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\title{Effective Reducibility of Quasiperiodic Linear
Equations close to Constant Coefficients}
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\author{\`Angel Jorba, Rafael Ram\'{\i}rez-Ros and Jordi Villanueva}
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\date{}
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\begin{document}
\maketitle
\begin{center}
Departament de Matem\`atica Aplicada I (ETSEIB),\\
Universitat Polit\`ecnica de Catalunya,\\
Diagonal 647, 08028 Barcelona, Spain.
\end{center}
\begin{abstract}
Let us consider the differential equation
$$
\dot{x}=(A+\varepsilon Q(t,\varepsilon))x, \;\;\;\;
|\varepsilon|\le\varepsilon_0,
$$
where $A$ is an elliptic constant matrix and $Q$ depends on time in a
quasiperiodic (and analytic) way. It is also assumed that the eigenvalues
of $A$ and the basic frequencies of $Q$ satisfy a diophantine condition.
Then it is proved that this system can be reduced to
$$
\dot{y}=(A^{*}(\varepsilon)+\varepsilon R^{*}(t,\varepsilon))y,
\;\;\;\; |\varepsilon|\le\varepsilon_0,
$$
where $R^{*}$ is exponentially small in $\varepsilon$, and
the linear change of variables that performs such reduction is
also quasiperiodic with the same basic frequencies than $Q$.
The results are illustrated and discussed in a practical example.
\end{abstract}
\section{Introduction}\label{sec0}
The well-known Floquet theorem states that any linear periodic system,
$\dot{x}=A(t)x$, can be reduced to constant coefficients, $\dot{y}=By$,
by means of a periodic change of variables. Moreover, this change of
variables can be taken, over $\c$, with the same period than $A(t)$.
A natural extension is to consider the case in which the matrix $A(t)$
depends on time in a quasiperiodic way. Before starting the discussion
of this issue, let us recall the definition and basic properties of
quasiperiodic functions.
\begin{defi}
A function $f$ is a quasiperiodic function with vector of basic
frequencies $\omega=(\omega_1,\ldots,\omega_r)$ if
$f(t)=F(\theta_1,\ldots,\theta_r)$, where $F$ is $2\pi$ periodic in
all its arguments and $\theta_j=\omega_jt$ for $j=1,\ldots,r$.
Moreover, $f$ is called analytic on a strip of width $\rho$ if $F$ is
analytical on an open set containing $|\mbox{\rm Im }\theta_j|\leq\rho$
for $j=1,\ldots,r$.
\end{defi}
It is also known that an analytic quasiperiodic function $f(t)$ on a
strip of width $\rho$ has Fourier coefficients defined by
$$
f_k=\frac{1}{(2\pi)^r}\int_{\t^r} F(\theta_1,\ldots,\theta_r)
e^{-(k,\theta)\sqrt{-1}}\, d\theta,
$$
such that $f$ can be expanded as
$$
f(t)=\sum_{k\in\z^r} f_k e^{(k,\omega)\sqrt{-1}t},
$$
for all $t$ such that $|\mbox{\rm Im }t|\le\rho/\|\omega\|_{\infty}$.
We denote by $\|f\|_\rho$ the norm
$$
\|f\|_\rho=\sum_{k\in\z^r}{|f_k|e^{|k|\rho}},
$$
and it is not difficult to check that it is well defined for any
analytical quasiperiodic function defined on a strip of width $\rho$.
Finally, to define an analytic quasiperiodic matrix, we note that all
these definitions hold when $f$ is a matrix-valued function. In this
case, to define $\|f\|_{\rho}$ we use the infinity norm (that will be
denoted by $|\cdot|_{\infty}$) for the matrices $f_k$.
After those definitions and properties, let us return to the problem
of the reducibility of a linear quasiperiodic equation,
$\dot{x}=\widehat{A}(t)x$, to constant coefficients. The approach of
this work is to assume that the system is close to constant
coefficients, that is, $\widehat{A}(t)=A+\varepsilon Q(t,\varepsilon)$,
where $\varepsilon$ is small. This case has already been considered in
many papers (see \cite{BMS}, \cite{JS92} and \cite{JS94} among others),
and the results can be summarized as follows: let $\lambda_i$ be the
eigenvalues of $A$, and $\alpha_{ij}=\lambda_i-\lambda_j$, for $i\ne j$.
Then, if all the values $\mbox{\rm Re }\alpha_{ij}$ are different from
zero, the reduction can be performed for $|\varepsilon|<\varepsilon_0$,
$\varepsilon_0$ sufficiently small (see \cite{BMS}). If some of the
$\mbox{\rm Re }\alpha_{ij}$ are zero (this happens, for instance, if
$A$ is elliptic, that is, if all the $\lambda_i$ are on the imaginary
axis) more hypothesis are needed. The usual one is a diophantine
condition involving
the $\alpha_{ij}$ and the basic frequencies of $Q(t,\varepsilon)$, and
to assume a nondegeneracy condition with respect to $\varepsilon$ on the
corresponding $\alpha_{ij}(\varepsilon)$ of the matrix
$A+\varepsilon\Bar Q(\varepsilon)$ ($\Bar Q(\varepsilon)$ denotes the
average of $Q(t,\varepsilon)$). This allows
to prove (see \cite{JS94} for the details) that there exists a
Cantorian set ${\cal E}$ such that the reduction can be performed for
all $\varepsilon\in{\cal E}$. Moreover, the relative measure of the set
$[0,\,\varepsilon_0]\setminus{\cal E}$ in $[0,\,\varepsilon_0]$
is exponentially small in $\varepsilon_0$.
Our purpose here is a little bit different: instead of looking for
a total reduction to constant coefficients (this seems to lead us to
eliminate a dense set of values of $\varepsilon$, see \cite{JS92}
or \cite{JS94}), we try to minimize the quasiperiodic part, without
taking out any value of $\varepsilon$. The result obtained is that
the quasiperiodic part can be made exponentially small. As all the
proof is constructive (and it can be carried out with a finite number
of steps), it can be applied to practical examples
in order to do an ``effective" reduction: if $\varepsilon$ is small
enough, the remainder will be so small that, for practical purposes,
it can be taken equal to zero. The error produced with this dropping
can be bounded easily, by means of the Gronwall lemma. Finally,
we want to stress that we have also eliminated the nondegeneracy
hypothesis of previous papers (\cite{JS92}, \cite{JS94}).
Before finishing this introduction, we want to mention some similar
results obtained when the dynamics of the system is slow:
$\dot{x}=\varepsilon (A+\varepsilon Q(t,\varepsilon))x$. This case
is contained in \cite{S}, which is an extension of \cite{N}. The
result obtained is also that the quasiperiodic part can be
made exponentially small in $\varepsilon$. Total reducibility
has been also considered in this case: in \cite{T} is stated that
the reduction can be performed except for a set of values of
$\varepsilon$ of measure exponentially small.
There are many other results for the reducibility problem. For instance,
in the case of the Schr\"odinger equation with quasiperiodic potential
we can mention \cite{C}, \cite{DS}, \cite{El}, \cite{MP}, \cite{MP2}
and \cite{R}. Another classical and remarkable paper is \cite{JS},
where the general case (that is, without asking to be close to constant
coefficients) is considered. Finally, the classical results for
quasiperiodic systems can be found in \cite{F}.
In order to simplify the reading, the paper has been divided
in sections as follows: Section \ref{sec01} contains the exposition
(without technical details) of the main ideas and methodology,
Section \ref{sec1} contains the main theorem, Sections \ref{sec2}
and \ref{sec3} are devoted to the proofs and, finally, Section
\ref{sec4} contains an example to show how these results can be
applied to a concrete problem.
\section{The method}\label{sec01}
The method used is based on the same inductive scheme that \cite{JS92}.
Let us write our equation as
\begin{equation}
\dot{x}=(A+\varepsilon Q(t,\varepsilon))x, \label{eq:ini}
\end{equation}
where $A$ is an elliptic $d\times d$ matrix and $Q(t,\varepsilon)$ is
quasiperiodic with $\omega=(\omega_1,\ldots,\omega_r)$ as vector of basic
frequencies, and analytic on a strip of width $\rho$. First of all,
let us rewrite this equation as
$$
\dot{x}=(A_0(\varepsilon)+\varepsilon\Til Q(t,\varepsilon)),
$$
where $A_0(\varepsilon)=A+\Bar Q(\varepsilon)$ and
$\Til Q(t,\varepsilon)=Q(t,\varepsilon)-\Bar Q(\varepsilon)$.
Now let us assume that we are able to find a quasiperiodic $d\times d$
matrix $P$ (with the same basic frequencies than $Q$) verifying
\begin{equation}
\dot{P}=A_0(\varepsilon)P-PA_0(\varepsilon)+\Til Q(t,\varepsilon),
\label{eq:P}
\end{equation}
such that $\|\varepsilon P(t,\varepsilon)\|_{\sigma}<1$, for some
$\sigma>0$. In this case, it is not difficult to check that the change
of variables $x=(I+\varepsilon P(t,\varepsilon))y$ transforms equation
(\ref{eq:ini}) into
\begin{equation}
\dot{y}=(A_0(\varepsilon)+\varepsilon^2(I+\varepsilon
P(t,\varepsilon))^{-1}\Til Q(t,\varepsilon)P(t,\varepsilon)).
\label{eq:unpas}
\end{equation}
As this equation is like (\ref{eq:ini}) but with $\varepsilon^2$
instead of $\varepsilon$, the inductive scheme seems clear: to average
the quasiperiodic part of (\ref{eq:unpas}) and to restart this process.
The main difficulty that appear in this process comes from equation
(\ref{eq:P}), because the solution contains the denominators
$\lambda_i(\varepsilon)-\lambda_j(\varepsilon)+\sqrt{-1}(k,\omega)$,
$1\le i,\, j\le d$, where $\lambda_i(\varepsilon)$ are the eigenvalues
of $A_0(\varepsilon)$ (this is shown inside the proof of Lemma
\ref{lem22}). This divisor appears in the $k$th Fourier coefficient
of $P$. Note that if the values $\lambda_i(\varepsilon)-
\lambda_j(\varepsilon)$ are outside the imaginary axis, the
(modulus of the) divisor can be bounded from below, being easy to prove
the convergence. On the other hand, the value
$\lambda_i(\varepsilon)-\lambda_j(\varepsilon)+\sqrt{-1}(k,\omega)$ can
be arbitrarily small giving rise to convergence problems.
\subsection{Avoiding the small divisors}
Let us start assuming that the eigenvalues $\lambda_i$ of the original
unperturbed matrix $A$ (see equation (\ref{eq:ini})) and the basic
frequencies of $Q$ satisfy the diophantine condition
\begin{equation}
|\lambda_i-\lambda_j+\sqrt{-1}(k,\omega)|\ge\frac{c}{|k|^{\gamma}},
\;\;\forall k\in\z^r\setminus\{ 0\}. \label{eq:dio}
\end{equation}
where $|k|=|k_1|+\cdots+|k_r|$.
Note that, in principle, we can not guarantee that in equation
(\ref{eq:P}) this condition holds, because the eigenvalues
of $A_0(\varepsilon)$ have been changed with respect to the ones of $A$
(in an amount of ${\cal O}(\varepsilon)$) and some of the
divisors can be very small or even zero.
The key point is to realize that, as the eigenvalues of $A$ move
in an amount of ${\cal O}(\varepsilon)$ at most, the quantities
$\lambda_i(\varepsilon)-\lambda_j(\varepsilon)$ are contained in a
(complex) ball $B_{i,j}(\varepsilon)$ centered
in $\lambda_i-\lambda_j$ and with radius ${\cal O}(\varepsilon)$.
As the centre of the ball satisfies condition (\ref{eq:dio}),
the values $(k,\omega)$ can not be inside that ball if $|k|$ is
less than some value $M(\varepsilon)$. This implies that it is possible
to cancel all the harmonics such that $0<|k|0$.
\item The vector $\omega$ satisfies the diophantine conditions
\begin{equation}
|\lambda_{j}-\lambda_{\ell}+i(k,\omega)|\ge
\frac{c}{|k|^{\gamma}}, \;\;\;\forall \, k\in\z^{r}\setminus\{0\},
\;\;\;\forall\,j,\,\ell\in\{1,\ldots,d\},\label{eq:dio2}
\end{equation}
for some constants $c>0$, $\gamma >r-1$. As usual, $|k|=|k_{1}|+\cdots+
|k_{r}|$.
\end{enumerate}
Then there exist positive constants $\varepsilon^{\ast}$, $a^{\ast}$,
$r^{\ast}$ and $m$ such that for all $\varepsilon$,
$|\varepsilon|\le\varepsilon^{\ast}$, the initial equation can be
transformed into
\begin{equation} \label{una}
\dot y=(A^{\ast }(\varepsilon )+\varepsilon R^{\ast }(t,\varepsilon
))y\, ,
\end{equation}
where:
\begin{enumerate}
\item $A^{\ast }$ is a constant matrix with $|A^{\ast }(\varepsilon)-
A|_{\infty}\leq a^{\ast}|\varepsilon |$.
\item $R^{\ast }(\cdot,\varepsilon )\in{\cal Q}_{d}(\rho ,\omega )$ and
$\|R^{\ast }(\cdot,\varepsilon )\|_{\rho-\delta }\le r^{\ast}
\exp\left(-\left(\frac{m}{|\varepsilon|}\right)^{1/\gamma}\delta
\right)$, $\forall \, \delta \in ]0,\rho ]$.
\end{enumerate}
Furthermore the quasiperiodic change of variables that performs this
transformation is also an element of ${\cal Q}_{d}(\rho ,\omega )$.
Finally, a general explicit computation of $\varepsilon^{\ast}$,
$a^{\ast}$, $r^{\ast}$ and $m$ is possible:
$$
\varepsilon^{\ast }=\min\left(\varepsilon_{0},\,\frac{\alpha}{eq\beta
(3d-1)}\right),\,\, a^{\ast}=\frac{eq\beta^{2}}{e-1},\,\,
r^{\ast}=ea^{\ast},\,\, m=\frac{c}{10eq\beta }
$$
where $e=\exp(1)$, $\alpha=\min_{j\not=\ell}
(|\lambda _{j}-\lambda_{\ell}|)$ and $\beta $ is the condition number
of a regular matrix $S$ such that $S^{-1}AS$ is diagonal, that is,
$\beta=C(S)=|S^{-1}|_{\infty }|S|_{\infty }$.
\end{th}
\begin{rem}\label{rem11}
For fixed values of $\lambda_{1},\dots,\lambda _{d}$ and $\gamma$
hypothesis 3 is not satisfied for any $c>0$ only for a
set of values of $\omega $ of zero measure if $\gamma>r-1$.
\end{rem}
\begin{rem}
In case that the eigenvalues of the perturbed matrices move on balls
of radius ${\cal O}(\varepsilon^p)$ (that is, if the nondegeneracy
hypothesis needed in \cite{JS92} or \cite{JS94} is not satisfied),
it is not difficult to show that the bound of the exponential can
be improved: $\|R^{\ast}(\cdot,\varepsilon)\|_{\rho-\delta}\le
r^{\ast}\exp(-(m/|\varepsilon|)^{p/\gamma}\delta)$. The proof is very
similar, but using $M(\varepsilon)=(m/|\varepsilon|)^{p/\gamma}$
instead of $(m/|\varepsilon|)^{1/\gamma}$.
\end{rem}
This last remark seems to show that this nondegeneracy hypothesis
is not necessary, and it is only used for technical reasons.
In fact, the results seem to be better when this hypothesis is not
satisfied.
\begin{rem}
If the unperturbed matrix $A$ has multiple eigenvalues (that is,
if hypothesis 1 is not satisfied) the theorem is still true, but the
exponent of $\varepsilon$ in the exponential of the remainder is
slightly worse. This happens because the (small) divisors are now
raised to a power that increases with the multiplicity of the
eigenvalues. The proof is not included, since it does not introduce new
ideas and the technical details are rather tedious.
\end{rem}
\begin{rem}
The values of $\varepsilon^{\ast}$, $a^{\ast}$, $r^{\ast}$ and $m$ given
in the theorem are rather pessimistic. In the proof, we have preferred
to use simple (but rough) bounds instead of cumbersome but more accurate
ones. If one is interested in realistic bounds for a given problem, the
best thing to do is to rewrite the proof for that particular case. We
have done this in Section \ref{sec4} where, with the help of a computer
program, we have applied some steps of the method to an example. This
allows not only to obtain better bounds, but also to obtain (numerically)
the reduced matrix as well as the corresponding change of variables.
\end{rem}
\section{Lemmas}\label{sec2}
We will use some lemmas to simplify the proof of the theorem.
\subsection{Basic lemmas}
\begin{lem}\label{lem12}
Let $Q(t)=\displaystyle{\sum_{k\in\z^{r}}} Q_{k}e^{i(k,\omega )t}$ be
an element of ${\cal Q}_{d}(\rho ,\omega )$ and $M>0$. Let us define
$\Bar Q=Q_{0}$, $\Til Q(t)=Q(t)-Q_{0}$,
$$
Q_{\geq M}(t)=\sum_{k\in\z^{r}\atop|k|\geq M} Q_{k}e^{i(k,\omega)t},
$$
and $\Til Q_{
\frac{c}{|k|^{\gamma}}-2q^{*}|\varepsilon|> \\
& > & \left(\frac{c}{m}-2q^{*}\right)|\varepsilon|=L|\varepsilon|,
\end{eqnarray*}
and hypothesis 3 of Lemma \ref{lem32} is verified.
In consequence the iterative process can be carried out and Lemma
\ref{lem42} ensures the convergence of the process.
The composition of all the changes $I+\varepsilon P_{n}$ is
convergent because $\|I+\varepsilon P_{n}\|_{\rho }\leq
1+q_{n}/2$. Then the final equation is
\begin{equation}\label{nou}
\dot x_{\infty }=(A_{\infty }(\varepsilon )+\varepsilon R_{\infty
}(t,\varepsilon))x_{\infty },\quad |\varepsilon|\le\varepsilon^{\ast},
\end{equation}
where $|A_{\infty}(\varepsilon)-D|_{\infty}\leq
q^{\ast }a_{\infty}|\varepsilon|\leq\frac{e\beta}{e-1}q|\varepsilon|$,
and
$$
\| R_{\infty }(\cdot,\varepsilon )\| _{\rho -\delta }\leq
q^{\ast }r_{\infty }e^{-M(\varepsilon )\delta }\leq \frac{e^{2}\beta
}{e-1}q\exp\left\{-\left(\frac{m}{|\varepsilon|}\right)^{1/\gamma
}\delta \right \}, \, \, \forall \, \delta \in ]0,\rho ].
$$
To end up the proof, the change $x_{\infty }=S^{-1}y$ transforms
equation (\ref{nou}) into equation (\ref{una}) with the bounds that we
were looking for.\hfill \Box
\section{An example}\label{sec4}
The results of this paper can be applied in many ways, according to
the kind of problem we are interested in. Let us illustrate this with
the help of an example.
Let us consider the equation
\begin{equation}
\ddot{x}+(1+\varepsilon q(t))x=0, \label{schro}
\end{equation}
where $q(t)=\cos(\omega_1 t)+\cos(\omega_2 t)$, being $\omega_1=
\sqrt{2}$ and $\omega_2=\sqrt{3}$. Defining $y$ as $\dot{x}$ we can
rewrite (\ref{schro}) as
\begin{equation}
\left(
\begin{array}{c}
\dot{x}\\ \dot{y}
\end{array}
\right) = \left[\left(
\begin{array}{rr}
0 & 1 \\
-1 & 0
\end{array}
\right)+\varepsilon\left(
\begin{array}{cr}
0 & 0 \\
-q(t) & 0
\end{array}
\right)\right]\left(
\begin{array}{c}
x\\ y
\end{array}
\right). \label{schro2}
\end{equation}
As $\lambda_{1,2}=\pm i$, the diophantine condition (\ref{eq:dio2})
is satisfied for $\gamma=1$ (because the frequencies are quadratic
irrationals). The value of $c$ will be discussed later.
For the sake of simplicity, let us take $\rho=2$
and $\delta=1$. This implies that $q=\|Q\|_{\rho}=2e^2$. It is not
difficult to derive $\beta=2$ and, finally,
$\varepsilon^{\ast}=4.9787\ldots\times 10^{-3}$ and
$r^{\ast}=2.5419\ldots\times 10^{2}$.
The value of $c$ might be calculated for all $k=(k_1,\,k_2)$, but
better (bigger) values can be used since
we only need to consider $|k|$ up to a finite order. For instance,
an easy computation shows that for $|k|\le 125$ $c$ is $0.149$.
If $|k|=126$, then $c$ must be $0.013$ at most, due to
the quasiresonance produced by $k=(70,-56)$. In the range
$126\le|k|\le 10^5$ there are no more relevant resonances, so the
value $c=0.013$ suffices.
To start the discussion, let us suppose that the value of
$\varepsilon$ in (\ref{schro2}) is $\varepsilon=
2\times 10^{-6}$. If we take $c=0.149$ we obtain that
$m=1.8545\ldots\times 10^{-4}$ and $M=93$ (recall that the process
cancels frequencies such that $|k|