Density functional theory (DFT) calculations and temperature programmed desorption (TPD) experiments were performed to study the adsorption of hydrogen on the Co(111) and Co(100) surfaces. On the Co(111) surface, hydrogen adsorption is coverage dependent and the calculated adsorption energies are very similar to those on the Co(0001) surface. The experimental adsorption saturation coverage on the Co(111)/(0001) surface is θ max ≈ 0.5 ML, although DFT predicts θ max ≈ 1.0 ML. DFT calculations indicate that preadsorbed hydrogen will kinetically impede the adsorption process as the coverage approaches θ = 0.5 ML, giving rise to this difference. Adsorption on Co(100) is coverage independent up to θ = 1.00 ML, contrasting observations on the Ni(100) surface. Hydrogen atoms have low barriers of diffusion on both the Co(111) and Co(100) surfaces. A microkinetic analysis of desorption, simulating the expected TPD experiments, indicated that on the Co(111) surface two TPD peaks are expected, while on the Co(100) only one peak is expected. Low coverage adsorption energies of between 0.97 and 1.1 eV are obtained from the TPD experiment on a smooth single crystal of Co(0001), in line with the DFT results. Defects play a important role in the adsorption process. Further calculations on the Co(211) and Co(221) surfaces have been performed to model the effects of step and defect sites, indicating that steps and defects will expose a broad range of adsorption sites with varying (mostly less favorable) adsorption energies. The effect of defects has been studied by TPD by sputtering of the Co crystal surface. Defects accelerate the adsorption of hydrogen by providing alternative, almost barrierless pathways, making it possible to increase the coverage on the Co(111)/(0001) surface to above θ = 0.50 ML. The presence of defects at a high concentration will give rise to adsorption sites with much lower desorption activation energies, resulting in broad low temperature TPD features.

A comprehensive density functional theory (DFT) study analysing the bulk and various low Miller index surfaces of Hägg Fe carbide (Fe(5)C(2)), considered to be the active phase in Fe-catalysed Fischer-Tropsch synthesis (FTS), has been carried out. The DFT determined bulk structure of Hägg Fe carbide (Fe(5)C(2)) is found to be in good agreement with reported monoclinic (C 2/c) XRD data, independently of whether a monoclinic (C 2/c) or triclinic ([Formula: see text]) bulk structure is used as input for calculations. Attention is focused on the construction of a surface energy stability trend with subsequent correlation with particular surface properties. It is found that a (010) Miller index plane results in the most stable surface (2.468 J m(-2)), while a (101) surface is the least stable (3.281 J m(-2)). The systematic comparison of calculated surface energies with surface properties such as the number of dangling bonds and surface atom density (within a broken bond model), as well as unrelaxed surface energies, relative ruggedness of surfaces, degree of surface relaxation upon optimization, total spin density changes of surfaces compared to the bulk, etc, result in only an approximate correlation with the surface stability trend in selected cases. From the results it is concluded that the relative surface energies fall in a narrow range and that a large number of additional surfaces may be defined, e.g. from higher Miller index planes, sharing similar surface energy values. The results serve to demonstrate the rich complexity and diverse nature of the Fe carbide phase responsible for FTS, effectively laying the foundation for further fundamental studies.

The harmonic map heat flow is a model for nematic liquid crystals and also has origins in geometry. We present an analysis of the asymptotic behaviour of singularities arising in this flow for a special class of solutions which generalises a known (radially symmetric) reduction. Specifically, the rate at which blowup occurs is investigated in settings with certain symmetries using the method of matched asymptotic expansions. We identify a range of blowup scenarios in both finite and infinite time, including degenerate cases.

In this paper we propose a rigorous numerical technique for the computation of symmetric connecting orbits for ordinary differential equations. The idea is to solve a projected boundary value problem (BVP) in a function space via a fixed point argument. The formulation of the projected BVP involves a high order parameterization of the invariant manifolds at the steady states. Using this parameterization, one can obtain explicit exponential asymptotic bounds for the coefficients of the expansion of the manifolds. Combining these bounds with piecewise linear approximations, one can construct a contraction in a function space whose unique fixed point corresponds to the wanted connecting orbit. We have implemented the method to demonstrate its effectiveness, and we have used it to prove the existence of a family of even homoclinic orbits for the Gray-Scott equation.1. Introduction. Equilibria, periodic orbits, connecting orbits, and, more generally, invariant manifolds are the fundamental components through which much of the structure of the dynamics of nonlinear differential equations is explained. Thus it is not surprising that there is a vast literature on numerical techniques for approximating these objects. In particular, the last 30 years have witnessed a strong interest in developing computational methods for connecting orbits [5,10,12,14,15,23]. As mentioned in [13], most algorithms for computing heteroclinic or homoclinic orbits reduce the question to solving a boundary value problem (BVP) on a finite interval where the boundary conditions are given in terms of linear or higher order approximations of invariant manifolds near the steady states. We adopt the same philosophy in this paper. The novelty of our approach is that our computational techniques provide existence results and bounds on approximations that are mathematically rigorous. We hasten to add that a variety of authors have already developed methods that involve a combination of interval arithmetic with analytical and topological tools and provide proofs for the existence of homoclinic and heteroclinic solutions to differential equations [28,22,31,6,32]. However, the combination of techniques we propose appears to be unique, perhaps because our approach is being developed with additional goals in mind. We return to this point later.

Abstract. In this paper, we present a new method for rigorously computing smooth branches of zeros of nonlinear operators f : R l 1 ×B 1 → R l 2 ×B 2 , where B 1 and B 2 are Banach spaces. The method is first introduced for parameter continuation and then generalized to pseudo-arclength continuation. Examples in the context of ordinary, partial and delay differential equations are given.

We prove that the stationary Swift-Hohenberg equation has chaotic dynamics on a critical energy level for a large (continuous) range of parameter values. The first step of the method relies on a computer assisted, rigorous, continuation method to prove the existence of a periodic orbit with certain geometric properties. The second step is topological: we use this periodic solution as a skeleton, through which we braid other solutions, thus forcing the existence of infinitely many braided periodic orbits. A semi-conjugacy to a subshift of finite type shows that the dynamics is chaotic.

We develop a rigorous numerical method to compare local minimizers of the Ohta-Kawasaki functional in two dimensions. In particular, we validate the phase diagram identifying regions of parameter space where rolls are favorable, where hexagonally packed spots have lowest energy and finally where the constant mixed state does. More generally, we present a method to rigorously determine such features in problems where optimal domain sizes are not known a priori.

MotivationNonlinear dynamics shape the world around us, from the harmonious movements of celestial bodies, via the swirling motions in fluid flows, to the complicated biochemistry in the living cell. Mathematically these beautiful phenomena are modelled by nonlinear dynamical systems, mainly in the form of ordinary differential equations (ODEs), partial differential equations (PDEs) and delay differential equations (DDEs). The presence of nonlinearities severely complicates the mathematical analysis of these dynamical systems, and the difficulties are even greater for PDEs and DDEs, which are naturally defined on infinite dimensional function spaces. With the availability of powerful computers and sophisticated software, numerical simulations have quickly become the primary tool to study the models. However, while the pace of progress increases, one may ask: just how reliable are our computations? Even for finite dimensional ODEs, this question naturally arises if the system under study is chaotic, as small differences in initial conditions (such as those due to rounding errors in numerical computations) yield wildly diverging outcomes. These issues have motivated the development of the field of rigorous numerics in dynamics.Rigorous numerics draws inspiration from the ideas in scientific computing, numerical analysis and approximation theory. In a nutshell, rigorous computations are mathematical theorems formulated in such a way that the assumptions can be rigorously verified on a computer. This requires an a priori setup that allows analysis and numerics to go hand in hand: the choice of function spaces, the choice of the basis functions and Galerkin projections, the analytic estimates, and the computational parameters must all work together to bound the errors due to approximation, rounding and truncation sufficiently tightly for the verification proof to go through. The goal is to provide a mathematically rigorous statement about the validity of a concrete numerical simulation (i.e. not in some asymptotic sense where for example the grid size tends to zero) as interpreted as an approximate solution of the original problem. This complements the field of scientific computing, where the goal is to achieve highly reliable results for very complicated problems. In rigorous computing one is after absolutely reliable results for somewhat less complicated (but still hard) problems.Outside of dynamics, computer-assisted proofs have been used to settle famous open problems. Two prominent examples are the four color theorem [1] and Kepler's densest sphere packing problem [2]. In dynamical systems, an early success is the demonstration of the universality of the Feigenbaum constant [3]. More recently, rigorous numerics were used to prove the existence of the strange attractor in the Lorenz system, which seemed, for decades, tentatively intuitive from computer simulations [4]. This settled the 14th problem in Smale's list of problems for the 21st century (the only other problem from the list that has been solved is the P...