DCDS-B
A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms
Àlex Haro Rafael de la Llave
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.
keywords: Quasi-periodic systems numerical methods. invariant manifolds invariant tori
DCDS
The parameterization method for one- dimensional invariant manifolds of higher dimensional parabolic fixed points
Inmaculada Baldomá Ernest Fontich Rafael de la Llave Pau Martín
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.
keywords: Parabolic point parameterization method. invariant manifold
DCDS-S
Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''
Xuemei Li Rafael de la Llave
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.
keywords: converse approximation. KAM theory Whitney differentiability
JMD
Renormalization and central limit theorem for critical dynamical systems with weak external noise
Oliver Díaz-Espinosa Rafael de la Llave
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.
keywords: transfer operators period doubling central limit theorem effective noise. renormalization critical circle maps
DCDS
Smooth dependence on parameters of solutions to cohomology equations over Anosov systems with applications to cohomology equations on diffeomorphism groups
Rafael de la Llave A. Windsor
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 $\eta \in C$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
Perturbation and numerical methods for computing the minimal average energy
Timothy Blass Rafael de la Llave
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.
keywords: Lindstedt series Minimal average energy Sobolev gradient descent Plane-like minimizers Cell problem quasiperiodic solutions of PDE.
DCDS
Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps
Rafael de la Llave Jason D. Mireles James
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.
keywords: quadratic convergence Invariant manifolds KAM theory volume preserving mappings. symplectic mappings
ERA-MS
A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems
Ernest Fontich Rafael de la Llave Yannick Sire
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.
keywords: coupled oscillators. Whiskered tori quasi-periodic breathers Hamiltonian systems quasi-periodic solutions
PROC
Fast iteration of cocycles over rotations and computation of hyperbolic bundles
Gemma Huguet Rafael de la Llave Yannick Sire
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.
keywords: quasi-periodic cocycles Quasi-periodic solutions numerical computation.
DCDS
Construction of response functions in forced strongly dissipative systems
Renato C. Calleja Alessandra Celletti Rafael de la Llave
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.
keywords: quasi--periodic solutions fixed point theorem. Strongly dissipative systems

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