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### Open Access Journals

DCDS-B

In this paper we develop several numerical algorithms for the
computation of invariant manifolds in quasi-periodically forced
systems. The invariant manifolds we consider are invariant tori and
the asymptotic invariant manifolds (whiskers) to these tori.

The algorithms are based on the parameterization method described in [36], where some rigorous results are proved. In this paper, we concentrate on numerical issues of algorithms. Examples of implementations appear in the companion paper [34].

The algorithms for invariant tori are based essentially on Newton method, but taking advantage of dynamical properties of the torus, such as hyperbolicity or reducibility as well as geometric properties.

The algorithms for whiskers are based on power-matching expansions of the parameterizations. Whiskers include as particular cases the usual (strong) stable and (strong) unstable manifolds, and also, in some cases, the slow manifolds which dominate the asymptotic behavior of solutions converging to the torus.

The algorithms are based on the parameterization method described in [36], where some rigorous results are proved. In this paper, we concentrate on numerical issues of algorithms. Examples of implementations appear in the companion paper [34].

The algorithms for invariant tori are based essentially on Newton method, but taking advantage of dynamical properties of the torus, such as hyperbolicity or reducibility as well as geometric properties.

The algorithms for whiskers are based on power-matching expansions of the parameterizations. Whiskers include as particular cases the usual (strong) stable and (strong) unstable manifolds, and also, in some cases, the slow manifolds which dominate the asymptotic behavior of solutions converging to the torus.

DCDS

We use the parameterization method to prove the existence
and properties of one-dimensional submanifolds of the center
manifold associated to the fixed point of $C^r$
maps with linear part equal to the identity.
We also provide some numerical experiments to test the method in these cases.

DCDS-S

In KAM theory and other areas of analysis, one is often led to consider sums
of functions defined in decreasing domains. A question of interest is whether
the limit function is differentiable or not.

We present examples showing that the answer cannot be based just on the size of the derivatives but that it also has to include considerations of the geometry of the domains.

We also present some sufficient conditions on the geometry of the domains that ensure that indeed the sum of the derivatives is a Whitney derivative of the sum of the functions.

We present examples showing that the answer cannot be based just on the size of the derivatives but that it also has to include considerations of the geometry of the domains.

We also present some sufficient conditions on the geometry of the domains that ensure that indeed the sum of the derivatives is a Whitney derivative of the sum of the functions.

JMD

We study the effect of weak noise on critical
one-dimensional maps; that is, maps with a renormalization
theory.

We establish a one-dimensional central limit theorem for weak noise and obtain Berry--Esseen estimates for the rate of this convergence.

We analyze in detail maps at the accumulation of period doubling and critical circle maps with golden mean rotation number. Using renormalization group methods, we derive scaling relations for several features of the effective noise after long periods. We use these scaling relations to show that the central limit theorem for weak noise holds in both examples.

We note that, for the results presented here, it is essential that the maps have parabolic behavior. They are false for hyperbolic orbits.

We establish a one-dimensional central limit theorem for weak noise and obtain Berry--Esseen estimates for the rate of this convergence.

We analyze in detail maps at the accumulation of period doubling and critical circle maps with golden mean rotation number. Using renormalization group methods, we derive scaling relations for several features of the effective noise after long periods. We use these scaling relations to show that the central limit theorem for weak noise holds in both examples.

We note that, for the results presented here, it is essential that the maps have parabolic behavior. They are false for hyperbolic orbits.

DCDS

We consider the dependence on parameters of the solutions of
cohomology equations over Anosov diffeomorphisms. We show that the
solutions depend on parameters as smoothly as the data. As a
consequence we prove optimal regularity results for the solutions of
cohomology
equations taking value in diffeomorphism groups. These results are
motivated by applications to rigidity theory, dynamical systems, and
geometry.

In particular, in the context of diffeomorphism groups we show: Let $f$ be a transitive Anosov diffeomorphism of a compact manifold $M$. Suppose that $\eta \in C$

In particular, in the context of diffeomorphism groups we show: Let $f$ be a transitive Anosov diffeomorphism of a compact manifold $M$. Suppose that $\eta \in C$

^{k+α}$(M,$Diff$^r(N))$ for a compact manifold $N$, $k,r \in \N$, $r \geq 1$, and $0 < \alpha \leq \Lip$. We show that if there exists a $\varphi\in C$^{k+α}$(M,$Diff$^1(N))$ solving$ \varphi_{f(x)} = \eta_x \circ \varphi_x$

then in fact $\varphi \in C$^{k+α}$(M,$Diff$^r(N))$.
The existence of this solutions for some range of regularities is
studied in the literature.

keywords:
Anosov diffeomorphisms
,
Cohomology equations
,
rigidity.
,
diffeomorphism groups
,
Livšic
theory

NHM

We investigate the differentiability of minimal average
energy associated to the functionals
$S_\epsilon (u) = \int_{\mathbb{R}^d} \frac{1}{2}|\nabla u|^2 + \epsilon V(x,u)\, dx$,
using numerical and perturbative methods. We use
the Sobolev gradient descent method as a numerical tool to
compute solutions of the Euler-Lagrange equations
with some periodicity conditions; this is
the cell problem in homogenization.
We use these solutions to determine the average minimal energy
as a function of the slope.
We also obtain a representation of the solutions to the Euler-Lagrange
equations as a Lindstedt series in the perturbation parameter
$\epsilon$, and use this to confirm our numerical results. Additionally, we
prove convergence of the Lindstedt series.

DCDS

We prove the existence of certain analytic invariant manifolds
associated with fixed points
of analytic symplectic and volume preserving
diffeomorphisms. The manifolds we discuss are
not defined in terms of either forward or backward
asymptotic convergence to the fixed point, and
are not required to be stable or unstable.
Rather, the manifolds we consider
are defined as being tangent to certain "mixed-stable" linear
invariant subspaces of the differential (i.e linear subspace
which are spanned by some combination of stable and
unstable eigenvectors). Our method
is constructive, but has to face small divisors.
The small divisors are overcome
via a quadratic convergent scheme
which relies heavily on the geometry of the problem
as well as assuming some Diophantine properties of the
linearization restricted to the invariant subspace.
The theorem proved has an

*a-posteriori*format (i.e. given an approximate solution with good condition number, there is one exact solution close by). The method of proof also leads to efficient algorithms.
ERA-MS

We describe a method to study the existence of
whiskered quasi-periodic solutions in Hamiltonian
systems.
The method applies to finite dimensional systems
and also to lattice systems and to PDE's including
some ill posed ones.
In coupled map lattices, we can also
construct solutions of infinitely many frequencies
which do not vanish asymptotically.

PROC

We present numerical algorithms that use small
requirements of storage and operations to compute the
iteration of cocycles over a rotation.
We also show that these algorithms can be used to compute
efficiently the stable and unstable bundles and the
Lyapunov exponents of the cocycle.

DCDS

We study the existence of quasi--periodic solutions of
the equation
\[
ε \ddot x + \dot x + ε g(x) = ε f(\omega t)\ ,
\]
where $x: \mathbb{R} \rightarrow \mathbb{R}$ is
the unknown
and we are
given $g:\mathbb{R} \rightarrow \mathbb{R}$, $f: \mathbb{T}^d \rightarrow \mathbb{R}$,
$\omega \in \mathbb{R}^d$ (without loss of generality we can assume that $\omega\cdot k\not=0$
for any $k \in \mathbb{Z}^d\backslash\{0\}$).
We assume that there is a $c_0\in \mathbb{R}$
such that $g(c_0) = \hat f_0$ (where $\hat f_0$ denotes the
average of $f$) and $g'(c_0) \ne 0$. Special cases of this
equation, for example when $g(x)=x^2$, are called the ``varactor
problem'' in the literature.

We show that if $f$, $g$ are analytic, and $\omega$ satisfies some very mild irrationality conditions, there are families of quasi--periodic solutions with frequency $\omega$. These families depend analytically on $ε$, when $ε$ ranges over a complex domain that includes cones or parabolic domains based at the origin.

The irrationality conditions required in this paper are very weak. They allow that the small denominators $|\omega \cdot k|^{-1}$ grow exponentially with $k$. In the case that $f$ is a trigonometric polynomial, we do not need any condition on $|\omega \cdot k|$. This answers a delicate question raised in [8].

We also consider the periodic case, when $\omega$ is just a number ($d = 1$). We obtain that there are solutions that depend analytically in a domain which is a disk removing countably many disjoint disks. This shows that in this case there is no Stokes phenomenon (different resummations on different sectors) for the asymptotic series.

The approach we use is to reduce the problem to a fixed point theorem. This approach also yields results in the case that $g$ is a finitely differentiable function; it provides also very effective numerical algorithms and we discuss how they can be implemented.

We show that if $f$, $g$ are analytic, and $\omega$ satisfies some very mild irrationality conditions, there are families of quasi--periodic solutions with frequency $\omega$. These families depend analytically on $ε$, when $ε$ ranges over a complex domain that includes cones or parabolic domains based at the origin.

The irrationality conditions required in this paper are very weak. They allow that the small denominators $|\omega \cdot k|^{-1}$ grow exponentially with $k$. In the case that $f$ is a trigonometric polynomial, we do not need any condition on $|\omega \cdot k|$. This answers a delicate question raised in [8].

We also consider the periodic case, when $\omega$ is just a number ($d = 1$). We obtain that there are solutions that depend analytically in a domain which is a disk removing countably many disjoint disks. This shows that in this case there is no Stokes phenomenon (different resummations on different sectors) for the asymptotic series.

The approach we use is to reduce the problem to a fixed point theorem. This approach also yields results in the case that $g$ is a finitely differentiable function; it provides also very effective numerical algorithms and we discuss how they can be implemented.

## Year of publication

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