Bifurcations of homoclinic tangency in two-dimensional area-preserving and reversible maps

Report for the scientific sojourn at the Research Institute for Applied Mathematics and Cybernetics, Nizhny Novgorod, Russia, from July to September 2006. Within the project, bifurcations of orbit behavior in area-preserving and reversible maps with a homoclinic tangency were studied. Finitely smooth normal forms for such maps near saddle fixed points were constructed and it was shown that they coincide in the main order with the analytical Birkhoff-Moser normal form. Bifurcations of single-round periodic orbits for two-dimensional symplectic maps close to a map with a quadratic homoclinic tangency were studied. The existence of one- and two-parameter cascades of elliptic periodic orbits was proved.

Non-hyperbolic boundary equilibrium bifurcations in planar Filippov systems: a case study approach

Boundary equilibrium bifurcations in piecewise smooth discontinuous systems are characterized by the collision of an equilibrium point with the discontinuity surface. Generically, these bifurcations are of codimension one, but there are scenarios where the phenomenon can be of higher codimension. Here, the possible collision of a non-hyperbolic equilibrium with the boundary in a two-parameter framework and the nonlinear phenomena associated with such collision are considered. By dealing with planar discontinuous (Filippov) systems, some of such phenomena are pointed out through specific representative cases. A methodology for obtaining the corresponding bi-parametric bifurcation sets is developed.

Rational periodic sequences for the Lyness recurrence

Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our attention to positive rational values of $a$ and positive rational initial conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of $a,$ positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien \& Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness map is a universal normal form for most birrational maps on elliptic curves.

Analysis of bulk and surface contributions in the neutron skin of nuclei

The neutron skin thickness of nuclei is a sensitive probe of the nuclear symmetry energy and has multiple implications for nuclear and astrophysical studies. However, precision measurements of this observable are difficult to obtain. The analysis of the experimental data may imply some assumptions about the bulk or surface nature of the formation of the neutron skin. Here we study the bulk or surface character of neutron skins of nuclei following from calculations with Gogny, Skyrme, and covariant nuclear mean-field interactions. These interactions are successful in describing nuclear charge radii and binding energies but predict different values for neutron skins. We perform the study by fitting two-parameter Fermi distributions to the calculated self-consistent neutron and proton densities. We note that the equivalent sharp radius is a more suitable reference quantity than the half-density radius parameter of the Fermi distributions to discern between the bulk and surface contributions in neutron skins. We present calculations for nuclei in the stability valley and for the isotopic chains of Sn and Pb.

Intraocular Straylight And Contrast Sensitivity Two Months After Lasik: Intraocular Straylight Measurement As A New Parameter In Visual Quality Assessment

PROPÒSIT: Estudiar l'efecte de la cirurgia LASIK en la llum dispersa i la sensibilitat al contrast. MÈTODES: Vint-i-vuit pacients van ser tractats amb LASIK. La qualitat visual es va avaluar abans de l'operació i dos mesos després. RESULTATS: La mitjana de llum dispersa i la sensibilitat al contrast abans de l'operació no va canviar en dos mesos després. Només un ull tenia un marcat augment en la llum dispersa. Nou ulls van presentar una lleugera disminució en la sensibilitat al contrast. S'han trobat dues complicacions. CONCLUSIÓ: Després de LASIK la majoria dels pacients (80%) no van tenir complicacions i van mantenir la seva qualitat visual. Uns pocs pacients (16%) van tenir una mica de qualitat visual disminuïda. Molt pocs (4%) van tenir complicacions clíniques amb disminució en la qualitat visual.

Two different 3D biogeochemical models for the NW mediterranean sea: validation with Meris data and intercomparison

Report for the scientific sojourn carried out at the University of New South Wales from February to June the 2007. Two different biogeochemical models are coupled to a three dimensional configuration of the Princeton Ocean Model (POM) for the Northwestern Mediterranean Sea (Ahumada and Cruzado, 2007). The first biogeochemical model (BLANES) is the three-dimensional version of the model described by Bahamon and Cruzado (2003) and computes the nitrogen fluxes through six compartments using semi-empirical descriptions of biological processes. The second biogeochemical model (BIOMEC) is the biomechanical NPZD model described in Baird et al. (2004), which uses a combination of physiological and physical descriptions to quantify the rates of planktonic interactions. Physical descriptions include, for example, the diffusion of nutrients to phytoplankton cells and the encounter rate of predators and prey. The link between physical and biogeochemical processes in both models is expressed by the advection-diffusion of the non-conservative tracers. The similarities in the mathematical formulation of the biogeochemical processes in the two models are exploited to determine the parameter set for the biomechanical model that best fits the parameter set used in the first model. Three years of integration have been carried out for each model to reach the so called perpetual year run for biogeochemical conditions. Outputs from both models are averaged monthly and then compared to remote sensing images obtained from sensor MERIS for chlorophyll.

Asynchronous growth and competition in a two-sex age-structured population model

Asynchronous exponential growth has been extensively studied in population dynamics. In this paper we find out the asymptotic behaviour in a non-linear age-dependent model which takes into account sexual reproduction interactions. The main feature of our model is that the non-linear process converges to a linear one as the solution becomes large, so that the population undergoes asynchronous growth. The steady states analysis and the corresponding stability analysis are completely made and are summarized in a bifurcation diagram according to the parameter R0. Furthermore the effect of intraspecific competition is taken into account, leading to complex dynamics around steady states.

Vacancy-driven ordering in a two-dimensional binary alloy

Domain growth in a system with nonconserved order parameter is studied. We simulate the usual Ising model for binary alloys with concentration 0.5 on a two-dimensional square lattice by Monte Carlo techniques. Measurements of the energy, jump-acceptance ratio, and order parameters are performed. Dynamics based on the diffusion of a single vacancy in the system gives a growth law faster than the usual Allen-Cahn law. Allowing vacancy jumps to next-nearest-neighbor sites is essential to prevent vacancy trapping in the ordered regions. By measuring local order parameters we show that the vacancy prefers to be in the disordered regions (domain boundaries). This naturally concentrates the atomic jumps in the domain boundaries, accelerating the growth compared with the usual exchange mechanism that causes jumps to be homogeneously distributed on the lattice.

Dynamics of the two-dimensional gonihedric spin model

In this paper, we study dynamical aspects of the two-dimensional (2D) gonihedric spin model using both numerical and analytical methods. This spin model has vanishing microscopic surface tension and it actually describes an ensemble of loops living on a 2D surface. The self-avoidance of loops is parametrized by a parameter ¿. The ¿=0 model can be mapped to one of the six-vertex models discussed by Baxter, and it does not have critical behavior. We have found that allowing for ¿¿0 does not lead to critical behavior either. Finite-size effects are rather severe, and in order to understand these effects, a finite-volume calculation for non-self-avoiding loops is presented. This model, like his 3D counterpart, exhibits very slow dynamics, but a careful analysis of dynamical observables reveals nonglassy evolution (unlike its 3D counterpart). We find, also in this ¿=0 case, the law that governs the long-time, low-temperature evolution of the system, through a dual description in terms of defects. A power, rather than logarithmic, law for the approach to equilibrium has been found.

Ground-state properties of a dilute homogeneous Bose gas of hard disks in two dimensions

The energy and structure of a dilute hard-disks Bose gas are studied in the framework of a variational many-body approach based on a Jastrow correlated ground-state wave function. The asymptotic behaviors of the radial distribution function and the one-body density matrix are analyzed after solving the Euler equation obtained by a free minimization of the hypernetted chain energy functional. Our results show important deviations from those of the available low density expansions, already at gas parameter values x~0.001 . The condensate fraction in 2D is also computed and found generally lower than the 3D one at the same x.

Potts fully frustrated model: Thermodynamics, percolation and dynamics in two dimensions

We consider a Potts model diluted by fully frustrated Ising spins. The model corresponds to a fully frustrated Potts model with variables having an integer absolute value and a sign. This model presents precursor phenomena of a glass transition in the high-temperature region. We show that the onset of these phenomena can be related to a thermodynamic transition. Furthermore, this transition can be mapped onto a percolation transition. We numerically study the phase diagram in two dimensions (2D) for this model with frustration and without disorder and we compare it to the phase diagram of (i) the model with frustration and disorder and (ii) the ferromagnetic model. Introducing a parameter that connects the three models, we generalize the exact expression of the ferromagnetic Potts transition temperature in 2D to the other cases. Finally, we estimate the dynamic critical exponents related to the Potts order parameter and to the energy.

Classical predictive electrodynamics of two charges with radiation: Energy and 3. momentum balance and scattering crossections-II

We deal with a classical predictive mechanical system of two spinless charges where radiation is considered and there are no external fields. The terms (2,2)Paa of the expansion in the charges of the HamiltonJacobi momenta are calculated. Using these, together with known previous results, we can obtain the paa up to the fourth order. Then we have calculated the radiated energy and the 3-momentum in a scattering process as functions of the impact parameter and the incident energy for the former and 3-momentum for the latter. Scattering cross-sections are also calculated. Good agreement with well known results, including those of quantum electrodynamics, has been found.

Two component systems: physiological effect of a third component

Signal transduction systems mediate the response and adaptation of organisms to environmental changes. In prokaryotes, this signal transduction is often done through Two Component Systems (TCS). These TCS are phosphotransfer protein cascades, and in their prototypical form they are composed by a kinase that senses the environmental signals (SK) and by a response regulator (RR) that regulates the cellular response. This basic motif can be modified by the addition of a third protein that interacts either with the SK or the RR in a way that could change the dynamic response of the TCS module. In this work we aim at understanding the effect of such an additional protein (which we call ‘‘third component’’) on the functional properties of a prototypical TCS. To do so we build mathematical models of TCS with alternative designs for their interaction with that third component. These mathematical models are analyzed in order to identify the differences in dynamic behavior inherent to each design, with respect to functionally relevant properties such as sensitivity to changes in either the parameter values or the molecular concentrations, temporal responsiveness, possibility of multiple steady states, or stochastic fluctuations in the system. The differences are then correlated to the physiological requirements that impinge on the functioning of the TCS. This analysis sheds light on both, the dynamic behavior of synthetically designed TCS, and the conditions under which natural selection might favor each of the designs. We find that a third component that modulates SK activity increases the parameter space where a bistable response of the TCS module to signals is possible, if SK is monofunctional, but decreases it when the SK is bifunctional. The presence of a third component that modulates RR activity decreases the parameter space where a bistable response of the TCS module to signals is possible.

Soluble models in two-dimensional dilaton gravity

A one-parameter class of simple models of two-dimensional dilaton gravity, which can be exactly solved including back-reaction effects, is investigated at both classical and quantum levels. This family contains the RST model as a special case, and it continuously interpolates between models having a flat (Rindler) geometry and a constant curvature metric with a nontrivial dilaton field. The processes of formation of black hole singularities from collapsing matter and Hawking evaporation are considered in detail. Various physical aspects of these geometries are discussed, including the cosmological interpretation.