Dynamical Systems, Stability, and Chaos

In this expository and resources chapter we review selected aspects of the mathematics of dynamical systems, stability, and chaos, within a historical framework that draws together two threads of its early development: celestial mechanics and control theory, and focussing on qualitative theory. From this perspective we show how concepts of stability enable us to classify dynamical equations and their solutions and connect the key issues of nonlinearity, bifurcation, control, and uncertainty that are common to time-dependent problems in natural and engineered systems. We discuss stability and bifurcations in three simple model problems, and conclude with a survey of recent extensions of stability theory to complex networks.Comment: 28 pages, 10 figures. 26/04/2007: The book title was changed at the last minute. No other changes have been made. Chapter 1 in: J.P. Denier and J.S. Frederiksen (editors), Frontiers in Turbulence and Coherent Structures. World Scientific Singapore 2007 (in press

The C-Theorem and Chiral Symmetry Breaking in Asymptotically Free Vectorlike Gauge Theories

We confront Cardy's suggested c-function for four-dimensional field theories with the spontaneous breaking of chiral symmetries in asymptotically free vectorlike gauge theories with fermions transforming according to different representations under the gauge group. Assuming that the infrared limit of the c-function is determined by the dimension of the associated Goldstone manifold, we find that this c-function always decreases between the ultraviolet and infrared fixed points.Comment: 8 pages, no figures, a few references adde

BFKL at NNLO

We present a recent determination of an approximate expression for the O(alpha_s^3) contribution chi_2 to the kernel of the BFKL equation. This includes all collinear and anticollinear singular contributions and is derived using duality relations between the GLAP and BFKL kernels.Comment: 8 pages. Talk presented at 12th International Conference on Elastic and Diffractive Scattering: Forward Physics and QCD, Hamburg, DESY, Germany, 21-25 May 200

Lattice deformations at martensite-martensite interfaces in Ni-Al

The atomic configurations at macrotwin interfaces between microtwinned martensite plates in $Ni_{65}Al_{35}$ material are investigated using high resolution transmission electron microscopy (HRTEM). The observed structures are interpreted in view of possible formation mechanisms of these interfaces. A distinction is made between cases in which the microtwins, originating from mutually perpendicular \{110\} austenite planes, enclose a final angle larger or smaller than $90^{\circ}$, measured over the boundary. Two different configurations, one with crossing microtwins and the other with ending microtwins producing a step configuration are described. The latter is related with the existence of microtwin sequences with changing variant widths. Although both features appear irrespective of the material’s preparation technique, rapid solidification seems to prefer the step configuration. Depending on the actual case, tapering, bending and tip splitting of the small microtwin variants is observed. Sever lattice deformations and reorientations occur in a region of 5 – 10 nm around the interface while sequences of single plane ledges gradually bending the microtwins are found up to 50 nm away form the interface. These structures and deformations are interpreted in view of the need to accommodate any remaining stresses

Incompatible sets of gradients and metastability

We give a mathematical analysis of a concept of metastability induced by incompatibility. The physical setting is a single parent phase, just about to undergo transformation to a product phase of lower energy density. Under certain conditions of incompatibility of the energy wells of this energy density, we show that the parent phase is metastable in a strong sense, namely it is a local minimizer of the free energy in an $L^1$ neighbourhood of its deformation. The reason behind this result is that, due to the incompatibility of the energy wells, a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus. We define and characterize incompatible sets of matrices, in terms of which the transition layer estimate at the heart of the proof of metastability is expressed. Finally we discuss connections with experiment and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis.Comment: Archive for Rational Mechanics and Analysis, to appea

Bs0→η(′)η(′)B_s^0 \to \eta^{(\prime)} \eta^{(\prime)} decays in the pQCD approach

We calculate the CP averaged branching ratios and CP-violating asymmetries for $B_s^0 \to \eta \eta, \eta \eta^\prime$ and $\eta^\prime \eta^\prime$ decays in the perturbative QCD (pQCD) approach here. The pQCD predictions for the CP-averaged branching ratios are Br(B_s^0 \to \eta \eta) = \left (14.2^{+18.0}_{-7.5}) \times 10^{-6}, Br(B_s^0 \to \eta \eta^\prime)= \left (12.4 ^{+18.2}_{-7.0}) \times 10^{-6}, and Br(B_s^0 \to \eta^{\prime} \eta^{\prime}) = \left (9.2^{+15.3}_{-4.9}) \times 10^{-6}, which agree well with those obtained by employing the QCD factorization approach and also be consistent with available experimental upper limits. The gluonic contributions are small in size: less than 7% for $B_s \to \eta \eta$ and $\eta \eta^\prime$ decays, and around 18% for $B_s \to \eta' \eta'$ decay. The CP-violating asymmetries for three decays are very small: less than 3% in magnitude.Comment: 11 pages, 1 ps figure, Revte

Improved analysis of B → πeν from QCD sum rules on the light-cone

We present a new calculation of the Brightarrowpi form-factor f+, relevant for the measurement of |Vub| from semileptonic Brightarrowpi transitions, from QCD sum rules on the light-cone. The new element is the calculation of radiative corrections to next-to-leading twist-3 accuracy. We find that these contributions are factorizable at O(alphas), which lends additional support to the method of QCD sum rules on the light-cone. We obtain f+(0) = 0.26+/-0.06+/-0.05, where the first error accounts for the uncertainty in the input-parameters and the second is a guesstimate of the systematic uncertainty induced by the approximations inherent in the method. We also obtain a simple parametrization of the form-factor which is valid in the entire kinematical range of semileptonic decays and consistent with vector-meson dominance at large momentum-transfer

Bd,s→ρ,ω,K*,ϕ decay form factors from light-cone sum rules reexamined

We present an improved calculation of B--> light vector form factors from light-cone sum rules, including one-loop radiative corrections to twist-2 and twist-3 contributions, and leading order twist-4 corrections. The total theoretical uncertainty of our results at zero momentum transfer is typically 10% and can be improved, at least in part, by reducing the uncertainty of hadronic input parameters. We present our results in a way which details the dependence of the form factors on these parameters and facilitates the incorporation of future updates of their values from, e.g., lattice calculations. We also give simple and easy-to-implement parametrizations of the q2 dependence of the form factors which are valid in the full kinematical regime of q2