Systematic Derivation of Amplitude Equations and Normal Forms for Dynamical Systems

We present a systematic approach to deriving normal forms and related amplitude equations for flows and discrete dynamics on the center manifold of a dynamical system at local bifurcations and unfoldings of these. We derive a general, explicit recurrence relation that completely determines the amplitude equation and the associated transformation from amplitudes to physical space. At any order, the relation provides explicit expressions for all the nonvanishing coefficients of the amplitude equation together with straightforward linear equations for the coefficients of the transformation. The recurrence relation therefore provides all the machinery needed to solve a given physical problem in physical terms through an amplitude equation. The new result applies to any local bifurcation of a flow or map for which all the critical eigenvalues are semisimple i.e. have Riesz index unity). The method is an efficient and rigorous alternative to more intuitive approaches in terms of multiple time scales. We illustrate the use of the method by deriving amplitude equations and associated transformations for the most common simple bifurcations in flows and iterated maps. The results are expressed in tables in a form that can be immediately applied to specific problems.Comment: 40 pages, 1 figure, 4 tables. Submitted to Chaos. Please address any correspondence by email to [email protected]

Radial sine-Gordon kinks as sources of fast breathers

We consider radial sine-Gordon kinks in two, three and higher dimensions. A full two dimensional simulation showing that azimuthal perturbations remain small allows to reduce the problem to the one dimensional radial sine-Gordon equation. We solve this equation on an interval $[r_0,r_1]$ and absorb all outgoing radiation. Before collision the kink is well described by a simple law derived from the conservation of energy. In two dimensions for $r_0 \le 2$, the collision disintegrates the kink into a fast breather while for $r_0 \ge 4$ we obtain a kink-breather meta-stable state where breathers are shed at each kink "return". In three and higher dimensions $d$ a kink-pulson state appears for small $r_0$. The three states then exist as shown by a study of the $(d,r_0)$ parameter space. On the application side, the kink disintegration opens the way for new types of terahertz microwave generators

Close binary evolution. III. Impact of tides, wind magnetic braking, and internal angular momentum transport

Massive stars with solar metallicity lose important amounts of rotational angular momentum through their winds. When a magnetic field is present at the surface of a star, efficient angular momentum losses can still be achieved even when the mass-loss rate is very modest, at lower metallicities, or for lower-initial-mass stars. In a close binary system, the effect of wind magnetic braking also interacts with the influence of tides, resulting in a complex evolution of rotation. We study the interactions between the process of wind magnetic braking and tides in close binary systems. We discuss the evolution of a 10 M$_\odot$ star in a close binary system with a 7 M$_\odot$ companion using the Geneva stellar evolution code. The initial orbital period is 1.2 days. The 10 M$_\odot$ star has a surface magnetic field of 1 kG. Various initial rotations are considered. We use two different approaches for the internal angular momentum transport. In one of them, angular momentum is transported by shear and meridional currents. In the other, a strong internal magnetic field imposes nearly perfect solid-body rotation. The evolution of the primary is computed until the first mass-transfer episode occurs. The cases of different values for the magnetic fields and for various orbital periods and mass ratios are briefly discussed. We show that, independently of the initial rotation rate of the primary and the efficiency of the internal angular momentum transport, the surface rotation of the primary will converge, in a time that is short with respect to the main-sequence lifetime, towards a slowly evolving velocity that is different from the synchronization velocity. (abridged).Comment: 11 pages, 13 figures, accepted for publication in Astronomy and Astrophysic

Kink propagation in a two-dimensional curved Josephson junction

We consider the propagation of sine-Gordon kinks in a planar curved strip as a model of nonlinear wave propagation in curved wave guides. The homogeneous Neumann transverse boundary conditions, in the curvilinear coordinates, allow to assume a homogeneous kink solution. Using a simple collective variable approach based on the kink coordinate, we show that curved regions act as potential barriers for the wave and determine the threshold velocity for the kink to cross. The analysis is confirmed by numerical solution of the 2D sine-Gordon equation.Comment: 8 pages, 4 figures (2 in color

Multi-Decadal Decline of Mercury in the North Atlantic Atmosphere Explained by Changing Subsurface Seawater Concentrations

[1] We analyze 1977–2010 trends in atmospheric mercury (Hg) from 21 ship cruises over the North Atlantic (NA) and 15 over the South Atlantic (SA). We find a steep 1990–2009 decline of −0.046 ± 0.010 ng m−3 a−1 (−2.5% a−1) over the NA (steeper than at Northern Hemispheric land sites) but no significant decline over the SA. Surface water Hg0 measurements in the NA show a decline of −5.7% a−1since 1999, and limited subsurface ocean data show an ∼80% decline from 1980 to present. We use a coupled global atmosphere-ocean model to show that the decline in NA atmospheric concentrations can be explained by decreasing oceanic evasion from the NA driven by declining subsurface water Hg concentrations. We speculate that this large historical decline of Hg in the NA Ocean could have been caused by decreasing Hg inputs from rivers and wastewater and by changes in the oxidant chemistry of the atmospheric marine boundary layer.Engineering and Applied Science

Isotope Effect in the Presence of Magnetic and Nonmagnetic Impurities

The effect of impurities on the isotope coefficient is studied theoretically in the framework of Abrikosov-Gor'kov approach generalized to account for both potential and spin-flip scattering in anisotropic superconductors. An expression for the isotope coefficient as a function of the critical temperature is obtained for a superconductor with an arbitrary contribution of spin-flip processes to the total scattering rate and an arbitrary degree of anisotropy of the superconducting order parameter, ranging from isotropic s-wave to d-wave and including anisotropic s-wave and mixed (s+d)-wave as particular cases. It is found that both magnetic and nonmagnetic impurities enhance the isotope coefficient, the enhancement due to magnetic impurities being generally greater than that due to nonmagnetic impurities. From the analysis of the experimental results on La-Sr-Cu-M-O high temperature superconductor, it is concluded that the symmetry of the pairing state in this system differs from a pure d-wave.Comment: 4 pages, 3 figure

Subharmonic Shapiro steps and assisted tunneling in superconducting point contacts

We analyze the current in a superconducting point contact of arbitrary transmission in the presence of a microwave radiation. The interplay between the ac Josephson current and the microwave signal gives rise to Shapiro steps at voltages V = (m/n) \hbar \omega_r/2e, where n,m are integer numbers and \omega_r is the radiation frequency. The subharmonic steps (n different from 1) are a consequence of the ocurrence of multiple Andreev reflections (MAR) and provide an unambiguous signature of the peculiar ac Josephson effect at high transmission. Moreover, the dc current exhibits a rich subgap structure due to photon-assisted MARs.Comment: Revtex, 4 pages, 4 figure