Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems

Many problems in Physics are described by dynamical systems that are conformally symplectic (e.g., mechanical systems with a friction proportional to the velocity, variational problems with a small discount or thermostated systems). Conformally symplectic systems are characterized by the property that they transform a symplectic form into a multiple of itself. The limit of small dissipation, which is the object of the present study, is particularly interesting. We provide all details for maps, but we present also the modifications needed to obtain a direct proof for the case of differential equations. We consider a family of conformally symplectic maps $f_{\mu, \epsilon}$ defined on a $2d$-dimensional symplectic manifold $\mathcal M$ with exact symplectic form $\Omega$; we assume that $f_{\mu,\epsilon}$ satisfies $f_{\mu,\epsilon}^*\Omega=\lambda(\epsilon) \Omega$. We assume that the family depends on a $d$-dimensional parameter $\mu$ (called drift) and also on a small scalar parameter $\epsilon$. Furthermore, we assume that the conformal factor $\lambda$ depends on $\epsilon$, in such a way that for $\epsilon=0$ we have $\lambda(0)=1$ (the symplectic case). We study the domains of analyticity in $\epsilon$ near $\epsilon=0$ of perturbative expansions (Lindstedt series) of the parameterization of the quasi--periodic orbits of frequency $\omega$ (assumed to be Diophantine) and of the parameter $\mu$. Notice that this is a singular perturbation, since any friction (no matter how small) reduces the set of quasi-periodic solutions in the system. We prove that the Lindstedt series are analytic in a domain in the complex $\epsilon$ plane, which is obtained by taking from a ball centered at zero a sequence of smaller balls with center along smooth lines going through the origin. The radii of the excluded balls decrease faster than any power of the distance of the center to the origin

Gravitational Collapse in Einstein dilaton Gauss-Bonnet Gravity

We present results from a numerical study of spherical gravitational collapse in shift symmetric Einstein dilaton Gauss-Bonnet (EdGB) gravity. This modified gravity theory has a single coupling parameter that when zero reduces to general relativity (GR) minimally coupled to a massless scalar field. We first show results from the weak EdGB coupling limit, where we obtain solutions that smoothly approach those of the Einstein-Klein-Gordon system of GR. Here, in the strong field case, though our code does not utilize horizon penetrating coordinates, we nevertheless find tentative evidence that approaching black hole formation the EdGB modifications cause the growth of scalar field "hair", consistent with known static black hole solutions in EdGB gravity. For the strong EdGB coupling regime, in a companion paper we first showed results that even in the weak field (i.e. far from black hole formation), the EdGB equations are of mixed type: evolution of the initially hyperbolic system of partial differential equations lead to formation of a region where their character changes to elliptic. Here, we present more details about this regime. In particular, we show that an effective energy density based on the Misner-Sharp mass is negative near these elliptic regions, and similarly the null convergence condition is violated then.Comment: 35 pages, 11 figures, edited to resemble journal versio

Implementation of standard testbeds for numerical relativity

We discuss results that have been obtained from the implementation of the initial round of testbeds for numerical relativity which was proposed in the first paper of the Apples with Apples Alliance. We present benchmark results for various codes which provide templates for analyzing the testbeds and to draw conclusions about various features of the codes. This allows us to sharpen the initial test specifications, design a new test and add theoretical insight.Comment: Corrected versio