Hybrid journal (It can contain Open Access articles) ISSN (Print) 0272-4960 - ISSN (Online) 1464-3634 Published by Oxford University Press[370 journals]

Authors:Wattis JD. Abstract: AbstractWe use asymptotic techniques to describe the bifurcation from steady-state to a periodic solution in the singularly perturbed delayed logistic equation $\epsilon\dot x(t) = - x(t) + \lambda f(x(t-1))$ with $\epsilon \ll 1$. The solution has the form of plateaus of approximately unit width separated by narrow transition layers. The calculation of the period two solution is complicated by the presence of delay terms in the equation for the transition layers, which induces a phase shift that has to be calculated as part of the solution. High order asymptotic calculations enable both the shift and the shape of the layers to be determined analytically, and hence the period of the solution. We show numerically that the form of transition layers in the four cycles is similar to that of the two cycle, but that a three cycle exhibits different behaviours. PubDate: 2017-05-04

Authors:Shariff MM; Bustamante RR, Merodio JJ. Abstract: AbstractIn the literature, residual stress problems are generally formulated using classical invariants despite most of them having an unclear physical meaning and not having experimental advantages. In this article, we give an alternative formulation for residual stress problems using a set of spectral invariants. These invariants have a clear physical meaning which may facilitate the design of a residual stress experiment. For the case of an energy function that depends on the right Cauchy tensor and the residual stress tensor, we show that only nine of the classical invariants are independent, not 10 as commonly assumed. Details of the spectral formulation are given and several boundary value problems are illustrated. PubDate: 2017-05-03

Authors:Zhang Y; Shi G. Abstract: AbstractThis work focuses on two main topics about the transmission eigenvalue problem defined on the unit interval $[0,1]$: (i) the distributions of transmission eigenvalues in a region ${\it{\Sigma}} _{\varepsilon }$ which is symmetric about the axes; (ii) the existence of real transmission eigenvalues under some conditions on acoustic profiles $n_{1}\left( x\right) $ and $n_{2}\left( x\right) $. These subjects are central to the so-called qualitative methods for inverse scattering involving penetrable obstacles. Specifically, in the case where $% n_{1}(x)=n_{2}(x)$ for $x=0 $, $1$, we show how to locate the transmission eigenvalues in ${\it{\Sigma}} _{\epsilon }$. The existence of infinitely many real transmission eigenvalues is proven in the following cases: (i) $n_{2}\left( 0\right) /n_{1}\left( 0\right) \neq n_{2}\left( 1\right) /n_{1}\left( 1\right) $ and $n_{1}\left( 0\right) n_{1}\left( 1\right) \neq n_{2}\left( 0\right) n_{2}\left( 1\right);$ (ii) $n_{1}\left( 0\right) =n_{2}\left( 0\right)\!,n_{1}\left( 1\right) =n_{2}\left( 1\right) $ and $n_{1}^{\prime }\left( 1\right) -n_{2}^{\prime }\left( 1\right) \neq n_{1}^{\prime }\left( 0\right) -n_{2}^{\prime }\left( 0\right) $. The asymptotics of the real eigenvalues, as the byproduct of their existence, is obtained in both cases. All the results are obtained under the assumptions on the values of $% n_{i}\left( x\right) $ and $n_{i}^{\prime }\left( x\right) $ for $i=1$, $2$, $x=0$, $1$. There are no more restrictions on the values of $n_{1}\left( x\right)\!,$$n_{2}\left( x\right) $ in the interval $(0,1)$. PubDate: 2017-04-29

Authors:Yang J; Chen Y, Kuniya T. Abstract: AbstractIn this article, we propose and analyse an epidemic model with relapse, infection age and a general nonlinear incidence rate. Roughly speaking, we establish a threshold dynamics determined by the basic reproduction number $\mathscr R_0$. If $\mathscr R_0Š1$ then the disease-free steady state is globally asymptotically stable while if $\mathscr R_0>1$ then the endemic steady state is globally asymptotically stable. The global attractivity for the steady states is obtained by employing the fluctuation lemma and the approach of Lyapunov functionals. Our results imply that decreasing the initial transmission rate and drawing up efficient prevention ways play a much more important role on controlling the disease spread than increasing the total treatment rate. The obtained theoretical results are illustrated with numerical simulations, which also indicate that infection age is an important factor affecting the epidemic spread. PubDate: 2017-04-24

Authors:Chugainova AP; Il’ichev AT, Kulikovskii AG, et al. Abstract: AbstractWe deal with front solutions of the Korteweg-de Vries-Burgers equation which is the simplest one describing long wave propagation in a dissipative media with dispersion in the weakly nonlinear limit. This equation represents a singular perturbation of the hyperbolic Hopf equation, and its nonlinearity is assumed to be presented by a non-convex potential having a concrete expression. Various types of stationary structures of the originating discontinuities are investigated, the domains of their existence are described, and also the analysis of their dynamic stability is undertaken. In the domain of absence of stationary structures, a numeric study of stability of the non-stationary structures has been conducted. We consider self-similar solutions to the problem of arbitrary discontinuity disintegration which consist only of stable discontinuities with a stationary or non-stationary structure and simple waves. The conclusive criterion has been formulated of selection of an admissible discontinuity which is a solution of the Cauchy problem of arbitrary discontinuity disintegration governed by the Hopf equation. PubDate: 2017-04-01

Authors:Murphy EA; Lee WT. Abstract: AbstractCompression moulded contact lenses are produced by placing fluid between two moulds and squeezing the fluid outwards to form the shape of the lens. A common problem seen in this process is that at times the fluid moves outwards asymmetrically, resulting in partially formed lenses. In this article, the system is modelled using the thin film equations and the results are analysed to find the optimal operating setup to reduce asymmetrical flow. A simple model with one curved surface and one flat surface is considered first. This assumption is verified by a more realistic model that investigates the effects of curvature on the dynamics of the fluid. The simple model is modified to include the effect of surface tension. The results of this model show that surface tension plays no role in the fluid dynamics for this particular fluid. A second modified model allows for lateral movement of the lower mould. The model shows that allowing the lower mould to slide hinders the symmetrical flow of the fluid. PubDate: 2017-03-28

Authors:Florio BJ; Bassom AP, Sakellariou K, et al. Abstract: AbstractConvection can occur in a confined saturated porous box when the associated Rayleigh number exceeds a threshold critical value: the identity of the preferred onset convection mode depends sensitively on the geometry of the box. Here we discuss examples for which the box dimensions are such that four modes share a common critical Rayleigh number. Perturbation theory is used to derive a system of coupled ordinary differential equations that governs the nonlinear interaction of the four modes and an analysis of this set is undertaken. In particular, it is demonstrated that as the Rayleigh number is increased beyond critical so a series of pitchfork bifurcations occur and multiple stable states are identified that correspond to the survival of just one of the four modes. The basins of attraction for each mode in the 4D phase space are described by a reduction to a suitable 3D counterpart. PubDate: 2017-02-27

Authors:Wihler TP. Abstract: AbstractWe develop a new class of mathematical ranking models for the purpose of quantifying the social structure in animal populations. Our approach is based on taking into account both the interaction between single individuals as well as their role within their community as a whole. From a mathematical point of view, these models are (possibly nonlinear) eigenvalue problems for column stochastic matrices. In order to provide a procedure for their computational treatment, we derive a suitable Newton iteration method on simplexes. PubDate: 2017-02-27

Authors:Carvalho T; Euzébio RD, Teixeira M, et al. Abstract: AbstractWe consider a piecewise smooth vector field in $\mathbb{R}^3$, where the switching set is on an algebraic variety expressed as the zero of a Morse function. We depart from a model described by piecewise constant vector fields with a non-usual center that is constant on the sliding region. Given a positive integer $k$, we produce suitable nonlinear small perturbations of the previous model and we obtain piecewise smooth vector fields having exactly $k$ hyperbolic limit cycles instead of the center. Moreover, we also obtain suitable nonlinear small perturbations of the first model and piecewise smooth vector fields having a unique limit cycle of multiplicity $k$ instead of the center. As consequence, the initial model has codimension infinity. Some aspects of asymptotical stability of such system are also addressed in this article. PubDate: 2017-02-27

Authors:Zhang Z; Zhou Z. Abstract: AbstractIn this article, we consider an inverse problem of recovering the potential term in a 1D time-fractional diffusion equation from the overdetermined final time data. We introduce a reconstruction operator and show its contractivity and monotonicity, which give the unique determination and an efficient algorithm. Further, for noisy data, we propose a regularized iterative algorithm based on mollification and derive error estimates for the approximation. Extensive numerical experiments for both smooth and nonsmooth potential data are provided to illustrate the efficiency and stability of the algorithm, and to verify the convergence theory. PubDate: 2017-02-27