
acting on $s_1^{j}s_2^{k}$ and its natural extension to a vector of 
functions). 
The equation (\ref{eq:Zss}) that we need to solve can be written as
$Z=\calC(Z)$, where
$$
\calC(Z)=D_\lambda^{-1}(AZ+B(s_1v+s_2\bar v+Z))\,.
$$
We will restrict our analysis to a small ball $B_r(Z_0)$ in $\calZ^3$,
centered at an approximate solution $Z_0$.
The map $B$ considered is essentially a polynomial,
expect for a factor $[1-L(Z-Z_0)]^{-1}$,
where $L$ is a continuous linear functional
that is bounded away from 1 in the ball $B_r(0)$.
Thus, by Lemma \ref{lem:Banach}, the map $\calC: B_r\to\calZ^3$ is analytic.
And its fixed point is the solution of (\ref{eq:Zss}) that we are looking for.

What is crucial for our computer-assisted proof is the following:
\begin{lem}\label{lem:c}
$\calC$ is a limit of finite rank operators.
\end{lem}
This follows immediately from the definition of
$\mathcal{X}$ and of $D_\lambda^{-1}$, together with the
continuity of the map
$Z(s_1,s_2)\mapsto(AZ(s_1,s_2)+B(s_1v+s_2\bar v+z(s_1,s_2)))$.

A common method for solving the fixed point problem
for a smooth map like $\calC$ is the Newton-like iteration
$z\mapsto \calC(z)-[D(\calC(z)-z)]^{-1}(\calC(z)-z)$.
Thanks to the above-mentioned property of $\calC$,
we can replace the the derivative $D(\calC(z)-z)$ by a finite rank approximation.
To this end, we consider an $m$-dimensional subspace $\complex_m$ of $\calZ^3$,
obtained via a projection $P:\calZ^3\to \complex_m$
that truncates high order Taylor coefficients.
Identifying $\complex_m$ with $\complex^m$,
we choose an invertible $m\times m$ matrix $M$ that approximates
$D(P\calC P)-I$,
where $D$ denotes the Jacobian and $I$ the $m\times m$ identity matrix.
Let $\calN:\mathcal{Z}^3\to\mathcal{Z}^3$ be defined by
$$
\calN(Z)=\calC(Z)-M^{-1}P(\calC(Z)-Z)\,;
$$
clearly, if $M-I$ is invertible, then fixed points of $\calN$
correspond to fixed points of $\calC$.
A direct consequence of the contraction theorem is the following
\begin{lem}
If there exist positive constants $\varepsilon,r,K$ and
$Z_0\in\mathcal{Z}^3$ such that
\begin{itemize}
\item $\|\calN(Z_0)-Z_0\|<\varepsilon$,
\item $\|D\calN(Z)\|\le K$ for all $Z\in B_r(Z_0)$
\item $\varepsilon+rK<1$,
\end{itemize}
then there exists a unique fixed point of $\calN$ in $B_r(Z_0)$.
\end{lem}

\section{The computer assisted proof}\label{sec:cap}

\subsection{The differential equation}

 If we set $y=(v,z,w)$,
\begin{equation}
A_c=\left(\begin{array}{ccc}
0 & 1 & 0\\
\alpha & -c\varepsilon & 1\\
-1/c & 0 & 1/(c\tau)\end{array}\right)
\text{ and }
B_c(v,z,w)=\left(\begin{array}{ccc}0\\b_2(v,z,w)\\b_3(v,z,w) \end{array}\right)
\label{eq:mat}
\end{equation}
with
\begin{align}
\begin{split}b_2(v,z,w)= & c\varepsilon\beta vz(3-2\beta
  v)-\frac{\beta z^{2}}{1-\beta v}
+(v^{5}\beta^{2}-v^{4}\alpha\beta^{2}-v^{4}\beta^{2}-2v^{4}\beta\\
 & +v^{3}\alpha\beta^{2}+2v^{3}\alpha\beta+2v^{3}\beta
+v^{3}-2v^{2}\alpha\beta-v^{2}\alpha-v^{2})-\beta vw\,,\\
b_3(v,z,w)=&-\frac{\beta}{c} v^2
\end{split}
\label{eq:nonlin}
\end{align}
then the system \eqref{eq:ode} takes the form
\begin{equation}
y'=A_cy+B_c(y)\,,\label{eq:system}
\end{equation}
so we can apply the technique developed in the previous sections.
The first step consists in computing the eigenvalues and eigenvectors of $A_c$;
a direct computation yields:
\begin{lem}
For all $c\in[c_0-\delta,c_0+\delta]$ the matrix $A_c$ admits a real
eigenvalue $\lambda_1(c)>0$ and a pair of complex conjugate
eigenvalues $\lambda_{2,3}(c)=\lambda_R(c)\pm i\lambda_I(c)$ with
$\lambda_R(c)<0$.
\end{lem}
We compute explicitely such eigenvalues and the corresponding
eigenvectors $v_j$.
The following lemmas, whose proof is computer assisted and is
described in Section \ref{sec:cap}, provide a
local parametrization of both the stable and the unstable manifold at $0$.
\begin{lem}\label{lem:unst}
There exist coefficients $z_k(c)$ depending on $c$ and a positive real number $E$ such that
$$
z_c(s)=\sum_{k=2}^{100}z_{k}(c)s^{k}+E_z(s)
$$
with $\|E_z\|<E$ is a solution of \cref{eq:z} for all
$c\in[c_0-\delta,c_0+\delta]$, where the matrix $A$ is
given in (\ref{eq:mat}), $B$ is given in (\ref{eq:nonlin}), $\lambda$
and $v$ are the real eigenvalue/vector of $A_c$.
\end{lem}

We set $y_0(c)=z_c(1/4)$, and by Theorem \ref{th:real} we know that $y_0(c)$
belongs to the unstable manifold at $0$ of (\ref{eq:system}). In order to follow the unstable
manifold, we solve the initial value problem
\begin{equation}
\left\{
\begin{array}{ll}
y_c'&=A_cy_c+B_c(y_c)\\
y_c(0)&=y_0(c)
\end{array}
\right.
\label{eq:cauchy}
\end{equation}
in a similar way, that is we define the operator
$D^{-1}_{y}$ by
$$
(D^{-1}_{y}z)(t)=\int_y^t z(s)\,ds\,,
$$
we set $C(z)=D^{-1}_{y_0(c)}(A_cz+B_c(z))$
and we look for a fixed point of $C$. In this case, we do not need a
Newton map, since the operator $C$ turns out to be a contraction.

\begin{lem}\label{lem:unst2}
There exist coefficients $z_k(c)$ depending on $c$ and a positive real number $E$ such that
$$
z_c(s)=\sum_{k=0}^{100}z_{k}(c)s^{k}+E_z(s)
$$
with $\|E_z\|<E$ is a solution of (\ref{eq:cauchy}).
\end{lem}

We set $y_1(c)=z_c(1)$ and we apply again Lemma \ref{lem:unst2}
iteratively, until we obtain $y_{45}(c)$. Then, we do the same again, but
with a time step $t=1/16$, until we have $\tilde y(c)=y_{50+3/4}(c)$.
Clearly, $\tilde y(c)$ lies on the unstable manifold.

Finally, we build a local parametrization of the stable manifold:

\begin{lem}\label{lem:stable}
There exist coefficients $z_{jk}(c)$ depending on $c$ as described
above, and a positive real number $E$ such that
$$
Z(s_1,s_2,c)=\sum_{j,k\ge0}^{j+k\le 30}z_{jk}(c)s_1^js_2^{k}+E_Z(s_1,s_2)\,,
$$
$\|E_Z\|<E$, is a solution of \cref{eq:z2}.
\end{lem}

Once we have accurate bounds on the the local stable
and unstable manifolds, we verify the following estimates.
\begin{lem}\label{lem:inter}
Let $\tilde y(c)$ be as above, $a_1=0.05$, $a_2=0.5$ and $a_3=3/128$.
Let $Z(s_1,s_2,c)$ be as in Lemma \ref{lem:stable}.
There exists a coordinate system in $\mathbb{R}^3$
such that, if $P_j$ is the projection on the $j$-th coordinate, then
\begin{enumerate}
\item $|P_1(Z(s,\bar s,c))|<a_1$ for all $|s|\le a_3$ and all $c$.
\item $P_1(\tilde y(c_0-\delta))<-a_1$ and $P_1(\tilde y (c_0+\delta))>a_1$.
\item $|P_2(\tilde y (c))|^2+|P_3(\tilde y (c))|^2< a_2$ for all $c$.
\item Let $C_c=\{x\in\mathbb{R}^3\,:\,x=Z(s,\bar s,c),\, |s|=a_3\}$
and $B=\{x\in\mathbb{R}^3\,:\,|P_2(x)|^2+|P_3(x)|^2\le a_2\}$.
For all $c$ we have $C_c\cap B=\emptyset$ and the set $C_c$
is not contractible to a point in $\mathbb{R}^3\setminus B$.
\end{enumerate}
\end{lem}

The above lemma imply that there exists $c\in(c_0-\delta,c_0+\delta)$
and $s\in\complex$, $|s|<3/128$ such that
$u(c)=Z(s,\bar s,c)$, therefore the manifolds intersect and Theorem \ref{th:hs} follows.

\subsection{More on the computer-assisted proof}\label{sec:cap2}

Here we describe some of the details of our computer-assisted proof
of Lemmas \ref{lem:unst}, \ref{lem:unst2},
\ref{lem:stable} and \ref{lem:inter}.
Given the Taylor polynomials and the matrices $M$
(obtained from purely numerical computations),
the proofs are clearly a sequence of trivial estimates,
assuming that there are no fundamental obstructions.
The sequence is finite, as one would expect from Lemma \ref{lem:c}.
But the steps are much too numerous to be carried out by hand,
so we enlist the help of a computer.
For the types of operations needed here,
the techniques are quite standard by now.
Thus, we will restrict our description
mainly to the problem-specific parts.

As with any lengthy task, proper organization is crucial.
We start by associating to a space $X$
a collection $\std(X)$ of subsets of $X$,
that are representable on the computer.
These sets will be referred to as ``standard sets'' for $X$.
A ``bound'' on an element $s\in X$
is then a set $S\in\std(X)$ containing $s$.
Each collection $\std(X)$ corresponds to a data type in our programs.
Unless stated otherwise,
$\std(X\times Y)$ is taken to be the collection of all sets $S\times T$
with $S\in\std(X)$ and $T\in\std(Y)$.

Our standard sets for $\real$ are associated with
a type {\tt Ball}, which consists of pairs {\tt S=(S.C,S.R)},
where {\tt S.C} is a representable number ({\tt Rep})
and {\tt S.R} a nonnegative representable number ({\tt Radius}).
The standard set defined by a {\tt Ball} {\tt S}
is the interval
$\BB({\tt S})=\{s\in\real: |s-{\tt S.C}|\le{\tt S.R}\}$.

For non-representable system parameters,
such as the constants appearing in the statement of Theorem 1
(with the exception of $\delta$),
we use objects of type {\tt Ball} that contain the given values.

We represent functions in $\chi^2$ as follows:
\begin{equation}
Z(s_1,s_2)=\sum_{j,k\ge0}^{j+k\le M}z_{jk}s_1^js_2^{k}+E_Z\,,
\label{eq:repr}
\end{equation}
where $E_Z$ is a function in $\calXX$ with all coefficients of degree less or
equal to $M$ equal to zero.
Our standard sets for $\calXX$ are represented by a type
{\tt Taylor2} consisting of a pair {\tt F=(F.C,F.E)},
where {\tt F.C} is an {\tt array(0..K,0..K) of Ball},
and {\tt F.E} is a {\tt Ball}.

For the representable numbers, we choose a data type
(renamed to {\tt Rep}) for which elementary operations
are available with controlled rounding.
This makes it possible to implement a bound {\tt Sum}
on the function $(s,t)\mapsto s+t$ on $\real\times\real$,
as well as bounds on other elementary functions on $\real$ or $\real^N$,
including things like the matrix product.

Here, a bound on a map $f: X\to Y$ is a map $F: D_F\to\std(Y)$,
with domain $D_F\subset\std(X)$,
such that $f(s)\in F(S)$ whenever $s\in S\in D_F\,$.
Such bounds are implemented as procedures or functions in our programs.
This can be done hierarchically.
Using e.g.~the {\tt Sum} for the type {\tt Ball},
it is straightforward to implement a bound {\tt Sum}
on the map $(g,h)\mapsto g+h$ from $\calZ\times\calZ$ to $\calZ$.
Similarly for maps like $u\mapsto\|u\|$ or $D^{-1}$.
Implementing a bound on the product $(g,h)\mapsto gh$
is a bit more tedious, but straightforward.

In order to estimate $\|D\NN(h)\|$ we use the following fact.
If $L$ is a continuous linear operator on $\chi^2$, then
$$
\|L\|=\sup_k\|L e_k\|\,,\qquad e_k=\|v_k\|^{-1}v_k\,,
$$
where $\{v_k\}$ is a basis of $\chi^2$.
This explicit expression for $\|L\|$
is our main reason for working with a weighted $\ell^1$ norm.
For the operator $L=D\NN(h)$,
it is easy to determine $k_0\,$, given $c>0$,
such that $\|L e_k\|\le c$ whenever $k\ge k_0\,$.
Thus, estimating the norm of $D\NN(h)$ reduces to a finite computation.
Choosing $\delta>0$ to be a representable number,
this estimate can be carried out simultaneously
for all functions $h\in B_\delta(0)$,
since $B_\delta(0)$ belongs to $\std(\calZ)$.

A crucial step of the proofs consists in making all computations and
estimates with a parameter that takes values in an interval.
This issue is usually addressed with interval arithmetics; in our
setting, we may define the parameter $c$ as a {\tt Ball} with center
$c_0$ and radius $\delta$. This approach is unfeasible in this
proof: if one attempts to follow the unstable manifold using a
parameter of finite width, however small, the errors accumulate very
rapidly, and the computation gets quickly out of hand.
Even a fine partition of the interval $[c_0-\delta,c_0+\delta]$ is not
feasible. Therefore, instead of an interval enclosure for $c$,
we use a type {\tt TBall}, which is a {\tt Taylor} of order 2 with coefficients
of type {\tt Ball}.
So every {\tt Scalar} is effectively a function of $c$.
We call $\xi$ the parameter normalized to the interval $[-1,1]$, so that 
$c$ is represented by the {\tt TBall}
$$c_0+\delta\xi\,.$$
By using {\tt TBalls} as coefficients for the previously discussed
algorithm, we obtain an explicit expression for a parametrization of
the invariant manifolds, depending on both a geometrical parameter and $c$.
 
For a precise and complete description of all definitions
and estimates, we refer to the source code
and input data of our computer programs \cite{aak}.
The source code is written in Ada2005.
For the type {\tt Rep} we use a MPFR floating point type,
with $128$ or $256$ mantissa bits, depending on the program.
MPFR is an open source multiple-precision floating-point library
that supports controlled rounding.
Our programs were run successfully on a standard desktop machine,
using a public version of the gcc/gnat compiler.


\section*{Acknowledgments}

This work has been supported by the ERC Advanced Grant \emph{Mathcard}
(number 227058).

\begin{thebibliography}{19}

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{\tt http://www.ma.utexas.edu/mp\_arc} 12-10

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\emph{Electromechanical Coupling in Cardiac Dynamics:
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SIAM J. Appl. Math., 71:605-621 (2011). 

\bibitem{ak}G. Arioli, H. Koch, \emph{Computer-assisted methods
for the study of stationary solutions in dissipative systems,
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Arch. Ration. Mech. Anal. 197 (2010), no. 3, 1033-1051.

\bibitem{cfd1}X. Cabr\'e, E. Fontich, R. de la Llave,
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\bibitem{cfd2}X. Cabr\'e, E. Fontich, R. de la Llave,
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\bibitem{cfd3}X. Cabr\'e, E. Fontich, R. de la Llave,
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 J. Differential Equations 218 (2005), no. 2, 444-515

\bibitem{teresi}C. Cherubini, S. Filippi, P. Nardinocchi, and L. Teresi,
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\bibitem{jt}T. Johnson, W. Tucker, \emph{A Note on the Convergence of
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\bibitem{keenersneyd} Keener, J. and Sneyd, J., \emph{Mathematical
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\bibitem{liu} I-Shih Liu, \emph{Continuum Mechanics}, Springer (2002). 

\bibitem{whiteley} Whiteley, J.P., Bishop, M.J. and Gavaghan, D.J.,
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\bibitem{wittig} A. Wittig, \emph{Rigorous High-Precision
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\end{thebibliography}

\end{document}
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