Global periodicity conditions for maps and recurrences via Normal Forms

We face the problem of characterizing the periodic cases in parametric families of (real or complex) rational diffeomorphisms having a fixed point. Our approach relies on the Normal Form Theory, to obtain necessary conditions for the existence of a formal linearization of the map, and on the introduction of a suitable rational parametrization of the parameters of the family. Using these tools we can find a finite set of values p for which the map can be p-periodic, reducing the problem of finding the parameters for which the periodic cases appear to simple computations. We apply our results to several two and three dimensional classes of polynomial or rational maps. In particular we find the global periodic cases for several Lyness type recurrences.Comment: 25 page

Cosmo-dynamics and dark energy with a quadratic EoS: anisotropic models, large-scale perturbations and cosmological singularities

In general relativity, for fluids with a linear equation of state (EoS) or scalar fields, the high isotropy of the universe requires special initial conditions, and singularities are anisotropic in general. In the brane world scenario anisotropy at the singularity is suppressed by an effective quadratic equation of state. There is no reason why the effective EoS of matter should be linear at the highest energies, and a non-linear EoS may describe dark energy or unified dark matter (Paper I, astro-ph/0512224). In view of this, here we study the effects of a quadratic EoS in homogenous and inhomogeneous cosmological models in general relativity, in order to understand if in this context the quadratic EoS can isotropize the universe at early times. With respect to Paper I, here we use the simplified EoS P=alpha rho + rho^2/rho_c, which still allows for an effective cosmological constant and phantom behavior, and is general enough to analyze the dynamics at high energies. We first study anisotropic Bianchi I and V models, focusing on singularities. Using dynamical systems methods, we find the fixed points of the system and study their stability. We find that models with standard non-phantom behavior are in general asymptotic in the past to an isotropic fixed point IS, i.e. in these models even an arbitrarily large anisotropy is suppressed in the past: the singularity is matter dominated. Using covariant and gauge invariant variables, we then study linear perturbations about the homogenous and isotropic spatially flat models with a quadratic EoS. We find that, in the large scale limit, all perturbations decay asymptotically in the past, indicating that the isotropic fixed point IS is the general asymptotic past attractor for non phantom inhomogeneous models with a quadratic EoS. (Abridged)Comment: 16 pages, 6 figure

Devil's Staircase in Magnetoresistance of a Periodic Array of Scatterers

The nonlinear response to an external electric field is studied for classical non-interacting charged particles under the influence of a uniform magnetic field, a periodic potential, and an effective friction force. We find numerical and analytical evidence that the ratio of transversal to longitudinal resistance forms a Devil's staircase. The staircase is attributed to the dynamical phenomenon of mode-locking.Comment: two-column 4 pages, 5 figure

Perturbation Energy Production in Pipe Flow over a Range of Reynolds Numbers using Resolvent Analysis

The response of pipe flow to physically realistic, temporally and spatially continuous(periodic) forcing is investigated by decomposing the resolvent into orthogonal forcing and response pairs ranked according to their contribution to the resolvent 2-norm. Modelling the non-linear terms normally neglected by linearisation as unstructured forcing permits qualitative extrapolation of the resolvent norm results beyond infinitesimally small perturbations to the turbulent case. The concepts arising have a close relationship to input output transfer function analysis methods known in the control systems literature. The body forcings that yield highest disturbance energy gain are identified and ranked by the decomposition and a worst-case bound put on the energy gain integrated across the pipe cross-section. Analysis of the spectral variation of the corresponding response modes reveals interesting comparisons with recent observations of the behavior of the streamwise velocity in high Reynolds number (turbulent) pipe flow, including the importance of very long scales of the order of ten pipe radii, in the extraction of turbulent energy from the mean flow by the action of turbulent shear stress against the velocity gradient

ASYMPTOTIC BEHAVIOR OF COMPLEX SCALAR FIELDS IN A FRIEDMAN-LEMAITRE UNIVERSE

We study the coupled Einstein-Klein-Gordon equations for a complex scalar field with and without a quartic self-interaction in a curvatureless Friedman-Lema\^{\i}\-tre Universe. The equations can be written as a set of four coupled first order non-linear differential equations, for which we establish the phase portrait for the time evolution of the scalar field. To that purpose we find the singular points of the differential equations lying in the finite region and at infinity of the phase space and study the corresponding asymptotic behavior of the solutions. This knowledge is of relevance, since it provides the initial conditions which are needed to solve numerically the differential equations. For some singular points lying at infinity we recover the expected emergence of an inflationary stage.Comment: uuencoded, compressed tarfile containing a 15 pages Latex file and 2 postscipt figures. Accepted for publication on Phys. Rev.

Surface critical behavior of bcc binary alloys

The surface critical behavior of bcc binary alloys undergoing a continuous B2-A2 order-disorder transition is investigated in the mean-field (MF) approximation. Our main aim is to provide clear evidence for the fact that surfaces which break the two-sublattice symmetry generically display the critical behavior of the NORMAL transition, whereas symmetry-preserving surfaces exhibit ORDINARY surface critical behavior. To this end we analyze the lattice MF equations for both types of surfaces in terms of nonlinear symplectic maps and derive a Ginzburg-Landau model for the symmetry-breaking (100) surface. The crucial feature of the continuum model is the emergence of an EFFECTIVE ORDERING (``staggered'') SURFACE FIELD, which depends on temperature and the other lattice model parameters, and which explains the appearance of NORMAL critical behavior for symmetry-breaking surfaces.Comment: 16 pages, REVTeX 3.0, 13 EPSF figures, submitted to Phys. Rev.

Low-density series expansions for directed percolation III. Some two-dimensional lattices

We use very efficient algorithms to calculate low-density series for bond and site percolation on the directed triangular, honeycomb, kagom\'e, and $(4.8^2)$ lattices. Analysis of the series yields accurate estimates of the critical point $p_c$ and various critical exponents. The exponent estimates differ only in the $5^{th}$ digit, thus providing strong numerical evidence for the expected universality of the critical exponents for directed percolation problems. In addition we also study the non-physical singularities of the series.Comment: 20 pages, 8 figure

Evolution of the Bianchi I, the Bianchi III and the Kantowski-Sachs Universe: Isotropization and Inflation

We study the Einstein-Klein-Gordon equations for a convex positive potential in a Bianchi I, a Bianchi III and a Kantowski-Sachs universe. After analysing the inherent properties of the system of differential equations, the study of the asymptotic behaviors of the solutions and their stability is done for an exponential potential. The results are compared with those of Burd and Barrow. In contrast with their results, we show that for the BI case isotropy can be reached without inflation and we find new critical points which lead to new exact solutions. On the other hand we recover the result of Burd and Barrow that if inflation occurs then isotropy is always reached. The numerical integration is also done and all the asymptotical behaviors are confirmed.Comment: 22 pages, 12 figures, Self-consistent Latex2e File. To be published in Phys. Rev.