A methodology for obtaining asymptotic estimates for the exponentially small splitting of separatrices to whiskered tori with quadratic frequencies

The aim of this work is to provide asymptotic estimates for the splitting of separatrices in a perturbed 3-degree-of-freedom Hamiltonian system, associated to a 2-dimensional whiskered torus (invariant hyperbolic torus) whose frequency ratio is a quadratic irrational number. We show that the dependence of the asymptotic estimates on the perturbation parameter is described by some functions which satisfy a periodicity property, and whose behavior depends strongly on the arithmetic properties of the frequencies.Comment: 5 pages, 1 figur

Acumulación e transferencia de contaminantes emerxentes nunha cadea trofica

Traballo fin de mestrado (UDC.CIE). Acuicultura. Curso 2017/2018[Resumen] Este estudio trata de investigar la potencial bioacumulación y transferencia de dos contaminantes emergentes, bisfenol- A (un retardante de llama) y triclosán (bactericida) a través de dos componentes del plancton, constituyentes de una cadena trófica (microalga Tetraselmis suecica y rotífero Brachionus plicatilis). Para ello se analizaron diferentes parámetros: crecimiento poblacional y cantidad de hembras ovadas y huevos por mililitro. Los resultados muestran una alteración significativa de estos parámetros respecto a los controles, evidenciando que el triclosán afecta en mayor grado que el bisfenol-A, y que la exposición a través del medio acuático, produce una mayor bioacumulación de contaminante en el rotífero, que a través del alimento.[Abstract] This study tries to research the potential bioaccumulation and transfer of two emerging pollutants, bisphenol-A (a flame retardant) and triclosan (bactericide) through two components of the plankton, constituents of a trophic chain (microalga Tetraselmis suecica and rotifer Brachionus plicatilis). To do this, different parameters were analyzed: population growth and number of ovate females and eggs per milliliter. The results show a significant alteration of these parameters with respect to the controls, showing that triclosan affects to a greater degree than bisphenol-A, and that the exposure through the aquatic environment, produces a greater bioaccumulation of contaminant in the rotifer, than through food.[Resumo] Este estudo trata de investigar a potencial bioacumulación e transferencia de dous contaminantes emerxentes, bisfenol- A (un retardante de chama) e triclosán (unha sustancia bactericida) a través de dous compoñentes do plancto, constituíntes dunha cadea trófica (a microalga Tetraselmis suecica e o rotífero Brachionus plicatilis). Para iso analizáronse diferentes parámetros: crecemento poboacional e cantidade de femias con ovos e número de ovos por mililitro. Os resultados mostran unha alteración significativa destes parámetros respecto dos controis, evidenciando que o triclosán afecta en maior grao que o bisfenol-A, e que a exposición a través do medio acuático, produce unha maior bioacumulación de contaminante no rotífero, que a través do alimento

Parentesco, poder y religiosidad en las fiestas públicas de la Buenos Aires Virreinal. 1780-1808.

A lo largo del siglo XVIII, las fiestas públicas se convirtieron en espacios de negociación y transformación de las estructuras y jerarquías sociales de grupos. Este trabajo se propone analizar las tensiones y conflictos de poder que se manifestaron en las elebraciones públicas, en el período comprendido entre la creación del Virreinato del Río de la Plata y la vacante regia de 1808, en Buenos Aires

Experimentación animal : problemática y legislación /

Treball presentat a l'assignatura de Deontologia i Veterinària Legal (21223

Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies

We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional toruswith a fast frequency vector¿/ve, with¿= (1,¿, ~¿) where ¿ is a cubic irrational number whose two conjugatesare complex, and the components of¿generate the fieldQ(¿). A paradigmatic case is the cubic golden vector,given by the (real) number ¿ satisfying ¿3= 1-¿, and ~¿ = ¿2. For such 3-dimensional frequency vectors,the standard theory of continued fractions cannot be applied, so we develop a methodology for determining thebehavior of the small divisors,k¿Z3. Applying the Poincaré-Melnikov method, this allows us tocarry outa careful study of the dominant harmonic (which depends one) of the Melnikov function, obtaining an asymptoticestimate for the maximal splitting distance, which is exponentially small ine, and valid for all sufficiently smallvalues ofe. This estimate behaves like exp{-h1(e)/e1/6}and we provide, for the first time in a system with 3frequencies, an accurate description of the (positive) functionh1(e) in the numerator of the exponent, showing thatit can be explicitly constructed from the resonance properties of the frequency vector¿, and proving that it is aquasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence ofthe estimates for the splitting on the arithmetic properties of the frequenciesPreprin

Exponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequency ratio

The splitting of invariant manifolds of whiskered (hyperbolic) tori with two frequencies in a nearly integrable Hamiltonian system, whose hyperbolic part is given by a pendulum, is studied. We consider a torus with a fast frequency vector $\omega/\sqrt\varepsilon$, with $\omega=(1,\Omega),$ where the frequency ratio $\Omega$ is a quadratic irrational number. Applying the Poincaré--Melnikov method, we carry out a careful study of the dominant harmonics of the Melnikov potential. This allows us to provide an asymptotic estimate for the maximal splitting distance and show the existence of transverse homoclinic orbits to the whiskered tori with an asymptotic estimate for the transversality of the splitting. Both estimates are exponentially small in $\varepsilon$, with the functions in the exponents being periodic with respect to $\ln\varepsilon$, and can be explicitly constructed from the continued fraction of $\Omega$. In this way, we emphasize the strong dependence of our results on the arithmetic properties of $\Omega$. In particular, for quadratic ratios $\Omega$ with a 1-periodic or 2-periodic continued fraction (called metallic and metallic-colored ratios, respectively), we provide accurate upper and lower bounds for the splitting. The estimate for the maximal splitting distance is valid for all sufficiently small values of $\varepsilon$, and the transversality can be established for a majority of values of $\varepsilon$, excluding small intervals around some transition values where changes in the dominance of the harmonics take place, and bifurcations could occur. Read More: http://epubs.siam.org/doi/10.1137/15M1032776Peer ReviewedPostprint (published version

Exponentially small lower bounds for the splitting of separatrices to whiskered tori with frequencies of constant type

We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearlyintegrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a fast frequency vector $\omega/v\epsilon$, with $\epsilon=(1,\Omega)$ where $\Omega$ is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincar´e–Melnikov method, we find exponentially small lower bounds for the maximal splitting distance between the stable and unstable invariant manifolds associated to the invariant torus, and we show that these bounds depend strongly on the arithmetic properties of the frequencies.Preprin

Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tort with quadratic and cubic frequencies

We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector omega = (1, Omega), where Omega is a quadratic irrational number, or a 3-dimensional torus with a frequency vector w = (1, Omega, Omega(2)), where Omega is a cubic irrational number. Applying the Poincare-Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which Q is the so-called cubic golden number (the real root of x(3) x - 1= 0), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases.Postprint (published version

Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies

We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector ¿/ev, with ¿=(1,O,O˜) where O is a cubic irrational number whose two conjugates are complex, and the components of ¿ generate the field Q(O). A paradigmatic case is the cubic golden vector, given by the (real) number O satisfying O3=1-O, and O˜=O2. For such 3-dimensional frequency vectors, the standard theory of continued fractions cannot be applied, so we develop a methodology for determining the behavior of the small divisors ¿k,¿¿, k¿Z3. Applying the Poincaré–Melnikov method, this allows us to carry out a careful study of the dominant harmonic (which depends on e) of the Melnikov function, obtaining an asymptotic estimate for the maximal splitting distance, which is exponentially small in e, and valid for all sufficiently small values of e. This estimate behaves like exp{-h1(e)/e1/6} and we provide, for the first time in a system with 3 frequencies, an accurate description of the (positive) function h1(e) in the numerator of the exponent, showing that it can be explicitly constructed from the resonance properties of the frequency vector ¿, and proving that it is a quasiperiodic function (and not periodic) with respect to lne. In this way, we emphasize the strong dependence of the estimates for the splitting on the arithmetic properties of the frequencies.Peer ReviewedPreprin