\section{Introduction}

Lattices of coupled dynamical systems have been studied in many
different contexts: as discrete
versions of partial differential equations of evolution type
\cite{BuSi}, \cite{Radu}, models of neuronal
networks \cite{aron}, \cite{huerta} and phase lock loops
\cite{nekor}, and in statistical mechanics \cite{fpu}.  The
particular problem of synchronization and emergence of coherent behavior in
lattices of diffusively coupled,
chaotic, continuous time dynamical systems has received
much attention recently due to its applications in neuroscience \cite{abarbanel},
\cite{huerta}, and chaotic synchronization \cite{josic}, \cite{heagy}.

Analytical results in the literature suggest that the coupling
strength necessary for the appearance of coherent behavior in networks
of chaotic systems grow with the size of the network
\cite{lin},\cite{nekor},\cite{hale}.  For instance,
estimates of Afraimovich and Lin \cite{lin} suggest that the coupling
strength necessary to synchronize a lattice of
generalized forced Duffing systems grows as a high power of the size
of the lattice.  Since stable, coherent states in lattices may
frequently be viewed as instances of generalized synchronization
\cite{josic}, one might guess that the appearance of dynamically
stable states of this type also requires that the coupling grows
with the number of oscillators in the system.
This would make it unlikely that such networks
are of importance in nature, since the coupling strength
in realistic systems cannot be
varied by orders of magnitude.

In contrast to these theoretical results, the
numerical experiments of Huerta, et. al. \cite{huerta} indicate that
partial synchronization may appear in certain networks of chaotic
systems at a nearly constant coupling strength, regardless of the size
of the network. In particular, \cite{huerta} considered a system
with about $10^4$ coupled neurons.  Each neuron was modeled
by a system of three nonlinear, ordinary differential equations which,
when uncoupled, underwent chaotic motion.  They observed a variety
of different stable, coherent motions including a number
of states that exhibited roughly periodic patterns.
 It is the goal of this paper to show rigorously that
such such stable, coherent states can occur in arbitrarily large
systems of coupled chaotic oscillators at a coupling strength
which is independent of the size of the system.

Since the Lorenz system is  one of the paradigms of
chaotic flows, we have chosen a ring of diffusively coupled Lorenz
equations for our investigations.  The systems in the ring evolve
according to the equations
\begin{eqnarray}\label{coupledlorenz}
x_i' = \sigma (y_i-x_i) + d_x \Delta x_i \nonumber \\
y_i' = r x_i - y_i - x_i z_i + d_y \Delta y_i \label{eq1} \\
z_i' = - \beta z_i + x_i y_i + d_z \Delta z_i \nonumber
\end{eqnarray}
where $\Delta x_i =  x_{i-1} - 2 x_i + x_{i+1}$ with the index $i$
taken modulo $n$ is the discretized Laplacian with periodic boundary
conditions.  The constants
$\sigma = 10, r = 27$, and $\beta = 8/3$ are chosen so that
each uncoupled system would be in the chaotic regime. Following the
work of \cite{huerta}, we will focus on the case $d_x \neq 0$,
$d_y = d_z = 0$.
As shown in \cite{josic}, such a ring
is synchronized identically for sufficiently large
values of $d_x$. We will be concerned with the case when $d_x$
is sufficiently large for coherent spatial structures to emerge in the
chain, but insufficient to synchronize the ring.

The paper is organized as follows:  Typical spatial
structures observed in numerical experiments are described in section
\ref{numerics}. In particular, we give examples of stable stationary
states, traveling waves, and breathers.
In section \ref{steadystate} a general framework for the
study of fixed states in chains and rings of dynamical systems is
discussed.  We show that the existence of such states can be
reduced to the study of periodic solutions of reversible dynamical
systems on ${\Bbb{R}}^n$.  We then apply these results to two different cases,
including the chains of coupled Lorenz oscillators (\ref{coupledlorenz}),
to show that a large number of fixed states
can be expected in such rings.  The conditions under which these
states are stable are investigated in section \ref{stabilityofsteady}.
We  then show how Floquet theory  can be used to reduce the
stability question in lattices of arbitrarily many oscillators
to the study of the spectrum of a matrix of fixed size (but depending
on a parameter -- the ``quasi-momentum''.)
Finally, in section \ref{example} we  implement this program to
rigorously construct an example of a spatially periodic
stable fixed state which remains stable for rings of arbitrary size.

\section{Numerical Experiments} \label{numerics}

Numerical experiments were performed on a Sun SuperSparc using the
numerical integration program XPP. Several different integration
methods were used to check the validity of the results. These included
 fixed and variable step Runge-Kutta methods of
varying order and the Gear method.

At very low values of the coupling constant $d_x$, the individual systems
in the ring behaved as if they were uncoupled.  As the coupling
strength was increased the behavior of the ring became more coherent.
Since the discrete Laplacian acts to synchronize
neighboring systems in the ring, these results agree with our
expectations.  As was emphasized in the introduction, the onset of
the coherent behavior occurred at coupling strengths which seemed to be
independent of the number of oscillators in the ring.
Some of the typical structures observed in these
numerical experiments are discussed below.

\subsection{Breathers}

If some of the systems in the ring are nearly stationary,
while others are undergoing oscillations we say that the oscillating systems
form a breather.  Since we are considering rings of
finite size, it is not expected that any single system in the ring is
stationary, unless the entire ring is stationary.

An example of a system with two breathers is shown
in figure~\ref{breath}.  It is interesting to note that in these states
each of the individual systems undergoes oscillations that are nearly
periodic, and that there is a strong spatial dependence on the motion.
Loosely speaking the motion of each individual system is trapped in a region
close to one of the `lobes'' of the Lorenz attractor.  No stable states in
which all systems in the ring are trapped in the vicinity of only
\emph{one}  ``lobe'' has been observed.
Similar breathers have been studied in coupled map lattices in
\cite{bunimovich}, while occurrences of nearly periodic behavior with
pronounced  spatial patterns in lattices of
chaotically bursting neurons have been observed numerically by
M.I. Rabinovich, et. al. \cite{rabinovich}.

Another interesting feature of breathers, which is also shared with
other types of solutions of this network, is that frequently
nonadjacent systems synchronize without
synchronizing with the oscillators that lie between them.
The synchrony is usually not exact, however this may be due to
numerical errors in the calculations.  Due to the many symmetries of
the system this sometimes seems to be due to the stability of an
invariant manifold corresponding to a partially synchronized state in the chain
as discussed \cite{josic}.  In other cases the state
does not appear to be symmetric, and it is unclear what the mechanism
behind this synchronization is.

It is important to note that some of the qualitative features of the
dynamics remain unaffected by the size, or the coupling strength in the
ring.  Figure \ref{timese} shows typical timeseries of the $x$
variable at four different coupling values in a ring of 32
oscillators.  Note the smooth envelope and regularity of the
oscillations, as well as the agreement of the timescales among the
different timeseries.  These general features remain unchanged as the
number of systems in the ring is varied, as long as the coupling is
below the synchronization threshold and above the value necessary for
the appearance of coherent behavior.

\subsection{Stable Stationary States}

In numerical experiments with rings of 8 to 52
systems stable steady states were observed for a variety of initial
conditions for $d_x > 10$.
Typical stable fixed states are shown in figure \ref{sensitive}.

For certain values of the coupling it appears that all
initial conditions lead to one of the stationary states of the
system.  The basins of attraction of the stationary states seem to be
intertwined in a complicated way in this case, as shown in figure
\ref{sensitive}.  The same is true for other stable states of the
system.  Small changes in the initial values of the parameters, and
changes in the integrating methods can result in very different
asymptotic behavior of the system.  This situation is similar to
that of attractors with riddled basins of attraction \cite{ashwin1},
and was also observed in numerical simulations of networks of
chaotically bursting neurons \cite{huerta}.

\subsection{Traveling Pulses}

Traveling pulses were observed in the ring only when $d_x \neq 0$ and
$d_y \neq 0$.  A typical traveling pulse is shown in figure
\ref{wave}.  The pulse  oscillates as it
propagates along the chain, and is thus a periodic, rather than a
fixed state in a moving coordinate frame. This can be seen in the timeseries
in figure \ref{wave}.

\section{A General Framework for the Study of Steady States}\label{steadystate}

System (\ref{coupledlorenz}) is a special case
of a lattice discrete or continuous time dynamical system
\begin{equation}\label{lattice}
(u_j)' = F(\{ u_j \}^s)
\end{equation}
where $j \in \Bbb{Z}^D$, $u_j \in \Bbb{R}^p$ and $\{u_j\}^s =
\{u_i : |i-j| \leq s\}$.  In the following discussion we will
use the convention that in $u_i^k$ the subscript $i \in \Bbb{Z}^D$
denotes the position in the lattice, while the superscript $1 \leq k \leq p$ denotes
the component of the vector $u_i$.  Such systems have been
studied by many authors \cite{AfChow}, \cite{BuSi}, \cite{Radu} as discrete
versions of partial differential equations of evolution type.

We will mainly be concerned with systems continuous in time  with $D=1$ and
$s=1$. These are simply chains of systems coupled to their nearest neighbors.
The special case of \emph{rings}, that is chains of finite size with periodic
boundary condition will be the focus of our attention.
A state in a ring of
$n$ systems corresponds to a state of period $n$ in the spatial variable in
an infinite chain.

A steady state of a chain given in (\ref{lattice}) is determined by
\begin{eqnarray}
F(\{u_j\}^s) = F(u_{j-1}, u_j, u_{j+1}) = 0 \label{continuous}\\
F(\{u_j\}^s) = F(u_{j-1}, u_j, u_{j+1}) = u_j\label{discrete}
\end{eqnarray}
in the continuous, respectively discrete case.  We will study systems
that satisfy the following condition:

{\bf Condition 1} If we write $F: \Bbb{R}^{3p} \rightarrow \Bbb{R}^p$
as $F(\chi, \eta, \zeta) = (F^1(\chi, \eta, \zeta),$ $  F^2(\chi, \eta, \zeta),$ $  \dots, $ $F^p(\chi, \eta, \zeta))$
then det $\!\left[ \frac{\partial F^i}{\partial \zeta^j}(\chi,\eta,\zeta) \right]_{i,j} \neq 0$.

By the Implicit Function Theorem in the case
of continuous time if det $\!\left[ \frac{\partial F^i}{\partial \zeta^j}(\chi,\eta,\zeta) \right]_{i,j}$ $ \neq 0$
for all values of $\chi,\eta,\zeta$ then either there exists a point $(a,b,c) \in \Bbb{R}^{3p}$ such that
$F(a,b,c) = 0$ and
therefore a function
$G(\eta,\chi)$ such that $F(\chi,\eta,G(\eta,\chi))=0$, or $F(\chi,\eta,\zeta) = 0$ has no solutions and there are no
fixed points of (\ref{lattice}).  If such a function $G$ exists and we define
$u_{j+1} = G(u_j, u_{j-1})$, $x_j = u_j$ and $y_j = u_{j-1}$ this leads to the following dynamical system
on $\Bbb{R}^{2p}$:
\begin{eqnarray}\label{fixed}
x_{j+1} = G(x_j , y_j) \\
y_{j+1} = x_j \nonumber
\end{eqnarray}
The steady states of (\ref{lattice}) are given by the $x$-coordinates
of the orbits of (\ref{fixed}).  In general, the function $G$ may not
be unique, in which case more than one system of the form
(\ref{fixed}) is needed to determine all the fixed points of the chain.
An equivalent argument holds in the case of
discrete time.

Since the function $G$ can assume any form, not much can be said about
such systems in general.  The subclass of chains of systems with the
following type of coupling is easier to analyze.
\begin{definition}
A nearest neighbor coupling of a chain of systems is said to
be \emph{symmetric} if  $F(u_{j-1}, u_j, $ $u_{j+1})$ $=  F(u_{j+1}, u_j, u_{j-1})$.
\end{definition}
This simply means that a system in the chain is coupled to its left
and right neighbor in the same way. For instance
the couplings $\Delta u_j =  u_{j-1} - 2 u_j + u_{j+1}$ and
$\Psi u_j =  u_{j-1} u_{j+1} + u_j$ are of this type.
The stationary states of symmetrically coupled chains are related to
the following class of
dynamical systems:
\begin{definition}\label{reversible}
Given a diffeomorphism
$\Phi:\Bbb{R}^{2p} \rightarrow \Bbb{R}^{2p}$ and an
involution $R: \Bbb{R}^{2p} \rightarrow \Bbb{R}^{2p}$ such that the
dimension of the fixed point set of $R$ is $p$ we say that
$\Phi$ is \emph{R-reversible} if
\begin{equation}
R \circ \Phi = \Phi^{-1} \circ R
\end{equation}
The dynamical system defined by
$x_{i+1} = \Phi(x_i)$ is also said to be R-reversible.
\end{definition}
The relation between symmetrically coupled chains and reversible systems is given in the following
\begin{theorem} \label{rever}
The fixed states of a symmetrically coupled chain of systems
satisfying condition 1 correspond to the orbits of a dynamical system
of the form (\ref{fixed}) which
is $R$-reversible and volume preserving.
For such a system $R(x,y) = (y,x)$, with $x,y \in \mathbb{R}^p$.
\end{theorem}

{\bf Proof:} The arguments for the continuous and discrete case are
virtually identical,
so only the first will be considered.  Equation (\ref{continuous}) together with the assumption that
the coupling is symmetric implies that
\begin{equation}
F(u_{j-1}, u_j, u_{j+1}) = F(u_{j+1}, u_j, u_{j-1})= 0
\end{equation}
Therefore this leads to the recursion equations
\begin{eqnarray}
u_{j+1} = G(u_j, u_{j-1}) & u_{j-1} = G(u_j, u_{j+1})
\end{eqnarray}
and the two dynamical systems
\begin{equation} \label{dyn}
\begin{aligned}
x_{j+1} &= G(x_j , y_j) \\
y_{j+1} &= x_j
\end{aligned} \qquad \qquad
\begin{aligned}
 z_{j-1} &= G(z_j , w_j)\\
 w_{j-1} &= z_j
\end{aligned}
\end{equation}
where $x_j = z_j = u_j$, $y_j = u_{j-1}$ and $w_j = u_{j+1}$.  The action of these
systems is shown schematically in figure \ref{action}.

Let $R(x,y) = (y,x)$ and $\Phi(x,y) = (G(x,y),x)$ so that the dynamical systems in (\ref{dyn}) is  generated
by $\Phi$. By the definition of this diffeomorphism
\begin{equation}
\begin{split}
R \circ \Phi \circ R \circ \Phi(x_i,y_i) & = R \circ \Phi (y_{i+1}, x_{i+1}) =
R \circ \Phi (z_i, w_i) \\
& = R (z_{i-1}, w_{i-1}) = R(y_i, x_i) = (x_i, y_i)
\end{split}
\end{equation}
Since $R$ is an involution with Fix$(R) = \{ (x,y) : x=y \}$ the diffeomorphism $\Phi$ and
the dynamical system it induces on the plane are $R$-reversible.

To prove that the map $\Phi$ is volume  preserving notice that
\begin{equation}
D\Phi =
\begin{bmatrix}
D_1G & D_2G \\
I & 0
\end{bmatrix}
\end{equation}
By the definition of the function G we know that $F(\chi,\eta,G(\eta,\chi))=0$ so that
differentiating with respect to $\chi$ leads to
\begin{equation}
D_1F + D_3F D_2G =0
\end{equation}
$D_3F$ is an invertible matrix by Condition 1, so that $D_2G = - D_1F
(D_3F)^{-1}$.  By assumption
the system is symmetric so that $D_1F =D_3F$ at all points in $\Bbb{R}^{3p}$ and hence
$D_2G = -I$. Since
\begin{equation}
\begin{bmatrix}
0 & I \\
-I & 0
\end{bmatrix}
\begin{bmatrix}
D_1G & -I \\
I & 0
\end{bmatrix} =
\begin{bmatrix}
I & 0 \\
-D_1G & I
\end{bmatrix}.
\end{equation}
and
\begin{equation}
\det \begin{bmatrix} 0 & I \\
-I & 0
\end{bmatrix} =
\det \begin{bmatrix}
I & 0 \\
-D_1G & I
\end{bmatrix}=1
\end{equation}
it follows that $\det D\Phi =1$ and the diffeomorphism $\Phi$ is
volume preserving.
$\diamondsuit$

$R$-reversible systems have many properties that facilitate their study.
We will make use of several of these summarized in the
following proposition adopted from \cite{Dev}

\renewcommand{\theenumi}{\alph{enumi}}

\begin{proposition}\label{devaney}
Let $R$ and $\Phi$ be as in Definition (\ref{reversible}).
\begin{enumerate}
\item If $p \in$ Fix($R$) and $\Phi^k(p) \in$ Fix($R$), then $\Phi^{2k}(p) = p$.
Such periodic points will be referred to as \emph{symmetric periodic points}.
\item Let $p \in$ Fix($R$) be a fixed point of $\Phi$, then $R(W^u(p)) = W^s(p)$
and $R(W^s(p)) = W^u(p)$ so that if $q \in W^u(p) \cap$ Fix($R$) then $q$ is
a homoclinic point.
\item If $p \in$ Fix($R$) is a fixed point of  $\Phi$ such that $W^u(p)$ intersects
Fix($R$) transversally at a point $q$, then there exist infinitely many symmetric
periodic points in any neighborhood of $p$.
\end{enumerate}
\end{proposition}

\section{Steady States In the Case of Discrete Laplacian Coupling} \label{section.steady}

For a chain of systems of the form $u' = f(u)$ coupled through a  discrete Laplacian $\Delta u_j =
(u_{j-1} - 2u_j + u_{j+1})$ the equation (\ref{lattice}) takes the form
\begin{equation}\label{lap}
(u_j)' = f(u_j) + d (u_{j-1} - 2u_j + u_{j+1})
\end{equation}
and the dynamical system (\ref{fixed}) takes the form
\begin{equation}\label{laplacefixed}
\begin{align}
x_{i+1} &= -\frac{1}{d} f(x_i) - y_i + (2 + \frac cd) x_i \\
y_{i+1} &= x_i \nonumber
\end{align}
\end{equation}
where $c=0$ in the case of continuous time and $c=1$ in the case of discrete time.
The following proposition follows immediately from these definitions
\begin{proposition}\label{lapfixed}
The fixed points of (\ref{laplacefixed}) are of the form $x_i = y_i = u_0$ where
$u_0$ is any solution of the equation $f(u) = 0$ in the continuous
time and $f(u)= u$ in the discrete time case.  Thus all fixed points of (\ref{laplacefixed})
lie on Fix$(R)= \{ (x,y) : x=y \}$ and are in 1-1 correspondence with the
steady states of an uncoupled system in the chain.
\end{proposition}

The fixed points of (\ref{fixed}) correspond to states of the
chain that are constant in space and time and proposition~\ref{lapfixed}
shows that the only such states $\{u_j\}_{j \in \Bbb{Z}}$ in the case of a discrete Laplacian coupling
are given by $u_j = u_0$ for all $j \in \Bbb{Z}$ where $u_0$ is
a fixed point of an uncoupled system of the chain $u' = f(u)$. Since the fixed points of (\ref{laplacefixed})
are $(u_0, u_0) \in$ Fix($R$)
this allows us to make use of proposition \ref{devaney}.c\footnote{The
chains under consideration are a special case of chains of the form
\begin{equation}\label{special}
(u_j)' = f(u_j) + g(u_{j-1}, u_j, u_{j+1}) \nonumber
\end{equation}
where $g(u_{j-1}, u_j, u_{j+1})$ is a symmetric coupling vanishing
 on the linear submanifold
of $\Bbb{R}^{3p}$ defined by $u_{j-1} = u_j = u_{j+1} =0$.  Such
 systems can be analyzed using the approach of this section} .

Moreover we can apply Theorem~\ref{rever} directly in this case to conclude that
the map defining the fixed points is $R$-reversible and volume preserving.

{\bf Remark:} It is interesting to note that if the fixed states
of a chain of one dimensional systems  are determined by a dynamical system
of the form (\ref{laplacefixed})
on $\Bbb{R}^2$ and $(x_0, y_0)$ is a fixed point of (\ref{laplacefixed}) then
the discriminant of the characteristic polynomial of $D\Phi(x_0,y_0)$ is
\begin{equation}
\sqrt{(-\frac{1}{d}f'(x_0)+ 2+c/d)^2 -4}.
\end{equation}
Since det~$D \Phi (x_0,y_0) =1$ we can conclude the following:
\begin{enumerate}
\item In the case of discrete time the eigenvalues of $D \Phi (x_0,y_0)$ are
on the unit circle when $f'(x_0) > 1$ and $d>(f'(x_0)-1)/4)$ or $f'(x_0) <1$ and $d<(f'(x_0)-1)/4)$.
\item In the case of continuous time, the eigenvalues of $D \Phi (x_0,y_0)$ are
on the unit circle for sufficiently large \emph{positive} $d$ when $f'(x_0) >0$ and
for sufficiently large \emph{negative} $d$ when $f'(x_0) <0$,
\end{enumerate}
By a theorem of Birkhoff (see for example \cite{moser})
since $\Phi$ is area preserving, the dynamical system
(\ref{laplacefixed}) will generically have
infinitely many periodic orbits in any neighborhood of $(x_0, y_0)$.  Any such
periodic orbit will correspond to a fixed state of (\ref{lap}) which is periodic
in space, i.e. in the variable $j$.

The condition  det $D \Phi (x_0,y_0) =1$ is not sufficient to guarantee that
$D \Phi (x_0,y_0)$ will have eigenvalues on the unit circle if  (\ref{laplacefixed})
is not a dynamical system on the plane.  Additional information needs to be considered
to conclude the existence of such periodic orbits in this case.
\vspace{.5cm}

{\bf Example 1:} Consider a chain of tent maps coupled through a
discrete Laplacian described by the equation $u_j' = f(u_j) + \Delta u_j$ with
\begin{equation}
f(u_j) = \begin{cases}
         2 u_j & \text{if $u_j < 1/2$}, \\
         -2 u_j +2 & \text{if $u_j \geq 1/2$}.
         \end{cases}
\end{equation}
This is a discrete dynamical system with fixed states determined by
a homeomorphism of the plane given in (\ref{laplacefixed})
with $c=1$.

The only fixed points of (\ref{laplacefixed}) are
(0,0) and (2/3, 2/3) by proposition~\ref{lapfixed}. Since in this case the dynamical system (\ref{laplacefixed}) is linear in each
of the two halves of the plane $x < 1/2$ and $x \geq 1/2$ the analysis of the dynamics
around the fixed points is straightforward.  A direct calculation
shows that in the region $x < 1/2$ there is a family of invariant
ellipses whose major axis lies on the diagonal $D=\{(x,y) \in
\mathbb{R}^2: x=y \}$ and on which the action of (\ref{laplacefixed}) is a rotation.  As
$d$ is increased, these ellipses become more eccentric.

The fixed states of the chain corresponding to these orbits are not
dynamically stable.
If $\hat{u}=\{\hat{u}_j\}_{j \in \Bbb{Z}}$ is a fixed state in the chain such that
$\hat{u}_j < 1/2$ for all $j$ then around this state the dynamics of the
chain are described by:
\begin{equation}
u_j' = 2u_j + d \Delta u_j
\end{equation}
The spectrum of the operator $2 + d \Delta$ is the interval
$[2-4d, 2]$ and so  $\hat{u}$ cannot be stable, and would not be
observed in numerical experiments.

Besides these invariant ellipses, this system will also typically have
infinitely many periodic orbits, and will exhibit complicated
behavior, as the following argument shows.
A simple calculation shows that the point $p = (2/3, 2/3)$ is hyperbolic
for all values $d>0$.  Since the dynamics
is piecewise linear the  stable and unstable manifold can be calculated
explicitly.  $W^u(p)$ will consist of a ray contained
in the half plane $x>1/2$ and a second more complicated part which is
constructed as follows.  The first section of $W^u(p)$ is a line $A$,
as shown in figure \ref{ellman}.
The part of $A$ contained in the half plane $x<1/2$ is
rotated along the invariant ellipses
around the origin to the line $A'$ under forward iteration.
The subsequent images  of $A'$ are lines that are rotated further.
Therefore $f^{n_0}(A')$ intersects the diagonal $D$ for some $n_0$.
Whenever the angle of this intersection is not a right angle
$W^u(p)$ will intersect $W^s(p)$ transversely since by
proposition \ref{devaney}.b the stable manifold is the reflection
of the unstable manifold through the diagonal $W^u(p)$.  This shows
that complicated dynamics of (\ref{laplacefixed})
can be expected and implies the existence of infinitely many periodic
as well as spatially chaotic states in the chain of tent maps.

{\bf Example 2:}  Next we consider the chain of diffusively coupled
Lorenz system (\ref{eq1}) with $d_y = d_z = 0$.  This system does not
satisfy condition 1, however after setting the right--hand side
of equations (\ref{eq1}) to 0 and
using the second and third equation in (\ref{eq1})
to eliminate the
variables $y_i$ and $z_i$  the following equation for a fixed
state is obtained
\begin{align}
\sigma (\hat{x}_i - \frac{\beta r \hat{x}_i}{\beta + (\hat{x}_i)^2}) -
&  d  (\hat{x}_{i-1}
- 2 \hat{x}_i + \hat{x}_{i+1})  = 0 \label{steady} \\
\hat{y}_i = & \frac{\beta r \hat{x}_i}{\beta + (\hat{x}_i)^2}  \\
\hat{z}_i = & \frac{r (\hat{x}_i)^2}{\beta + (\hat{x}_i)^2}.
\end{align}
The function on the left hand side of equation (\ref{steady})
satisfies condition 1, and we can proceed as in the previous example.

Equation (\ref{steady}) defines the following dynamical system on
the plane
\begin{equation} \label{lorenzfixed}
\begin{bmatrix}
x_{i+1} \\  y_{i+1}
\end{bmatrix} = F \left(
\begin{bmatrix}
x_i \\ y_i
\end{bmatrix} \right) =
\begin{bmatrix}
\frac{\sigma}{d} (x_i -  \frac{\beta r x_i}{\beta + (x_i)^2}) + 2 x_i - y_i \\
x_i
\end{bmatrix}
\end{equation}
Orbits of period $n$ of this system are in 1--1 correspondence with
the  steady states in the ring of $n$
Lorenz systems. The two fixed points of (\ref{lorenzfixed}) are (0,0)
and $(\pm \sqrt{\beta (r-1)},\pm \sqrt{\beta (r-1)}) = (\pm
8.32\bar{6},\pm 8.32\bar{6})$ which both lie on
lie on Fix($R$).

The following definition from \cite{Dev} will be used in the remainder
of the argument.

\begin{definition}
A compact region $H$ in the plane is called \emph{overflowing} in the
$y$-direction for a function $F$ if the image of any point $(x_0, y_0) \in$ int($H$)
lies strictly above the line $y=y_0$.
\end{definition}

Notice that any point in the interior of $H$ either leaves $H$ or is asymptotic
to a periodic point on the boundary of $H$.

\begin{theorem}
The map (\ref{lorenzfixed}) has a homoclinic point and infinitely many
periodic points for all $0<d \leq 20$.
\end{theorem}

{\bf Proof:}  Let $p$ denote the fixed point $(-\sqrt{ \beta (r-1)},$ $-\sqrt{ \beta (r-1)})$
of (\ref{lorenzfixed}), $G$ the ray $y = -\sqrt{ \beta (r-1)}$ with $x > -\sqrt{ \beta (r-1)}$,
and $D = \{(x,y): x=y\}$ the diagonal in $\Bbb{R}^2$.  The image of the
triangular region $W$ bounded by $D$ and $G$ lies between the
two cubics
\begin{eqnarray}
F(D) = \{(x,y): x = \frac{\sigma}{d} (y - \frac{\beta r y}{\beta + y^2}) + y\} \\
F(G) = \{(x,y): x = \frac{\sigma}{d} (y - \frac{\beta r y}{\beta + y^2}) + 2 y  + \sqrt{ \beta (r-1)} \}
\end{eqnarray}
Consider the region $F(W) \cap W = H$ depicted in figure \ref{wedge2}.
Let $(x_0, y_0) \in$ int($H$)
and consider the vertical line segment $l$ passing through $(x_0, y_0)$ and connecting $G$
and $D$. The image of $l$ is a horizontal line segment connecting $F(G)$ and
$F(D)$ contained in the line $y = x_0$.  Any point of this line segment
lies above the line $y = y_0$.
Therefore $H$ is overflowing in the $y$ direction.

Since int($F(H) -$ int($H$)) and int($H$) lie on opposite sides of $D$, any
point in $H$ must have an iterate which either lies on $D$ or crosses
$D$. A direct calculation shows that one branch of $W^u(p)$ enters
$H$. Since $H$ is overflowing in the $y$ direction, $W^u(p)$ must either
cross $D$ or be asymptotic to the fixed point $(0,0)$ creating
a saddle-saddle connection.  The second
possibility can be excluded as follows:

The eigenvector of the linearization of F around (0,0) corresponding to the stable direction
is $\mathbf{v}_1= (1, -\frac{d}{d-130+ 2 \sqrt{65^2 -d}})$.  The vector $\mathbf{v}_2=(1, \frac{d-260}{d})$ is tangent
to $F(D)$ at (0,0).  A direct computation shows that for $0<d<20$ the vector $\mathbf{v}_1$ points to the left of
$\mathbf{v}_2$ and so the tangent to $W^s(0,0)$ at (0,0) does not
point into $H$.  This situation is depicted schematically in figure
\ref{vectors} and excludes the
possibility that $W^u(p)$ is asymptotic to (0,0).

By proposition~\ref{devaney}.b the fact that $W^u(p)$ meets $D$ implies the existence of a homoclinic
point.

By proposition \ref{devaney}.c in the case of $R$-reversible
systems an infinite number of periodic points exists whenever $W^u(p)$ crosses
$D$ transversely at a homoclinic point. If this point of intersection
is not transverse then a transverse intersection of $W^u(p)$
and $D$ can be produced nearby by an argument given in
\cite[p. 261]{Dev}. $\diamondsuit$

Numerical investigations suggest that the theorem remains true for
arbitrarily large values of d.

This theorem shows that rings of Lorenz systems coupled through the
first variable can be expected to have many periodic stationary
states. Numerical computations of  $W^u(p)$ suggest that even more
complicated stationary states can be expected (see figure
\ref{stablemanofp}).  However
it is unclear whether any of them are stable.
Numerical investigations show that not only are they stable, but
their basins of attraction occupy a large portion of phase space.
We next address the problem of stability of these fixed states.

\section{Stability of Steady States in a Ring} \label{stabilityofsteady}

The study of stable fixed states in a chain of oscillators is not new.
They are described in \cite{erm}, \cite{aron} as instances of oscillator
death in a chain of neural oscillators, in \cite{nekor} as synchronous
states of phase lock loops, and  conditions for the stability
of complex stationary states in general lattices are given in
\cite{AfrChow2}.  The situation presented here is different in
that an explicit periodic stationary state is analyzed in a case where
the methods from the theory of parabolic partial differential
equations used in  \cite{erm} are not applicable, and the conditions
proposed in \cite{AfrChow2} cannot be verified.  In addition, since
the coupling strength is neither very large nor very small, there is no
obvious perturbative approach to the problem.

Even if some of the fixed states in a ring of $n$ systems are
stable it can not be concluded that stable equilibria exist in
arbitrarily long rings of such systems. If a stable fixed
state is viewed as a special case of synchronization, one might
guess from results in Afraimovich and Lin \cite{lin} that the
coupling necessary to have stable steady states in longer rings
increases as a power of the size of the ring.  It has already been
argued that this is not realistic from a physical viewpoint, and we
will show below that it is not the case.

In the following sections we will concentrate on the specific example of
a ring of Lorenz systems coupled in the $x$ variable discussed in example
2 of the last section.  However the ideas presented can be applied
to any chain of symmetrically coupled systems for which condition 1 holds.

The linearization of each of the Lorenz equations
 around a point $\mathbf{\hat{v}_i} = (\hat{x}_i, \hat{y}_i, \hat{z}_i )$ when
$d = 0$ is
\begin{equation}\label{linlorenz}
D_{\mathbf{\hat{v}_i}}f = \begin{bmatrix}
-\sigma & \sigma & 0 \\
r - \hat{z}_i & -1 & - \hat{x}_i \\
\hat{y}_i & \hat{x}_i & - b
\end{bmatrix}
\end{equation}
and hence linearizing the entire ring around the steady state
$\hat{\mathbf{v}} = ( {\mathbf{\hat{v}_i}} )_{i=1}^n = (\hat{x}_1,
\hat{y}_1, \hat{z}_1, \dots, $ $\hat{x}_n, $ $\hat{y}_n, $ $\hat{z}_n
)$ leads to the $3n \times 3n$ matrix
\begin{equation}
D_{\mathbf{\hat{v}}}F = \begin{bmatrix}
D_{\mathbf{\hat{v}_1}}f - 2\Gamma & \Gamma & 0 & \dots & 0 & \Gamma \\
\Gamma  & D_{\mathbf{\hat{v}_2}}f - 2\Gamma& \Gamma & \dots & 0 & 0 \\
\hdotsfor[3]{6}                                   \\
\Gamma   & 0 & 0 & \dots &  \Gamma & D_{\mathbf{\hat{v}_n}}f -2\Gamma
\end{bmatrix}
\end{equation}
where the matrix $\Gamma$ is defined as:
\begin{equation}
\Gamma = \begin{bmatrix}
d & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}
\end{equation}

It is in general not possible to compute the eigenvalues of this
matrix analytically.
As mentioned above, we are specifically interested in coherent states
in networks with arbitrarily many oscillators.  Given a periodic
stationary state in a chain of $n$ oscillators, an easy way to obtain
a periodic stationary state in arbitrarily long chains is to repeat
this state.
In particular, if $\mathbf{\hat{v}}$
is a fixed state in a ring of $n$ oscillators
then repeating these values $K$ times to obtain
$\mathbf{\hat{v}}^{(K)} = \{\mathbf{\hat{v}}, \mathbf{\hat{v}}, \dots, \mathbf{\hat{v}} \}$ produces a steady
state in a chain of $Kn$ oscillators.  We refer to such a state
as the $K$-{\em multiple} of the steady state $\mathbf{\hat{v}}$. An
example of a stationary state and its 4-multiple is given in figure
\ref{kmultiples}.
 We next
derive conditions under which all $K$-multiples of a stable steady
state are themselves stable and demonstrate that these conditions hold in a
particular case.

The  linearization
around the steady state $\mathbf{\hat{v}}^{(K)}$ leads to the following stability
matrix:
\begin{equation*}
D_{\mathbf{\hat{v}}^{(K)}}F = \begin{bmatrix}
M & E_d & 0 & \dots & 0 & E_u \\
E_u & M & E_d & \dots & 0 & 0 \\
\hdotsfor[3]{6}                                   \\
E_d & 0 & 0 & \dots & E_u & M
\end{bmatrix}
\end{equation*}
where
\begin{equation*}
M  = \begin{bmatrix}
D_{\mathbf{\hat{v}_1}}f - 2\Gamma & \Gamma  & 0 & \dots & 0 & 0 \\
\Gamma  & D_{\mathbf{\hat{v}_2}}f - 2\Gamma & \Gamma & \dots & 0 & 0 \\
\hdotsfor[3]{6}                                   \\
0   & 0 & 0 & \dots &  \Gamma & D_{\mathbf{\hat{v}_n}}f -2\Gamma
\end{bmatrix}
\end{equation*}
and $E$ is an $3n \times 3n$ matrix of the form
\begin{align*}
E_u = \begin{bmatrix}
0 & \dots & \Gamma \\
\hdotsfor[3]{3} \\
0 & \dots & 0
\end{bmatrix} \qquad  &
E_d= \begin{bmatrix}
0 & \dots & 0\\
\hdotsfor[3]{3} \\
\Gamma & \dots & 0
\end{bmatrix}
\end{align*}

In general it is impossible to compute the spectrum of this $3Kn
\times 3Kn$ matrix.  However in the present instance
Bloch wave theory,
provides a way of expressing the eigenfunctions of a larger system
with a periodic dependency as the eigenfunctions
of a smaller system.  As usual, this reduction is accompanied by the
introduction of an additional parameter into the equations.
Here the Bloch wave  approach will be used to reduce the problem of computing the
eigenvalues of the $3Kn \times 3Kn$ matrix $D_{\mathbf{\hat{v}}^{(K)}}F$ for arbitrary $K$ to the computation of
the eigenvalues of an $3n \times 3n$ matrix dependent on a parameter $t$.

Let $\mathbf{e} = (\Psi(j), \Phi(j), \eta(j) )_{j = 1}^{Kn} = (\Psi(1), \Phi(1), \eta(1), \dots, \Psi(Kn), \Phi(Kn), \eta(Kn)) $
 where
\begin{eqnarray}
\Psi(j) = \exp \left( \frac{2 \pi i q}{nK} j \right)  \tilde{\Psi}_q (j) \nonumber \\
\Phi(j) = \exp \left( \frac{2 \pi i q}{nK} j \right)  \tilde{\Phi}_q (j) \\
\eta(j) = \exp \left( \frac{2 \pi i q}{nK} j \right)  \tilde{\eta}_q (j) \nonumber
\end{eqnarray}
for $1 \leq q \leq Kn$ and $\tilde{\Psi}_q, \tilde{\Phi}_q,\tilde{\eta}_q$
are assumed to be $n$-periodic. We will show that the vector $\mathbf{e}$ is an eigenvector of
$D_{\mathbf{\hat{v}}^{(K)}}$ whenever the vector
\begin{equation} \label{mathi}
\mathbf{i} = (\tilde{\Psi}(j), \tilde{\Phi}(j), \tilde{\eta}(j) )_{j = 1}^n
\end{equation}
is an eigenvector of a particular  $3n \times 3n$ matrix
derived from $D_{\mathbf{\hat{v}}^{(K)}}F$.
 A direct calculation leads to:
\begin{align}\label{bloch}
\left[D_{\mathbf{\hat{v}}^{(K)}}F\right] \mathbf{e} \, (3j) = &
e^{ \frac{2 \pi i q}{nK} j }  ( d e^{ \frac{2 \pi i q}{nK} }
 \tilde{\Psi}_q (j+1 \!\!\!\!\!\mod n) + \nonumber
de^{- \frac{2 \pi i q}{nK} } \tilde{\Psi}_q (j-1 \!\!\!\!\!\mod n) - \\
& 2 \tilde{\Psi}_q (j) + v_1(j) \tilde{\Psi}_q (j) + w_1(j) \tilde{\Phi}_q (j) +
u_1(j) \tilde{\eta}_q (j) ) \nonumber \\
\left[D_{\mathbf{\hat{v}}^{(K)}}F\right] \mathbf{e} \, (3j+1)= &
e^{ \frac{2 \pi i q}{nK} j } (v_2(j) \tilde{\Psi}_q (j) + w_2(j) \tilde{\Phi}_q (j) +
u_2(j) \tilde{\eta}_q (j))\\
\left[D_{\mathbf{\hat{v}}^{(K)}}F\right] \mathbf{e} \, (3j+2)= &
e^{ \frac{2 \pi i q}{nK} j } (v_3(j) \tilde{\Psi}_q (j) + w_3(j) \tilde{\Phi}_q (j) +
u_3(j) \tilde{\eta}_q (j)) \nonumber
\end{align}
for $1 \leq j \leq Kn$.  Here $v_i(j), w_i(j), u_i(j)$ are given by equation (\ref{linlorenz}) as
\begin{align*}
\begin{bmatrix}
v_1(j) &  w_1(j) & u_1(j)\\
v_2(j) &  w_2(j) & u_2(j)\\
v_3(j) &  w_3(j) & u_3(j)
\end{bmatrix}  \quad = \quad &
\begin{bmatrix}
-\sigma & \sigma & 0 \\
r - \hat{z}_{j \!\!\!\!\! \mod n} & -1 & - \hat{x}_{j \!\!\!\!\!\mod n} \\
\hat{y}_{j \!\!\!\!\!\mod n} & \hat{x}_{j \!\!\!\!\!\mod n} & - b
\end{bmatrix}
\end{align*}
and are thus $n$-periodic.

We want to check when $\mathbf{e}$ is an eigenvector of $D_{\mathbf{\hat{v}}^{(K)}}F$
and so we set $D_{\mathbf{\hat{v}}^{(K)}}F \mathbf{e} = \lambda \mathbf{e}$.  Using the
expressions obtained in
(\ref{bloch}) we get
\begin{equation}\label{bloch2}
\begin{split}
 & \exp \left( \frac{2 \pi i q}{nK} j \right) \biggl( d e^{ \frac{2 \pi i q}{nK} }
 \tilde{\Psi}_q (j+1 \!\!\!\!\!\mod n) +
d e^{- \frac{2 \pi i q}{nK} } \tilde{\Psi}_q (j-1 \!\!\!\!\!\mod n) - \\
& 2 \tilde{\Psi}_q (j) + v_1(j) \tilde{\Psi}_q (j) + w_1(j) \tilde{\Phi}_q (j) +
u_1(j) \tilde{\eta}_q (j), \;\;
v_2(j) \tilde{\Psi}_q (j) + w_2(j) \tilde{\Phi}_q (j) +
u_2(j) \tilde{\eta}_q (j), \\
& v_3(j) \tilde{\Psi}_q (j) + w_3(j) \tilde{\Phi}_q (j) +
u_3(j) \tilde{\eta}_q (j) \biggr)_{j = 1}^{Kn} =
\lambda \exp \left( \frac{2 \pi i q}{nK} j  \right) \biggl(
\tilde{\Psi}_q(j),
\tilde{\Phi}_q(j), \tilde{\eta}_q(j) \biggr)_{j = 1}^{Kn}.
\end{split}
\end{equation}

Since all the functions on the right hand side of (\ref{bloch2}) are
$n$-periodic we can express equation (\ref{bloch2}) as $\mathcal{D}(q)
\mathbf{i} = \lambda \mathbf{i}$ where $\mathbf{i}$ is defined in
equation (\ref{mathi}) and
\begin{equation}
\mathcal{D}(q) = \begin{bmatrix}\label{dq}
D_{\mathbf{\hat{v}_1}}f - 2\Gamma & E(q)^+ & 0 & \dots & 0 & E(q)^- \\
E(q)^-  & D_{\mathbf{\hat{v}_2}}f  - 2\Gamma& E(q)^+ & \dots & 0 & 0 \\
\hdotsfor[3]{6}                                   \\
E(q)^+   & 0 & 0 & \dots &  E(q)^- & D_{\mathbf{\hat{v}_n}}f -2\Gamma
\end{bmatrix}
\end{equation}
with the matrices $E(q)^{\pm}$ defined as:
\begin{equation}
E(q)^{\pm} = \begin{bmatrix}
d e^{ \pm \frac{2 \pi i q}{nK}} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}
\end{equation}
This implies that the  eigenvalues of $D_{\mathbf{\hat{v}}^{(K)}}F$
are the same as the eigenvalues of the $3n \times 3n$ matrices $\mathcal{D}(q)$ for
$1 \leq q \leq Kn$. The $3n \times 3n$ matrix $\mathcal{D}(q)$ is  easier to handle than
the $3Kn \times 3Kn$ matrix $D_{\mathbf{\hat{v}}^{(K)}}F$.

To conclude that any $K$-multiple of a stable fixed state
$\mathbf{v}_0$ of a ring
of $n$ oscillator is stable it is sufficient to show that the eigenvalues
of the matrix $\mathcal{D}(q)$ have negative real part for all values of
$q$ and $K$.
To simplify the calculations we can replace the argument $q \in \Bbb{Z}$
by a continuous parameter $t \in \Bbb{R}$ by replacing the matrices
$E(q)^{\pm}$ with the matrices
\begin{equation*}
E_t^{\pm} = \begin{bmatrix}
d e^{ \pm it} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}
\end{equation*}
in the expression for $\mathcal{D}$.  If we can show that $\mathcal{D}(t)$
has eigenvalues with negative real part for all $t \in \Bbb{R}$ then the same
is true for $\mathcal{D}(q)$ for any values of $K$ and $q$ in $\Bbb{Z}$.

The following lemma about the class of matrices of type (\ref{dq})
will be used in the next section.

\begin{lemma} \label{char}
Let $E(t)^{\pm}$ be as in the previous lemma.
The characteristic polynomial of a $Km \times Km$  matrix
\begin{equation*}
B = \begin{bmatrix}
M_1  & E(t)^+ & 0 & \dots & 0& 0 & E(t)^- \\
E(t)^-  & M_2 & E(t)^+ & \dots 0& & 0 & 0 \\
\hdotsfor[3]{6}  \\
 0 & 0 & 0 & \dots &  E(t)^- & M_{K-1} &  E(t)^+   \\
E(t)^+   & 0 & 0 & \dots & 0 & E(t)^- & M_K
\end{bmatrix}
\end{equation*}
is of the form
\begin{equation}\label{charpoly}
p_M(t, \lambda) = \sum_{i=0}^{(K-1)m} (\alpha_i + \beta_i \cos Kt)  \lambda^i
+ \sum_{i=(K-1)m+1}^{Km} \alpha_i \lambda^i .
\end{equation}
\end{lemma}

A proof of this lemma is given in the appendix.

\section{An Example} \label{example}

This section gives a computer assisted proof that any $K$-multiple
of a particular stable fixed state in a ring of four Lorenz system is
stable.  The proof involves the use of interval arithmetic and the
following theorems  about the convergence of the
Newton-Raphson-Kantorovich method.  The computations were performed
using Mathematica's implementation of interval arithmetic.

\begin{theorem} \label{newton}
Let $f(z)$ be a complex analytic function and
assume $f(z_0) f'(z_0) \neq 0$ for some $z_0$.  Define $h_0 = -f(z_0)/f'(z_0)$,
the disc $K_0 = \{z : |z - z_0| \leq |h_0|\}$ and
$M = \max_{K_0} |f''(z)|$.  If $2|h_0| M \leq |f'(z_0)|$ then
there is exactly one root of $f$ in the closed disc $K_0$.
\end{theorem}

\begin{theorem}[Kantorovich's Convergence Theorem] \label{kantorovich}
Given a twice differentiable
function $f : \Bbb{R}^n \rightarrow \Bbb{R}^n$ which is nonsingular
at a point $x_0 = ( x_1^0, x_2^0, \dots, x_n^0 )$ let
 $[\Gamma_{ik}] = [\frac{\partial f_i}{\partial x_k}(x_0)]^{-1}$.
 Let $A, B, C$ be positive real numbers such that
\begin{align}
\max_i \sum_{k=1}^{n} |\Gamma_{ik}| \leq A \qquad &
\max_i \sum_{k=1}^{n} |\Gamma_{ik} f_k(x_0)| \leq B \qquad
C \leq \frac{1}{2 A B}.
\end{align}
Define the region $R = \{x \in \Bbb{R}^n : \max_i |x_i - x_i^0|
\leq (AC)^{-1}(1 - \sqrt{1-2ABC}) \}$.  If
\begin{align}
\max_{x \in R} \sum_{j=1}^{n} \sum_{k=1}^{n} \left|
\frac{\partial^2 f_i}{ \partial x_j \partial x_k}
\right| \leq C \qquad & i = 1, \dots,n
\end{align}
then the equation $f(x) = 0$ has a solution in $R$.
\end{theorem}

The argument will proceed as follows: First a steady state of the ring
is found
using interval arithmetic.  Next it is shown that the characteristic polynomial
$p_{\mathcal{D}}(\lambda, t)$ of the matrix $\mathcal{D}(q)$ in
(\ref{dq}) corresponding to this steady state has roots with
negative real part for $t = \frac{\pi}{4}$.  Finally it is proved that
$p_{\mathcal{D}}(\lambda, t)$ does not have roots on the imaginary
axis for any $t \in \mathbb{R}$.
Since the roots of a polynomial depend continuously on its
coefficients this implies that the roots of $p_{\mathcal{D}}(\lambda,
t)$ cannot cross into right half of $\mathbb{C}$.  By the results of
section \ref{stabilityofsteady} this implies that all $K$-multiples of
the fixed state under consideration must be stable.

In the following intervals of numbers will be denoted with overbars to distinguish
them from real numbers, and make the notation less cumbersome.
For instance $\bar{\lambda}$ denotes
an interval and the interval  $[3.16524, 3.16533]$ is denoted  $\overline{3.165}$.

Numerical investigations with the program XPP
show that a stable state in a chain of four
oscillators coupled with $d=1.5$ occurs close to
$ \hat{\mathbf{x}}= (\hat{x}_1,\hat{x}_2, \hat{x}_3, \hat{x}_4 )$ =
(-6.4114408, 7.3696656, 8.1984129, 7.3696656). Notice that
the orbit of (\ref{lorenzfixed}) describing this state is not symmetric
in the sense of proposition \ref{devaney} although it is mapped into
itself under $R$.  This numerically obtained solution is used as an
initial guess in Mathematica's implementation of Newton's method
to determine the roots $x^i_a$ of the set of equations
\begin{equation} \label{stachain}
\sigma(x_i - \frac{\beta r x_i}{\beta+ (x_i)^2}) - 1.5 ( x_{(i-1)
\!\!\!\!\!\mod 4 +1} - 2 x_i + x_{(i+1)
\!\!\!\!\!\mod 4 +1}) \qquad i = 1,2,3,4.
\end{equation}
Although Mathematica can be instructed to find roots to an arbitrary
precision, so far only floating point arithmetic was used so
that none of the results are rigorous yet.

At this point all the quantities in the computations are redefined as
intervals rather than floating point numbers and using interval
arithmetic and theorem
\ref{kantorovich} we find an interval around
each $x^i_a$ in which the roots of equation (\ref{stachain})
must lie.  These bounds are now rigorous.

In section \ref{stabilityofsteady}  it was shown that
a fixed state and all of its $K$-multiples are stable if
the polynomial $p_{\mathcal{D}}(\lambda, t)$ corresponding to that
state
has roots with
negative real part for all $t \in [0, \pi/2]$.  Using interval
arithmetic it is shown that
\begin{equation}
\begin{split}
\bar{p}_{\mathcal{D}}(\lambda, t)= &
\overline{3.61524}\times{{10}^{12}} - \overline{5.02753}\times10^7 \cos 4t +
(\overline{9.49583}\times10^{11} - \overline{1.33199}\times10^7 \cos 4t ) \,\lambda + \\
& (\overline{2.67802}\times10^{11} - \overline{4.94338}\times10^6 \cos 4t ) \,\lambda^2+
(\overline{4.90382}\times10^{10} - \overline{771725} \cos 4t) \,\lambda^3 + \\
& (\overline{7.551}\times10^9 - \overline{144880} \cos 4t) \, \lambda^4 +
(\overline{9.81388}\times10^8 - \overline{13673.3} \cos 4t) \lambda^5 + \\
& (\overline{1.03403}\times10^8 - \overline{1560.66} \cos 4t) \lambda^6 +
(\overline{9.39525}\times10^6 - \overline{74.25} \cos 4t) \lambda^7 + \\
& (\overline{703005} - \overline{5.0625} \cos 4t) \lambda^8 +
\overline{42026.7}\,\lambda^9 + \overline{2023.92}\,\lambda^{10} +
  \overline{66.6667}\lambda^{11} + \lambda^{12}
\end{split}
\end{equation}
for the fixed state under consideration.

In the next step the roots of the polynomial
$\bar{p}_{\mathcal{D}}(\lambda, \frac{\pi}{2})$ are found.
To employ Mathematica's implementation of Newton's method we need
a polynomial with coefficients that are floating point numbers rather
than intervals. Floating point numbers inside the intervals which
determine the coefficients of the
polynomial $\bar{p}_{\mathcal{D}}(\lambda, \frac{\pi}{2})$ can be chosen
to define a polynomial
$p^a_{\mathcal{D}}(\lambda, \frac{\pi}{2})$
which approximates $p_{\mathcal{D}}(\lambda, \frac{\pi}{2})$.
The roots of $\{r^a_i\}_{i=1}^{12}$ of
$p^a_{\mathcal{D}}(\lambda,\frac{\pi}{2} )$ are now found using
Newton's method.

The complex intervals\footnote{There are several
ways in which complex intervals can be defined.  In this case we use regions of the
form $\bar{z} = \bar{x} + i \bar{y}$ where $\bar{x}$ and $\bar{y}$ are real intervals.
These are simply rectangular regions in $\Bbb{C}$.} containing the roots of
$\bar{p}_{\mathcal{D}}(\lambda,\frac{\pi}{2} )$ are found by
using theorem \ref{newton}.  We
set $z_0$ equal to complex intervals around $r^a_i$ and use complex
interval arithmetic to find the radius $K_0$ and check the conditions
of the theorem for each root  $r^a_i$.
The 4 real roots and 4 complex pairs
obtained by this procedure are given in the table below.

\vspace{5mm}
\centerline{
\begin{tabular}{||c|c||} \hline
$\overline{-18.481}$ & $\overline{-0.2463} \pm i \overline{9.769}$  \\ \hline
$\overline{-17.173}$ & $\overline{-0.1961} \pm i \overline{9.317}$ \\ \hline
$\overline{-15.437}$ & $\overline{-0.1345} \pm i \overline{9.164}$ \\ \hline
$\overline{-14.238}$ & $\overline{-0.09054} \pm i \overline{8.622}$ \\ \hline
\end{tabular}
}
\vspace{5mm}

Since interval arithmetic was used in these calculations, these
estimates are now rigorous.
The remainder of the argument shows that these roots will not cross the imaginary
axis as $t$ varies.

By lemma \ref{char} the characteristic polynomial $p_{\mathcal{D}}(\lambda, t)$ takes
the following form when evaluated on the imaginary axis:
\begin{equation}
\begin{split}
 p_{\mathcal{D}}(i\mu, t) & = \sum_{j=0, j \text{ even}}^{12} (\alpha_j + \beta_j \cos 4t)
\mu^j (-1)^{\frac{j}{2}} + i \sum_{j=1, j \text{ odd}}^{11} (\alpha_j + \beta_j \cos 4t)
\mu^j (-1)^{\frac{j-1}{2}} = \\
& = p_{\mathcal{D}}^{\text{R}}(\mu, t) + i p_{\mathcal{D}}^{\text{I}}(\mu, t) \nonumber
\end{split}
\end{equation}
Since the roots of $p_{\mathcal{D}}(\lambda, t)$ depend continuously on the parameter $t$,
if $p_{\mathcal{D}}(\lambda, t)$ has roots with positive real part for some $t_1$ then
there must exist a $t_0$ such that $p_{\mathcal{D}}(\lambda, t_0)$ has a root on the
imaginary axis.  In other words there must exist a $\mu_0$ such that
\begin{equation} \label{inequality}
p_{\mathcal{D}}^{\text{R}}(\mu_0, t_0) =p_{\mathcal{D}}^{\text{I}}(\mu_0, t_0)=0.
\end{equation}
We will use interval arithmetic to show that this cannot happen. The polynomial
$p_{\mathcal{D}}(\lambda, t)$ is split  into a real and complex part to avoid
using complex interval arithmetic in the numerical calculations since complex
interval arithmetic leads to much rougher estimates than real interval arithmetic.

By Gre\v{s}gorin's theorem the eigenvalues of the matrix $\mathcal{D}(t)$ lie
in the union of discs $C_i= \{ z: |z-\mathcal{D}_{ii}| < R_i \}$ where
$\mathcal{D}_{ij}$ are the entries in the matrix $\mathcal{D}(t)$ and
$R_i = \sum_{j=1, j \neq i}^{12} |\mathcal{D}_{ij}|$. A direct computation
shows that the intersection of $\cup_{i=1}^{12} C_i$ with the imaginary
axis is contained in the interval $[-17 i, 17i]$ and it is therefore
sufficient to show that $p_{\mathcal{D}}^{\text{R}}(\mu, t_0)$ and
$p_{\mathcal{D}}^{\text{I}}(\mu, t_0)$ are not zero simultaneously for
any value of $t$ and $\mu \in [-17,17]$.

Since the coefficients of $\bar{p}_{\mathcal{D}}(\lambda,
t)$ are $\frac{\pi}{2}$ periodic in $t$, it is sufficient to show that
(\ref{inequality}) is not satisfied for any $\lambda \in [-17,17]$ and
$t \in [0, \frac{\pi}{2}]$.  This is shown by subdividing these intervals into
a sufficient number of subintervals $\bar{\mu}_n$ and $\bar{t}_m$ so that
$[-17,17] \subset \cup_{i=1}^{N} \bar{\mu}_n$ and
$[0,\frac{\pi}{2}] \subset \cup_{m=1}^{M} \bar{t}_m$ with the property that
the intervals
$\bar{p}_{\mathcal{D}}^{\text{R}}(\bar{\mu}_n, \bar{t}_m)$ and
$\bar{p}_{\mathcal{D}}^{\text{I}}(\bar{\mu}_n, \bar{t}_m)$ do not
contain zero simultaneously for any given pair of subintervals
$\bar{\mu}_n$ and $\bar{t}_m$.
Therefore the roots of $p_{\mathcal{D}}(\lambda, t)$ stay in the left half plane
for all $t \in \mathbb{R}$ and all
$K$-multiples of the fixed state under consideration are also stable.

The paths of the roots of $p_{\mathcal{D}}(\lambda, t)$ as $t$ is
varied are shown in figure \ref{eigenvalues}.

The \emph{Mathematica} code used in these calculations is available at

\emph{http://math.bu.edu/people/josic/research/code}.

{\bf Acknowledgment:} The authors research was supported in part by
NSF grant DMS-9803164.  K.J. would like to thank the Department of
Mathematics at Boston University for its hospitality while he
conducted his research. The authors also thank J.P. Eckmann for
useful discussions concerning this problem.


\section{Introduction}

The study of tunneling in nanostructures has assumed an important role in
the last few years in connection with advances in nanoelectronics. The
problem of tunneling of wave packets through a potential barrier arises in
many cases, for example, in the study of the action of femtosecond light
pulses on coupled wells. This problem is also important because of possible
applications of scanning tunneling microscopes irradiated with femtosecond
pulses for simultaneously high-spatial and high-temporal resolution study of
nanostructures.\cite{r1} Other interesting questions are the tunneling time
in the ionization of a hydrogen atom by ultrashort laser pulses and the
tunneling time for tunneling induced by the action of a laser pulse on
low-lying nuclear energy levels. In the present paper we investigate the no
less interesting question of the time duration of tunneling in nanostructures.
The tunneling time is of practical interest in this case because it permits
to estimate the response time of semiconductor components. In this connection
we shall study the following problem: Let a laser pulse produce a wave
packet of an excited electron near a tunneling barrier. The question is:
When will the tunneling portion of the packet appear behind the barrier? The
arrival of the wave packet can be detected by studying local variations of
the optical properties using ultrashort probe pulses.

It is interesting that a number of effects which are absent in the
stationary case are observed when a packet passes through a tunneling
barrier. The tunneling time of a packet is determined in general not by the
reciprocal of the probability of stationary tunneling, but rather it is
related with quite complicated processes --- a change in the shape and
behavior of the packet inside the barrier. Moreover, the transmission time
through a barrier depends on the measured quantities, i.e. on
the type of experiment.

The investigation of the question of the presence time of a tunneling
particle under a barrier started quite a long time ago,\cite{r2,r3,r4} and
many theoretical and experimental methods for measuring the tunneling time
have been proposed. For example, there exist approaches where the peak of
the packet or the average coordinate (the ``centroid'') is chosen as the
observed quantity while the reflection or transmission time is determined by
their evolution. However, it has been shown in Refs. \onlinecite{r5} and \onlinecite{r6} and subsequent
works that a wave-packet peak incident on a potential barrier does not pass
into the peak of the transmitted wave. In Ref. \onlinecite{r7} it was shown that
on account of dispersion of the wave packet in momentum space the high-energy
components reached the barrier before the other components. Since the
tunneling probability increases with energy, these components made the main
contribution to the transmitted part of the packet. The initial parameters
could be chosen in a manner so that the transmitted part of the packet left
the barrier long before the main peak, chosen as the observed quantity,
appears. This example demonstrates the breakdown of the causality principle,
which is the basis of this method, and therefore it limits the applicability
of the method. Moreover, it is difficult to conceive of an experimental
method for measuring the arrival time of a packet according to its peak or
``centroid.''

There also exists a class of approaches that employ an ensemble of dynamical
trajectories to determine the tunneling time. These dynamical trajectories
arise as a necessary apparatus of the description in the Feynman and Bohm
interpretations of quantum mechanics. When Feynman trajectories were used,%
\cite{r8} the transmission time through a barrier was determined as a path
integral over all possible trajectories that start from a prescribed point
to the left of the barrier and arrive at a certain time at a point located
to the right of the barrier. The integrated function in the path integral
contained the product of a classical presence time of a trajectory inside
the barrier by a weighting factor $\exp \{iS(x(t^{\prime }))/\hbar ]$, where
$S(x(t^{\prime }))$ is the action related with the trajectory $x(t^{\prime })
$ under consideration. The calculated times possess real and imaginary parts
because of the multiplication by a complex weighting factor, and the
question of how these times should be associated to the physically
observable quantities, which are always real, arose. To explain the complex
times, which also arise in other methods (for example, in the method of
``physical clocks'' (see below)), Sokolovski and Connor\cite{r9} examined
so-called direct and indirect measurements. In indirect measurements as in
the case of Feynman trajectories method the quantities can be complex.

Approaches employing physical clocks have found quite wide application.
Physical clocks are various additional degrees of freedom in the system that
make it possible to determine the presence time of a particle in a given
region. Three types of clocks have been investigated in theoretical works.
Baz' and Rybachenko\cite{r10,r11} used spin precession in a weak uniform
magnetic field applied inside a barrier. At first spin precession in a single plane
was considered. Then Buttiker and Landauer\cite{r12} extended the analysis
to three dimensions. During the tunneling a spin of a particle acquires a component
along the direction of motion and along the magnetic field. It is obvious
that the intensities of the detected components with spin polarization in
these two directions will be proportional to the presence time of the
particle in the region with the magnetic field, i.e. in the region of the
barrier. It turned out that for a square barrier the tunneling times
determined in this manner are identical to the real and imaginary parts of
the complex tunneling time introduced via Feynman path integrals.\cite{r8}
The extension of this method to the case of an arbitrary potential barrier
was made in Ref. \onlinecite{r13}. Buttiker and Landauer\cite{r14} considered as physical
clocks an oscillating barrier in which the amplitude of the oscillations of
the temporal component was much smaller than the barrier height. At low
frequencies particles see an effective static barrier, since the
transmission time through the barrier is much shorter than the period of the
oscillations of the temporal component of the barrier. As the frequency
increases, the delayed particles or wave-packet components see a slightly
modified potential barrier. Finally, for some frequencies one or several
periods of the oscillations influence the tunneling particles. The frequency
at which a substantial difference from the adiabatic case corresponding to a
stationary barrier appears will determine the reciprocal of the
interaction time with the barrier or the transmission time through the
barrier. Martin and Landauer\cite{r15} chose as physical clocks the
oscillating amplitude of the incident wave. For this, a wave function
consisting of a superposition of two plane waves with different energies was
chosen to the left of the barrier. It is obvious that in this case the wave
function to the right of the barrier will also be a superposition of the
tunneled parts of the plane waves, which, however, possess a different
transmission amplitude, since the amplitude depends on the energy. The
transmitted wave function will reproduce the incident wave function if the
amplitudes of the tunneled plane waves differ very little; this corresponds
to the adiabatic case. The energy difference between the initial plane waves
for which the wave function behind the barrier does not reproduce the incident
wave function makes it possible to determine the transmission time through a
potential barrier. The main advantage of this method is that it is
applicable for all types of potentials, but it employs two values of the
energy, so that it is not clear to which energy the tunneling time obtained
should be ascribed.

Do all clocks give the same measurement result? Of course, no. However, in
many cases these results are close to one another or identical.\cite
{r13,r16,r17,r18} The main advantage of the approaches using physical clocks
is that they strive to determine the tunneling time in terms of possible
measurements in physical expeirments.

The search for ``time operators'' and the study of their properties is no
less popular.\cite{r19,r20,r21,r22,r23,r24} As first noted by Pauli,\cite
{r25} the main difficulty is that a measurable hermitian time operator
for a system Hamiltonian with a bounded spectrum does not exist. Various
attempts have been made to construct operators that would describe the
necessary properties of physical times. In order that the constructed
operator satisfy the correspondence principle, relations from classical
mechanics were taken as the basis for the operator construction. However, it
is well known that the construction of an operator expression corresponding
to a classical quantity is not unique, and its relation with the measurement
process requires additional analysis.

In the present work the tunneling time was determined as the difference of
the average ``arrival'' and ``presence'' times of a wave packet (see Sec.
3) before and after the barrier. The method of quantum molecular dynamics
was used to calculate these times and to investigate the dynamics of a
tunneling wave packet.\cite{r26,r27,r28}

It is well known that molecular dynamics investigates the properties of
classical systems in phase space. Therefore it is natural to extend
this method to quantum systems in phase space. The evolution of a system in
phase space can be described, for example, on the basis of the Wigner
formalism of quantum mechanics by the Wigner--Liouville equation. To solve
the Wigner--Liouville equation written in integral form it is convenient to
rewrite the equation in the form of an iterational series. Each term of this
series can be treated as a weighted contribution of a trajectory consisting
of segments of classical trajectories separated by finite disturbances of
the momentum. In what follows we shall call such a trajectory a quantum
trajectory. The statistical ensemble of quantum trajectories makes it
possible to calculate the sum of all terms in the series. The Monte Carlo
method is used to take account of only the trajectories making the main
contribution. In the classical limit the quantum trajectories turn into
classical trajectories, and the method of generalized molecular dynamics
becomes identical to usual molecular dynamics. The principles of the
method are presented in Sec. 2. The expressions for calculation of the
distributions of the arrival and presence times of a wave packet are
presented in Sec. 3 on the basis of the Wigner formalism of quantum
mechanics. The simulation results are discussed in Sec. 4. The
one-dimensional case is considered in this paper, but the method employed
makes it possible to perform similar calculations for multidimensional and
multiparticle systems, where it has serious advantages from the standpoint
of computer time over, for example, the solution of the nonstationary
Schr\"{o}dinger equation.

\section{Computational method}

To calculate the quantum-mechanical average of a quantity $A$ for a
nonstationary state $\left| \psi \right\rangle $ in the Wigner formulation
of quantum mechanics it is necessary to calculate an integral in phase space%
\cite{r29}

\begin{equation}
A\left( t\right) =\left\langle \psi \left| \hat{A}\left( t\right) \right|
\psi \right\rangle =\int \!\!\!\int dqdpA\left( q,p\right) W\left(
q,p,t\right) ,
\end{equation}
where, by definition, the Weyl symbol $A(q,p)$ is introduced for the
operator $\hat{A}$ and $W(q,p,t)$ is the Wigner function, which is the
Fourier transform of the off-diagonal density-matrix element:

\begin{eqnarray}
A\left( q,p\right)  &=&\int d\xi \exp \left( \frac{ip\xi }\hbar \right)
\left\langle q+\frac \xi 2\left| \hat{A}\right| q-\frac \xi 2\right\rangle ,
\\
W\left( q,p,t\right)  &=&\frac 1{2\pi \hbar }\int d\xi \exp \left( -\frac{%
ip\xi }\hbar \right) \psi ^{*}\left( q-\frac \xi 2,t\right) \psi \left( q+%
\frac \xi 2,t\right) .
\end{eqnarray}
Differentiating the distribution function with respect to time, substituting
it for the time derivative of the function $\psi $ on the right-hand side of
the Schr\"{o}dinger equation, and integrating by parts, we obtain the
Wigner--Liouville integrodifferential equation\cite{r30}

\begin{equation}
\frac{\partial W}{\partial t}+\frac pm\,\frac{\partial W}{\partial q}%
+F\left( q\right) \frac{\partial W}{\partial p}=\int\limits_{-\infty
}^\infty dsW\left( p-s,q,t\right) \omega \left( s,q\right) .
\end{equation}
In this equation

\begin{equation}
\omega \left( s,q\right) =\frac 2{\pi \hbar ^2}\int dq^{\prime }V\left(
q-q^{\prime }\right) \sin \left( \frac{2sq^{\prime }}\hbar \right) +F\left(
q\right) \frac{d\delta \left( s\right) }{ds}
\end{equation}
takes account of the nonlocal contribution of the potential, and $%
F(q)=-\partial V(q)/\partial q$ is a classical force. In the classical
limit, $\hbar \rightarrow 0$, Eq. (4) becomes the classical Liouville
equation

\begin{equation}
\frac{\partial W}{\partial t}+\frac pm\,\frac{\partial W}{\partial q}=-F
\left( q\right) \frac{\partial W}{\partial p}.
\end{equation}

The equation (4) can be written in an integral form. For this, one
introduces the dynamical trajectories $\{\bar{q}_\tau (\tau ;p,q,t),\bar{p}%
_\tau (\tau ;p,q,t)\},$ $\tau \in [0,t]$, starting from the point $(p,q)$ at
time $\tau =t$:

\begin{eqnarray}
d\bar p/d\tau &=&F(\bar p(\tau )),\ \bar
p_t(t;p,q,t)=p \nonumber
\\ d\bar q/d\tau &=&\bar q(\tau )/m,\ \bar
q_t(t;p,q,t)=q
\label{s8}
\end{eqnarray}

An integral equation is obtained by substituting the right-hand sides of
these equations into the Wigner--Liouville equation, whose left-hand side
becomes a total differential, and integrating over time one have

\begin{equation}
W\left( p,q,t\right) =W^0(\bar{p}_0,\bar{q}_0)+\int\limits_0^td\tau
\int\limits_{-\infty }^\infty dsW\left( \bar{p}_\tau -s,\bar{q}_\tau ,\tau
\right) \omega \left( s,\bar{q}_\tau \right) .
\end{equation}
Here $W^0(\bar{p}_0,\bar{q}_0)=W(p,q,0)\,\,$is the Wigner distribution
function at zero time. The solution of Eq. (8) can be represented as an
iterational series. For this we introduce the notation $\tilde{W}^{\tau _1}$
for the distribution function, which evolves classically in the interval $%
[0,\tau _1]$, and the integral operator $K_{\tau _i}^{\tau _{i+1}}$
describing the evolution between the times $\tau _i$ and $\tau _{i+1}$. Now
Eq. (8) can be represented in the form

\begin{equation}
W^t=\tilde{W}^t+K_\tau ^tW^\tau ,
\end{equation}
where $\tilde{W}^t=W^0(\bar{p}_0,\bar{q}_0)$. The corresponding iterational
series solving this equation can be written as

\begin{equation}
W^t=\tilde{W}^t+K_{\tau _1}^t\tilde{W}^{\tau _1}+K_{\tau _2}^tK_{\tau
_1}^{\tau _2}\tilde{W}^{\tau _1}+K_{\tau _3}^tK_{\tau _2}^{\tau _3}K_{\tau
_1}^{\tau _2}\tilde{W}^{\tau _1}+...
\end{equation}
Now, to calculate the quantum-mechanical average (1) it is necessary to
calculate a linear functional of the Wigner distribution function

\begin{eqnarray}
A\left( t\right) &=&\int \!\!\!\int dqdpA\left( q,p\right) W\left(
q,p,t\right)  \nonumber \\
&=&\left( A|\tilde{W}^t\right) +\left( A|K_{\tau _1}^t\tilde{W}^{\tau
_1}\right) +\left( A|K_{\tau _2}^tK_{\tau _1}^{\tau _2}\tilde{W}^{\tau
_1}\right) +\left( A|K_{\tau _3}^tK_{\tau _2}^{\tau _3}K_{\tau _1}^{\tau _2}%
\tilde{W}^{\tau _1}\right) +...
\end{eqnarray}
Here the brackets $(...\mid ...)$ for the functions $A=A(p,q)$ and $\tilde{W}%
^t$ or $K_{\tau _i}^tK_{\tau _{i-1}}^{\tau _i}...K_{\tau _1}^{\tau _2}\tilde{%
W}^{\tau _1}$ indicate averaging over the entire phase space $\{p,q\}$.

The first term on the right-hand side of Eq. (10) gives the classically
evolving initial distribution $W^0(\bar{p}_0,\bar{q}_0)$, i.e. the evolution
of the distribution function without quantum corrections. However, even this
first term of the iterational series describes not classical but rather
quantum effects and can contain arbitrary powers of the Planck constant,
since a quantum initial state of the system is taken as the initial data for
Eq. (10). The rest of the terms in the iterational series describe quantum
corrections to evolution. Each term of the iterational series (10) is a
multiple integral. This multiple integral can be replaced by an integral
sum, and each term of the integral sum can be represented as a contribution
of trajectories of a definite topological type. These trajectories consist
of segments of classical trajectories --- solutions of Eqs. (7) ---
separated from one another by random perturbations of the momentum.

All terms of the iterational series can be calculated in accordance with the
theory of Monte Carlo methods for solving linear integral equations. For
this purpose the Monte Carlo scheme was developed, which provides
the essential sampling of the iteration series terms (10). This essential
sampling also decreases the computer time required to calculate the rest of the
integrals appearing in each term of the iterational series. Let us consider
the second term of the series (10). This term can be rewritten as

\begin{eqnarray}
K_{\tau _1}^t\tilde{W}^{\tau _1} &=&\int\limits_0^1d\tau _1\int ds_1\omega
\left( s_1,\bar{q}_1\right) W^0(\bar{p}_0^1,\bar{q}_0^1)=  \nonumber \\
&=&\int\limits_0^1d\tau _1\left[ B(\bar{q}_2)\left( 1+Q(\bar{q}_2)\right)
\right] \theta \left( 1-\tau _2\right) r\left( \tau _2\right) \int ds_1P(s_1,%
\bar{q}  \nonumber \\
&\times &\left\{ \sigma \left( s_1,\bar{q}_1^2\right) t\tilde{Q}(\bar{q}%
_1^2)\theta \left( \tau _2-\tau _1\right) /C(\bar{q}_1^2)r(\tau _1)\right\}
W^0(\bar{p}_0^1,\bar{q}_0^1),
\end{eqnarray}
where the substitution of variables $\tau \rightarrow \tau t$ was made for
all terms of the iterational series (10). The quantity $r(\tau _1)$ is the
probability of choosing a random time $\tau _1$ and $\theta $ is the theta
function.

Once the second term of the series (10) is written in the form (12), it can
be given the following probabilistic interpretation. We will take advantage
of the time-reversibility of the equations of classical dynamics (7) and start the
construction of a trajectory at time $\tau =0$. At time $\tau _1$ for a
trajectory representing an arbitrary term in the iterational series a
perturbation of the momentum of the trajectory by an amount $s_1$ can occur
with probability $C(\bar{q}_1^2)$, and the probability of rejecting a
momentum perturbation is $B(\bar{q}_2)$ $(C(\bar{q}_1^2)+B(\bar{q}_2)=1)$ .
The probability $B$ for rejecting momentum jumps was introduced to make the
algorithm more flexible, so that depending on the degree of quantization of
the system a transition from quantum to classical trajectories would occur
automatically.

Since we are considering a trajectory representing the second term in the
iterational series, a perturbation of the momentum at the time $\tau _1$ was
accepted. Now it is necessary to choose in the time interval $[\tau _1,1]$ a
random value $\tau _2$ which is the time of the next attempt to perturb the
momentum. After a perturbation of the momentum by an amount $s$ we must
continue the generation of the trajectory  up to the time $\tau _2$  in
accordance with Hamilton's equations. At this time an attempt to perturb the
momentum for the second term of the iterational series must be rejected, and
we continue the generation of the trajectory up to the time $\tau =1$. The
rejected attempt of the perturbation of the momentum must be taken into account by
multiplying the weighting function of the trajectory by a compensating
factor, which stands in the braces on the right-hand side of the expression
(12). The product of the Weyl symbol of the operator under consideration and
the weighting function at different points along the trajectory gives the
time dependence of the computed quantities. Averaging over a large ensemble
of trajectories of this type gives the contribution of the second term of
the iterational series.

Similar expressions but with a large number of intermediate times on
classical trajectories when a perturbation of the momentum occurs can also
be written for the other terms in the series (10). The number of the term in
the iterational series (10), described by the given trajectory, determines
the number of momentum perturbations along the trajectory.

The final expression used to calculate the linear functional (11) is
\begin{eqnarray}
\label{b34}
A\left( t\right)&=&M\left\{ \alpha \left( A;T_{i}\right) \right\}=
\sum_{p,q}\left( \triangle p\triangle q\right) \sum_{i=0}^\infty
\sum_{j=0}^i\sum_{\tau_j}\sum_{s_j}\alpha \left( A;T_{i}\right)
\times P\left(T_{i}\right) ;\nonumber\\
\alpha \left( A;T_{i}\right)&=&A\left(p,q\right)W^0(\bar{p}_0^1,\bar{q}%
_0^1)\Omega \left(T_{i}\right),
\end{eqnarray}
where the functions $P$ and $\Omega $ are, respectively, the probability of
generating a quantum trajectory $T_i$ and the weighting function of this
trajectory.

\section{Measured quantities}

The study of the evolution of a wave packet can be taken as the starting
point for studying the temporal aspects of tunneling. The probability of
observing a wave packet or particle at an arbitrary point $X$ is determined
by the squared modulus $\left| \psi (X,t)\right| ^2\,$ of the wave function.
In a nonstationary problem this probability depends on the time and
determines the characteristic times of the wave-packet dynamics. If an ideal
detector (i.e. measurement by the detector does not disturb the wave
function), sensitive to the presence of particles, is used in the
experiment, then the average presence time measured by the detector at the
point $X$ is

\begin{equation}
\tilde{t}_X=\frac{\displaystyle \int\limits_0^\infty dt\,t\left| \psi \left(
X,t\right) \right| ^2}{\displaystyle \int\limits_0^\infty dt\left| \psi
\left( X,t\right) \right| ^2}.
\end{equation}
A description of these times can be found in Refs. \onlinecite{r31}--\onlinecite{r32}. The distribution
of presence times at the point $X$ is

\begin{equation}
\tilde{P}\left( t_X\right) =\frac{\left| \psi \left( X,t\right) \right| ^2}{%
\displaystyle \int\limits_0^\infty dt\left| \psi \left( X,t\right) \right| ^2%
}.
\end{equation}
To find the squared wave function $\left| \psi (X,t)\right| ^2\,$ it is
sufficient to calculate a quantum-mechanical average of an operator

\begin{eqnarray*}
\langle \psi \left( t\right) \left| \delta \left( \hat{q}-X\right) \right|
\psi \left( t\right) \rangle =\int dq\delta \left( q-X\right) \left| \psi
\left( q,t\right) \right| ^2=\left| \psi \left( X,t\right) \right| ^2.
\end{eqnarray*}
In the Wigner representation this is equivalent to calculation
of the integral

\begin{equation}
\langle \psi \left( t\right) \left| \delta (\hat{q}-X)\right| \psi \left(
t\right) \rangle =\int \!\!\!\int dqdp\delta \left( q-X\right) W\left(
q,p,t\right) =\int dp\,W(X,p,t).
\end{equation}
If the point $X$ is chosen to the right of the barrier, then this integral
makes it possible to calculate the squared wave function which has tunneled
through the barrier. The distribution of the ``presence'' times can be
rewritten, in accordance with Eq. (16), as

\begin{equation}
P_X\left( t\right) =\frac{\left| \psi \left( X,t\right) \right| ^2}{%
\displaystyle \int\limits_0^\infty dt\left| \psi \left( X,t\right) \right| ^2%
}=\frac{\int dpW\left( X,p,t\right) }{\displaystyle \int\limits_0^\infty
dt\int dpW\left( X,p,t\right) }.
\end{equation}
To determine the average time when the wave packet passes through a detector
at the point $X$ it is necessary to calculate the integral

\begin{eqnarray}
\left\langle t\left( X\right) \right\rangle =\int\limits_0^\infty
dt\,tP_X\left( t\right) ,
\end{eqnarray}
and the average transition time of a packet from the point $X_i$ to the
point $X_f$ will be

\begin{equation}
\left\langle t_T\left( X_i,X_f\right) \right\rangle =\left\langle t\left(
X_f\right) \right\rangle -\left\langle t\left( X_i\right) \right\rangle .
\end{equation}
If the points $X_i$ and $X_f$ are chosen on different sides of the potential
barrier, then the expression (19) can be used to estimate the tunneling time.

The main drawback of the definition (17) is that, as a rule, detectors
sensitive to a flux density and not a probability density are used in
physical experiments. Therefore a different quantity must be considered in
order to compare theory and experiment. For this, the distribution of
arrival times of a wave packet at a prescribed point in terms of the
probability flux density was introduced:\cite{r34}

\begin{equation}
P_X\left( t\right) =\frac{\left\langle \psi \left( t\right) \left| \hat{J}%
\left( X\right) \right| \psi \left( t\right) \right\rangle }{\displaystyle %
\int\limits_0^\infty dt\langle \psi \left( t\right) \left| \hat{J}\left(
X\right) \right| \psi \left( t\right) \rangle },
\end{equation}
where

\begin{equation}
\hat{J}\left( X\right) =\frac 12\left[ \hat{p}\delta \left( \hat{q}-X\right)
+\delta \left( \hat{q}-X\right) \hat{p}\right] .
\end{equation}
Of course, the definition (20) is not a real distribution function from
probability theory, since this function can assume negative values at some
points. Nonetheless the definition (20) will be a distribution function if
there is no reverse flux through the point $X$ or the flux is negligibly
small. For this the point $X$ is chosen at a sufficiently far from the
barrier. Measuring the distribution of the arrival times of a packet in
front of and beyond the barrier, the transition time through a region much
larger than the region of the potential barrier can be calculated. This time
is analogous to the asymptotic phase times\cite{r35} and, besides the
tunneling time and the packet--barrier interaction time, it also contains
the transmission time through the region where the potential barrier is
zero. These two times cannot be separated. Despite continuing discussions,
this tunneling-time problem has still not been finally solved.\cite
{r19,r20,r21,r22,r23,r24,r36,r37}

Another problem concerns the physical implementation of an experiment in
which simultaneous detection of a packet in front of and beyond a barrier
would not lead to substantial reduction of the wave function. For this
reason, ordinarily, a different quantity --- the ``time delay'' --- is
measured in experiments.\cite{r38,r39,r40,r41,r42} A time delay arises
because of the presence of a barrier and is defined as the difference of the
average arrival times of the tunneling and free packets:

\begin{eqnarray}
\Delta \tau _{arrival}(X)=\langle t_X\rangle ^{tun}-\langle t_X\rangle
^{free}.
\end{eqnarray}
The definition (20) for calculation the average arrival times gives a
reasonable estimate of the time delays measured in an experiment.

The distribution of arrival times (20) can be rewritten in the Wigner
formulation of quantum mechanics as

\begin{equation}
P_X\left( t\right) =\frac{\displaystyle \int \!\!\!\displaystyle \int
dqdpJ_X\left( q,p\right) W\left( q,p,t\right) }{\displaystyle %
\int\limits_0^\infty dt\displaystyle \int \!\!\!\displaystyle \int
dqdpJ_X\left( q,p\right) W\left( q,p,t\right) },
\end{equation}
where the Weyl symbol of the current operator $\hat{J}(X)$ is

\begin{eqnarray}
J_X\left( q,p\right) =\frac \hbar 2\sin \left( \frac{2p\left( X-q\right) }%
\hbar \right) \frac \partial {\partial q}\delta \left( q-X\right) .
\end{eqnarray}
Substituting into Eq. (20) the expression (24) and calculating the integral
over the variable $q$ by parts we obtain the expression

\begin{equation}
P_X\left( t\right) =\frac{\displaystyle \int dppW(X,p,t)}{\displaystyle %
\int\limits_0^\infty dt\displaystyle \int dppW(X,p,t)}.
\end{equation}
Comparing the expressions (17) and (25), it is easy to see that they differ
by the fact that the momentum $p$ appears in the numerator and denominator
in Eq. (25). This momentum appeared in the last expression because the
probability flux density is measured there.

\section{Simulation results}

We shall examine a series of experiments on the tunneling of an electron
with the wave function

\begin{equation}
\psi (x,0)=\frac 1{\left( 2\pi \sigma _x\right) ^{1/4}}\exp \left[ -\left(
\frac{x-x_0}{2\sigma _x}\right) ^2+ik_0x\right]
\end{equation}
through a Gaussian potential barrier

\[
V\left( x\right) =V_0\exp \left[ -\frac{(x-d)^2}{\sigma ^2}\right] .
\]
The Wigner distribution function (3) corresponding to the initial wave
function of the electron can be written as

\begin{equation}
W(p,q,0)=2\exp \left[ -\frac{(q-x_0)^2}{2\sigma ^2}\right] \exp \left[ -%
\frac{2\sigma ^2(p-\hbar k_0)^2}{\hbar ^2}\right] .
\end{equation}
The center $x_0=\left\langle \psi (x,0)\right| \hat{x}\left| \psi
(x,0)\right\rangle $ of the wave packet at zero time was chosen far enough
from the left-hand boundary of the barrier so that the probability density
beyond the barrier would be negligibly small compared with the transmission
probability $\left| T\right| ^2$ through the barrier. Tunneling occurred
through a ``wide'' $(\sigma =2.5$ nm --- this parameter of the barrier is
characteristic for Al$_x$Ga$_{1-x}$As structures) and a ``narrow'' $(\sigma
=0.5$ nm) Gaussian barrier of height $V_0=0.3$ eV centered at $d=0$. The
electron kinetic energy was $E_0=\hbar ^2k_0^2/2m=V_0/2=0.15$ eV. We used
the system of units where $\hbar =m=V_0=1$. Distances were measured in units
of the reduced de Broglie wavelength $\lambda =1/k_0$. In this system of
units the parameters of the wave packet and barrier are: $E_0=0.5,$ $\Delta
k=0.04$ $(0.125),$ $\sigma _x=1/2\Delta k=12.5$ $(4),$ $x_0=-92.5$ $(-43),$ $%
\sigma =5$ $(2.5$ nm$),$ and $\sigma =1$ $(0.5\,$nm$)$.

\subsection{Evolution of the wave packet}

The interaction of a wave packet $(\hbar \Delta k=0.125)$ with a narrow
potential barrier $(\sigma =1$ $(0.5$ nm$))\,$ is shown in Figs. \ref{f1}a
and b. These figures show the probability density $\left| \psi (x,t)\right|
^2$ (curves 1--5) of reflected (Fig. \ref{f1}a) and tunneled (Fig. \ref{f1}%
b) wave packets at successive times $t=114-239$ fs. The probability density
was calculated using Eq. (16), i.e. in terms of the Wigner distribution
function. This integral was calculated along quantum and classical
trajectories. In the calculation over classical trajectories only the
high-energy components of a packet could pass classically above the barrier.
This calculation corresponds to the curve 1 in Fig. \ref{f1}c, and the
evolution of the Wigner function can be described only by the first term of
the series (10). In the formalism of quantum trajectories the passage of the
components of a packet beyond the barrier is associated with random
perturbations of the momentum, i.e. with a virtual change in energy. The
results of this calculation correspond to the curve 2 in Fig. \ref{f1}c. Now
the quantum corrections introduced by all terms in the series (10) are taken
into account in the evolution of the Wigner function.

Of course, the calculation over quantum trajectories also takes account of
the high-energy components that pass above the barrier, since they
describe the contribution of the first term in the series (10). However,
comparing the curves 1 and 2 in Fig. \ref{f1}c shows that their role is
negligible for a narrow barrier and most of the packet passes above the
barrier on account of the virtual change in energy, described as random
perturbations of the momentum of the quantum trajectories. A study of
tunneling through a wide barrier leads to the opposite conclusion. The
curves 1 and 2 in Fig. \ref{f1}d approximately coincide. This means that most
of the packet has passed above the barrier, and the contribution of all
terms in the series (10), except for the first term, is negligibly small. To
avoid such a situation and to restore the importance of quantum effects, it
is necessary to decrease the uncertainty of the momentum of the initial wave
packet. In what follows all calculations for a wide barrier are presented
for momentum uncertainty $\hbar \Delta k=0.04$.

\subsection{Average coordinate, average momentum, and their variances}

Figure \ref{f2}a shows the evolution of the average coordinate $\left\langle
\psi (t)\right| \hat{X}\left| \psi (t)\right\rangle $ of the wave packet for
calculation according to classical (curve 1) and quantum (curve 2)
trajectories. In these two methods for calculation the average coordinate $%
\bar{X}$ no differences are observed before interaction with the barrier
(curves 1 and 2 are coincident). This result can be explained as follows. In
the method under discussion the quantum-mechanical properties appear at two
points: in the properties of the initial state of a wave packet and in the
evolution of the packet. Since the same initial data were chosen for the
quantum and classical trajectories, the fact that $\bar{X}$ is the same must
be explained by the evolution of the wave packet. Specifically, while the
packet moves freely in front of the barrier, it is correctly described by
classical trajectories also. In this case the first term in the series (10)
is sufficient to describe the evolution of the Wigner function. This result
can also be obtained analytically, estimating the right-hand side of the
Wigner--Liouville equation (4). For the initial Wigner function (27) and
Gaussian barrier which we have chosen it is easy to show that the integral
on the right-hand side of Eq. (4) decays exponentially with increasing
distance from the barrier. In this case Eq. (4) becomes the classical
Liouville equation, whose characteristics are ordinary classical
trajectories.

A difference in the behavior of the curves 1 and 2 appears after the packet
interacts with a barrier. Now the classical trajectories are no longer
characteristics and do not describe the evolution of the wave packet
correctly. In Figs. \ref{f2}a and b the average coordinate and the momentum
of the calculation over quantum trajectories (curve 2) are greater than for
classical trajectories (curve 1). This is due to the following
circumstances. In the first place, since most of the packet is reflected, as
one can see from Fig. \ref{f2}b the average momentum changes sign after
being scattered by the barrier. In the second place, the classical
trajectories (curve 1) do not take account of tunneling; they only take
account of the negligible above-barrier transmission, arising because of the
uncertainty in the momentum of a Gaussian wave packet. At the same time it
is obvious that the tunneling part of the packet has positive momentum and
moves in the opposite direction relative to the reflected part. Therefore
its contribution to $\bar{X}$ and $\bar{P}\,$ has a different sign. This is
the explanation of the difference between the curves 1 and 2.

In addition, the motion of the tunneling and reflected packets on different
sides of the barrier also explains the more rapid increase of the coordinate
variance in the quantum calculation (curve 2, Fig. \ref{f2}c) as compared
with the classical calculation (curve 1), which takes into account only the
spreading of the wave packet. The behavior of the packet width on scattering
by a barrier is shown in greater detail in the upper left-hand part of Fig.
\ref{f2}c.

The interaction of a packet with the barrier also leads to an interesting
behavior of the momentum variance in Fig. \ref{f2}d. The constant values
(curve 1) on the initial and final sections show the momentum variance in
the incident and reflected wave packets, i.e. before and after interaction
with the barrier. The observed peak is due to the change in the sign of the
momentum of the packet and to the fact that different components reach the
barrier and are reflected from it at different times. The increase in the
momentum variance (curve 2) on the final section is explained by the
appearance of a tunneling packet with positive momentum in the quantum
computational method, while the total average momentum is negative.

\subsection{Distribution of arrival and presence times. Momentum
distribution function}

The results of the calculation of the unnormalized presence time distribution
(17) at different points in front of the barrier, inside the barrier,
and beyond barrier are presented in Figs. \ref{f3}a and b (curves 1--5).
Figures \ref{f4}a and b show the analogous results for the unnormalized
distribution (20) of the arrival times. The curves 1 in Figs. \ref{f3}a and
\ref{f4}a show the behavior of the probability density and flux,
corresponding to the fact that the incident and reflected wave packets pass
through the detector at different times. Curve 2 in Fig. \ref{f4}a shows the
behavior of the flux measured at a certain point to the left of barrier
center. The tunneling and high-energy components present in the initial
packet can classically reach this point. An
interesting result is obtained for the probability flux density in Fig. \ref
{f4}b (curves 3--5). The flux measured at barrier center (curve 3) is much
less than the flux on the right-hand boundary of the barrier (curve 4) and
far to the right of the barrier (curve 5). This means that interference of
the tunneling components of the wave packet which move in opppostie
directions occurs inside the barrier. Some of these components pass
completely through the barrier, while others are reflected inside the
barrier and do not reach its right-hand boundary. Interference of the
reflected and transmitted components leads to the observed decrease in the
flux amplitude at barrier center (curve 3) and at the right-hand boundary
(compare curves 4 and 5). It is interesting that the investigation of tunneling
using classical trajectories in complex time also reveals the similar effect.%
\cite{r37} It was found that transition through a barrier occurs as a
series of attempts, many of which are unsuccessful because of reflections in
different regions under the barrier.

The comparison of the presence and arrival times distributions in Figs. \ref
{f3}b and \ref{f4}b shows that they are almost identical. The computed
average presence and arrival times (18) are also identical (the difference
is less than 1 fs). As we have already stated, the distribution of the
arrival times (20) is not a true distribution function and, as one can see
from Fig. \ref{f4}a (curve 2), it is not suitable for calculation of the
average arrival time of a packet in front of the barrier. This makes it
impossible to calculate the tunneling time as the difference (19) of the
average arrival times of the packet in front of and beyond the barrier.
Nonetheless the expression (19) can be used to estimate the tunneling time,
if the average presence time (14) is used instead of the average arrival
time in front of the barrier. Then the tunneling time through the potential
barrier is $\tau _T(-0.67\sigma ,+0.67\sigma )=12$ fs, i.e. it is almost
equal to the transmission time of a free packet through a similar region $%
\tau _T^{class}(-0.67\sigma ,+0.67\sigma )=13.5$ fs.

The time delays were measured at the points $x_4=0.67\sigma $ $(1.6$ nm) and
$x_5=5\sigma $ (12 nm) and were found to be $\Delta \tau _{arrival}(x_4)=8$
fs and $\Delta \tau _{arrival}(x_5)\le 0.5$ fs. If these measurements were
performed even farther to the right of the barrier, then $\Delta \tau
_{arrival}(x)$ would become negative. Thus an interesting behavior is discerned:
Even though the tunneling wave packet is delayed by the barrier ($\Delta
\tau _{arrival}(x_4)=8$ fs) and passes through the barrier approximately in
the same time as a free packet, it appears earlier at a definite distance to
the right of the barrier. This effect can be explained by the fact that the
transmission probability through a Gaussian barrier increases with energy,
so that packet components with a larger momentum have a higher probability
of ending up beyond the barrier. These components move more rapidly than a
free packet and eventually overtake a free packet. Then the time delays can
only be negative. This confirms the momentum distribution function

\begin{equation}
\frac{\langle \psi \left( t\right) \left| \delta \left( \hat{p}-p\right)
\right| \psi \left( t\right) \rangle }{\langle \psi \left( t\right) |\psi
\left( t\right) \rangle },
\end{equation}
calculated for narrow (Fig. \ref{f5}a) and wide (Fig. \ref{f5}b) barriers,
respectively, at times $t=218$ and $385$ fs. At these characteristic times
the distribution function no longer changes, since the interaction with the
barrier has finished. It is evident from Fig. \ref{f5} that the average
momentum of the tunneled wave packet (curve 2) is greater than the average
momentum of the wave packet initially (curve 1). The peak observed in the
momentum distribution function (curve 2 in Fig. \ref{f5}a) is due to the
packet components that had a quite large momentum and passed above the
barrier. It is evident that tunneling through a narrow potential barrier
leads to a larger variance of the distribution function, while tunneling
through a wide barrier substantially shifts the center of the distribution
in the direction of large momenta (curve 2 in Fig. \ref{f5}b).

\section{Conclusions}

The quantum generalization of classical molecular dynamics was used to solve
the Wigner--Liouville integral equation in the Wigner formulation of quantum
mechanics. The method discussed for solving this equation does not require a
large increase of computer time and makes it possible to avoid the
computational difficulties that arise when solving the nonstationary
Schr\"{o}dinger equation.

This approach was used to solve the nonstationary problem of tunneling of a
finite wave packet, i.e. a problem in which it is important to take account
of exponentially small quantum effects. The evolution of a wave packet, the
behavior of the average and the variances of the coordinate and momentum and
the distributions of the presence and arrival times for the wave packet at
different positions of an ideal detector were analyzed. The following
results were obtained: 1) The tunneling time through a potential barrier is
approximately of the same order of magnitude as the transmission time of a
free wave packet over a similar distance; 2) the tunneling wave packet is
delayed by the potential barrier, so that after the barrier the time delay
should be positive; 3) measurement of negative time delays is possible only
at sufficiently large distances from the barrier and is associated with a
shift of the momentum distribution function; 4) a Gaussian barrier transmits
mainly the high-energy components of a packet, interaction with the
barrier shifts the center of the momentum distribution function so that the
average momentum of the transmitted packet is larger than the initial
average momentum of the entire packet; 5) tunneling through a narrow
potential barrier leads to a larger variance of the momenta of the tunneled
components, while tunneling through a wide barrier leads to an appreciable
increase in the average momentum; and, 6) the computational results for the
probability flux density showed that the tunneling wave packet does not pass
completely through the barrier, a portion of the packet under the barrier
is reflected and does not reach its right boundary.

This work was partically supported by grants from the Russian Fund for
Fundamental Research and the Program ``Physics of Solid-State
Nanostructures.''