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One-dimensional piecewise-smooth discontinuous maps (maps with gaps) are known to have surprisingly rich dynamics, including periodic orbits with very high period and bifurcation diagrams showing period-adding or period-incrementing behavior. In this paper we study a new class of maps, which we refer to as regularized one-dimensional discontinuous maps, because they give very similar dynamics to discontinuous maps and closely approximate them, yet are continuous. We show that regularized discontinuous maps arise naturally as limits of higher dimensional continuous maps when we study the global dynamics of two nonsmooth mechanical applications: the cam-follower system and the impact oscillator. We review the dynamics of discontinuous maps, study the dynamics of regularized discontinuous maps, and compare this to the behavior of the two mechanical applications which we model. We show that we observe period-adding and period-incrementing behavior in these two systems respectively but that the effect of the regularization is to lead to a progressive loss of the higher period orbits.

Many time-dependent partial differential equations have solutions which evolve to have features with small length scales. Examples are blow-up singularities and interfaces. To compute such features accurately it is essential to use some form of adaptive method which resolves fine length and time scales without being prohibitively expensive to implement. In this paper we will describe an r-adaptive method (based on moving mesh partial differential equations) which moves mesh points into regions where the solution is developing singular behaviour. The method exploits natural symmetries which are often present in partial differential equations describing physical phenomena. These symmetries give an insight into the scalings (of solution, space and time) associated with a developing singularity, and guide the adaptive procedure. In this paper the theory behind these methods will be developed and then applied to a number of physical problems which have (blow-up type) singularities linked to symmetries of the underlying PDEs. The paper is meant to be a practical guide towards solving such problems adaptively and contains an example of a Matlab code for resolving the singular behaviour of the semi-linear heat equation.

In this article we survey r-adaptive (or moving grid) methods for solving time-dependent partial differential equations (PDEs). Although these methods have received much less attention than their h- and p-adaptive counterparts, particularly within the finite element community, we review the substantial progress that has been made in developing more robust and reliable algorithms and in understanding the basic principles behind these methods, and we give some numerical examples illustrative of the wide classes of problems for which these methods are suitable alternatives to the traditional ones.More specifically, we first examine the basic geometric properties of moving meshes in both one and higher spatial dimensions, and discuss the discretization process for PDEs on such moving meshes (both structured and unstructured). In particular, we consider the issues of mesh regularity, equidistribution, alignment, and associated variational methods. An overview is given of the general interpolation error analysis for a function or a truncation error on such an adaptive mesh. Guided by these principles, we show how to design effective moving mesh strategies. We then examine in more detail how these strategies can be implemented in practice. The first class of methods which we consider are based upon controlling mesh density and hence are called position-based methods. These make use of a so-called moving mesh PDE (MMPDE) approach and variational methods, as well as optimal transport methods. This is followed by an analysis of methods which have a more Lagrange-like interpretation, and due to this focus are called velocity-based methods. These include the moving finite element method (MFE), the geometric conservation law (GCL) methods, and the deformation map method. Finally, we present a number of specific types of examples for which the use of a moving mesh method is particularly effective in applications. These include scale-invariant problems, blow-up problems, problems with moving fronts and problems in meteorology. We conclude that, whilst r-adaptive methods are still in their relatively early stages of development, with many outstanding questions remaining, they have enormous potential and indeed can produce an optimal form of adaptivity for many problems.

Images are widely used to visualise physical processes. Models may be developed which attempt to replicate those processes and their effects. The technique of coupling model output to images, which is here called "image-model coupling", may be used to help understand the underlying physical processes, and better understand the limitations of the models. An information theoretic framework is presented for image-model coupling in the context of communication along a discrete channel. The physical process may be regarded as a transmitter of images and the model as part of a receiver which decodes or recognises those images. Image-model coupling may therefore be interpreted as image recognition. Of interest are physical processes which exhibit "memory". The response of such a system is not only dependent on the current values of driver variables, but also on the recent history of drivers and/or system description. Examples of such systems in geophysics include the ionosphere and Earth's climate. The discrete channel model is used to help derive expressions for matching images and model output, and help analyse the coupling.

This article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations (PDEs) in several spatial dimensions. The mesh is obtained by taking the gradient of a (scalar) mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation. This method gives a new technique for performing $r$-adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a (time-varying) scalar monitor function (used successfully in moving mesh methods in one-dimension). Detailed analysis of this new method is presented in which the convergence, regularity, and stability of the mesh is studied. Additionally, this new method is shown to be straightforward to program and implement, requiring the solution of only one simple scalar time-dependent equation in arbitrary dimension, with adaptivity along the boundaries handled automatically. We discuss three preexisting methods in the context of this work. Examples are presented in which either the monitor function is prescribed in advance, or it is given by the solution of a partial differential equation.

A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous-time piecewise-smooth dynamical systems. Motivated by applications, a pragmatic approach is taken to defining a discontinuity-induced bifurcation (DIB) as a nontrivial interaction of a limit set with respect to a codimension-one discontinuity boundary in phase space. Only DIBs that are local are considered, that is, bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows. Three classes of systems are considered, involving either state jumps, jumps in the vector field, or jumps in some derivative of the vector field. A rich array of dynamics are revealed, involving the sudden creation or disappearance of attractors, jumps to chaos, bifurcation diagrams with sharp corners, and cascades of period adding. For each kind of bifurcation identified, where possible, a kind of "normal form" or discontinuity mapping (DM) is given, together with a canonical example and an application. The goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators, impact oscillators, DC-DC converters, and problems in control theory.

We examine both the evolutionary structural optimisation (ESO) and solid isotropic microstructure with penalisation (SIMP) methodologies by investigating a cantilever tie–beam. Initially, both ESO and SIMP produce designs with higher objective function values relative to a previously published ‘intuitive’ design. However, after careful investigation of the numerical parameters such as the initial design domain and the mesh size, both methods obtain designs that have lower objective function values relative to the intuitive design. Thus, a clearer understanding of the numerical parame- ters and their influence on optimisation methods is achieved.

Previous multi-layered folding models have struggled to describe the geometry of the interfaces between layers. In the level set method (LSM), a single function ϕ ( x , y , t ) can be used to encode all the information of a multi-layer material in which all layers are in contact and thus is a very natural way of dealing with the geometry. This paper shows the potential for the LSM (in multi-layer problems) by demonstrating that it can describe the geometry of multi-layer folding patterns including those with singularities. The method is then applied to describe the mechanics of the modelling of parallel folding in multi-layered structures.

We study the asymptotic behaviour as t → T–, near a finite blow-up time T > 0, of decreasing-in-x solutions to the following semilinear heat equation with a non-local term:with Neumann boundary conditions and strictly decreasing initial function u0(x) with zero mass. We prove sharp estimates for u(x, t) as t → T–, revealing a non-uniform global blow-up:uniformly on any compact set [δ, 1], δ ∈ (0, 1).

In this chapter I will address the issue that whilst mathematics is vital to all of our lives, playing a vital role in modern technology and even helping us to understand the brain, it is often perceived to be a dry, boring, and useless subject. The chapter will explore various ways that mathematics can be presented to the general public in a way that makes it seem to be exciting and relevant, and captures its essence without dumbing it down. In particular it will look at strategies that have been shown to work well with the public including the use of careful real life examples and relating maths to people, hands on maths at science fairs, and maths in the media and on the Internet. The chapter includes some case studies of what does and does not work in the field of maths communication.

We consider an impacting mechanical system in which a particle at position u ( t ) impacts with a periodically moving obstacle at position z ( t ) , the motion of which is non-smooth. In particular we look at corner events when u impacts with z very close to a point where z loses smoothness. We show that this leads, through a corner bifurcation, to complex dynamics in u which can include periodic orbits of arbitrary period and period-adding cascades. By analysing associated maps close to the corner event, we show that this dynamics can be understood in terms of the iterations of a two-dimensional, piecewise linear, discontinuous map. We also show some links between this analysis and the difficult problem of understanding the motion of three objects which may have simultaneous impacts.

This paper constructs and analyses an adaptive moving mesh scheme for the numerical simulation of singular PDEs in one or more spatial dimensions. The scheme is based on computing a Legendre transformation from a regular to a spatially non-uniform mesh via the solution of a relaxed form of the Monge–Ampere equation. The method is shown to preserve the inherent scaling properties of the PDE and to identify natural computational coordinates. Numerical examples are presented in one and two dimensions which demonstrate the effectiveness of this approach.

We discuss the numerical computation of self-similar blow-up solutions of the classical nonlinear Schrodinger equation in three space dimensions. These solutions become unbounded in finite time at a single point at which there is a growing and increasingly narrow peak. The problem of the computation of this self-similar solution profile reduces to a nonlinear, ordinary differential equation on an unbounded domain. We show that a transformation of the independent variable to the interval [0,1] yields a well-posed boundary value problem with an essential singularity. This can be stably solved by polynomial collocation. Moreover, a Matlab solver developed by two of the authors can be applied to solve the problem efficiently and provides a reliable estimate of the global error of the collocation solution. This is possible because the boundary conditions for the transformed problem serve to eliminate undesired, rapidly oscillating solution modes and essentially reduce the problem of the computation of the physical solution of the problem to a boundary value problem with a singularity of the first kind. Furthermore, this last observation implies that our proposed solution approach is theoretically justified for the present problem.

An earlier nonlinear model for two-layer parallel folding with bedding plane slip is extended to embrace serial buckling behaviour. Approximating the folds using two cubic B-splines, a quasi-energy formulation admits both synchronous and serial-type buckling under conditions of both controlled load and controlled end-shortening. In the early stages of evolution, non-periodic saddle points, corresponding to localized folds, are found to provide the preferred solution. However, as the end-shortening increases, the saddle points converge with unstable maxima representing synchronous folding, until only periodic solutions exist. This shift from localized to two-hump periodic behaviour is seen as a primitive exposition of the more general theme of serial or sequential fold formation.

We discuss a new approach for the numerical computation of self-similar blow-up solutions of certain nonlinear partial differential equations. These solutions become unbounded in finite time at a single point at which there is a growing and increasingly narrow peak. Our main focus is on a quasilinear parabolic problem in one space dimension, but our approach can also be applied to other problems featuring blow-up solutions. For the model we consider here, the problem of the computation of the self-similar solution profile reduces to a nonlinear, second-order ordinary differential equation on an unbounded domain, which is given in implicit form. We demonstrate that a transformation of the independent variable to the interval [0,1] yields a singular problem which facilitates a stable numerical solution. To this end, we implemented a collocation code which is designed especially for implicit second order problems. This approach is compared with the numerical solution by standard methods from the literature and by well-established numerical solvers for ODEs. It turns out that the new solution method compares favorably with previous approaches in its stability and efficiency. Finally, we comment on the applicability of our method to other classes of nonlinear PDEs with blow-up solutions.

A new asymptotic multiple scale expansion is used to derive envelope equations for localized spatially periodic patterns in the context of the generalized Swift-Hohenberg equation. An analysis of this envelope equation results in parametric conditions for localized patterns. Furthermore, it yields corrections for wave number selection which are an order of magnitude larger for asymmetric nonlinearities than for the symmetric case. The analytical results are compared with numerical computations which demonstrate that the condition for localized patterns coincides with vanishing Hamiltonian and Lagrangian for periodic solutions. One striking feature of the choice of scaling parameters is that the derived condition for localized patterns agrees with the numerical results for a significant range of parameters which are an O ( 1 ) distance from the bifurcation, thus providing a novel approach for studying these localized patterns.

We consider the problem of computing blow-up solutions of chemotaxis systems, or the so-called chemotactic collapse. In two spatial dimensions, such solutions can have approximate self-similar behaviour, which can be very challenging to verify in numerical simulations [cf. Betterton and Brenner, Collapsing bacterial cylinders, Phys. Rev. E 64 (2001) 061904]. We analyse a dynamic (scale-invariant) remeshing method which performs spatial mesh movement based upon equidistribution. Using a suitably chosen monitor function, the numerical solution resolves the fine detail in the asymptotic solution structure, such that the computations are seen to be fully consistent with the asymptotic description of the collapse phenomenon given by Herrero and Velazquez [Singularity patterns in a chemotaxis model, Math. Ann. 306 (1996) 583-623]. We believe that the methods we construct are ideally suited to a large number of problems in mathematical biology for which collapse phenomena are expected.

We consider adaptive geometric integrators for the numerical integration of Hamiltonian systems with greatly varying time scales. A time regularization is considered using either the Sundman or the Poincare transformation. In the latter case, this gives a new Hamiltonian which is usually separable, but with one of the parts not always exactly solvable. This system can be numerically integrated with a splitting scheme where each part can be computed using a symplectic implicit or explicit method, preserving the qualitative properties of the exact solution. In this case, a backward error analysis for the numerical integration is presented. For a one-dimensional near singular problem, this analysis reveals a strong dependence of the performance of the method with the choice of the monitor function g, which is also observed when using other symmetric nonsymplectic integrators. We also show how this dependence greatly increases with the order of the numerical integrator used. The optimal choice corresponds to the function g, which nearly preserves the scaling invariance of the system. Numerical examples supporting this result are presented. In some cases a canonical transformation can also be considered, making the system more regular or easy to compute, and this is also illustrated with some examples.

In this article we construct, both asymptotically and numerically, multibump, blow-up, self-similar solutions to the complex Ginzburg--Landau equation (CGL) in the limit of small dissipation. Through a careful asymptotic analysis, involving a balance of both algebraic and exponential terms, we determine the parameter range over which these solutions may exist. Most intriguingly, we determine a branch of solutions that are not perturbations of solutions to the nonlinear Schrodinger equation (NLS); moreover, they are not monotone, but they are stable. Furthermore, these axisymmetric ring-like solutions exist over a broader parameter regime than the monotone profile.

We study the Cauchy problem in $\re \times \re_+$ for one-dimensional 2mth-order, m>1, semilinear parabolic PDEs of the form ($D_x=\partial/\partial x$) \[ u_t = \Dx u + |u|^{p-1}u, \,\,\,\mbox{where} \,\, \,\,\,p > 1, \quad \mbox{ and } \quad u_t = \Dx u + e^u \] with bounded initial data u0 (x). Specifically, we are interested in those solutions that blow up at the origin in a finite time T. We show that, in contrast to the solutions of the classical second-order parabolic equations ut = uxx + up and ut = uxx + eu from combustion theory, the blow-up in their higher-order counterparts is asymptotically self-similar. In particular, there exist exact nontrivial self-similar blow-up solutions, u* (x,t) = (T-t)-1/(p-1)f (y) in the case of the polynomial nonlinearity and u(x,t) = -ln(T-t) + f(y) for the exponential nonlinearity, where y= x/(T-t)1/2m is the backward higher-order heat kernel variable. The profiles f(y) satisfy related semilinear ODEs that share the same non--self-adjoint higher-order linear dif...

We present a generalisation of the Levi-Civita and Kustaanheimo-Stiefel regularisation. This allows the use of more general time rescalings. In particular, it is possible to find a regularisation which removes the singularity of the equations and preserves scaling invariance. In addition, these equations can, in certain cases, be integrated with explicit symplectic Runge-Kutta-Nystrom methods. The combination of both techniques gives an explicit adaptive symplectic (EASY) integrator. We apply those methods to some perturbations of the Kepler problem and illustrate, by means of some numerical examples, when scaling invariant regularisations are more efficient that the LC/KS regularisation.

A basic model is presented for parallel folding of two flexible layers under compression, obliged by the presence of overburden pressure to remain in contact along their length. The nonlinear effect of friction is fully explored via a quasi‐energy formulation and the calculus of variations, leading to representation as an ordinary differential equation. The outcomes of both theoretical and numerical modelling are compared with a simple Galerkin approximation. The significance of friction‐induced jamming is explored. Comparisons between the linearized differential equation under the nonlinear boundary conditions and the full nonlinear formulation indicate that the linearization captures most of the significant behaviour.

We use a variety of careful numerical and semi-analytical methods to investigate two outstanding conjectures on the solutions of the parametrised semi-linear elliptic equation Δu + λu + u5=0, u > 0, where u is defined to be zero on the boundary of a three dimensional domain. This equation is important in analysis and in studies of combustion and polytropic gases. It is known that there is a value λ0 > 0 such that no solutions exist for λ < λ0. McLeod and Schoen, and Bandle and Flucher have given different estimates for λ0; both of which have been conjectured to be exact. We perform a semi-analytic study of solutions on cylindrical domains and construct numerical approximations on cuboid domains using the finite element method combined with a careful post-processing step to reduce the otherwise significant errors. We compute these estimates for cylindrical and cuboid domains, and show that on these domains the estimates do not agree. We conclude that the conjecture of Bandle and Flucher is false. Our numerical computations on cuboid domains are consistent with McLeod's conjecture being true.

The bifurcation structure of a simple power-system model is investigated, with respect to changes to both the real and reactive loads. Numerical methods for this bifurcation analysis are presented and discussed. The model is shown to have a Bogdanov-Takens bifurcation point and hence homoclinic orbits; these orbits can be of Sil'nikov type with many coexisting periodic solutions. We may use the bifurcation calculations to divide the two-parameter plane into a number of regions, for which there are qualitatively different dynamics. We classify and further investigate the dynamical behavior in each of these regions, using a Monte Carlo method to investigate basins of attraction of various stable states. We then show how this classification can be used to denote each regions as either safe or unsafe with respect to the likelihood of voltage collapse.

This paper examines blow-up self-similar solutions of the cubic nonlinear Schrodinger equation close to the critical dimension $d=2$. It gives a formal asymptotic theory for self-similar solutions with multiple maxima, in which the solution close to each maximum takes the form of a rescaled one-dimensional soliton. As $d \rightarrow 2$, the maxima move to infinity and are centered close to the point $-\log(d-2)/( 2\pi/3 - \sqrt{3/4})$. However, the shape of the solution close to each maxima changes little in this limit, leading to an interesting nonuniform bifurcation. The formulae derived from the asymptotic theory are strongly supported by some numerical calculations.

The principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods, and a metric tensor M and mesh metric are useful theoretical tools for understanding a methods level of mesh alignment, or anisotropy. We consider a mesh redistribution method based on the Monge-Ampere equation, which combines equidistribution of a given scalar density function with optimal transport. It does not involve explicit use of a metric tensor M, although such a tensor must exist for the method, and an interesting question to ask is whether or not the alignment produced by the metric gives an anisotropic mesh. For model problems with a linear feature and with a radially symmetric feature, we derive the exact form of the metric M, which involves expressions for its eigenvalues and eigenvectors. The eigenvectors are shown to be orthogonal and tangential to the feature, and the ratio of the eigenvalues (corresponding to the level of anisotropy) is shown to depend, both locally and globally, on the value of the density function and the amount of curvature. We thereby demonstrate how the optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function. Numerical results are given to verify these results and to demonstrate how the analysis is useful for problems involving more complex features, including for a non-trivial time dependant nonlinear PDE which evolves narrow and curved reaction fronts., arXiv admin note: substantial text overlap with arXiv:1402.5453

With the increasing penetration of distributed generation (DG), a methodology that is able to recognize its contribution at the transmission level becomes crucial. However, tailored for distribution network operators (DNOs), the existing P2/6 approach fails to consider the characteristics of the distribution networks, DG penetration levels and locations. Consequently, its suggested DG contribution values tend to be oversimplified and too optimistic. In this paper, a Monte Carlo simulation based approach to evaluate DG contribution has been presented. Compared with the original P2/6, the new approach for the first time incorporates a series of potential influencing factors, such as network reliability, system capacity constraint, DG penetration level and DG location, into DG contribution evaluation, making the results valuable for transmission-level analysis. In addition, through a series of tests on an IEEE 30-bus system, it has been clearly shown that these previously ignored factors could make a huge difference to the final results.

We discuss an initial value problem for an implicit second order ordinary differential equation which arises in models of flow in saturated porous media such as concrete. Depending on the initial condition, the solution features a sharp interface with derivatives which become numerically unbounded. By using an integrator based on finite difference methods and equipped with adaptive step size selection, it is possible to compute the solution on highly irregular meshes. In this way it is possible to verify and predict asymptotical theory near the interface with remarkable accuracy.

We study the deformation of an elastic strut on a nonlinear Winkler foundation subjected to an axial compressive load P . Using multi–scale analysis and numerical methods we describe the localized, cellular, post–buckled state of the system when P is removed from the critical load P = 2. The solutions, and their modulation frequencies, differ significantly from those predicted by weakly nonlinear analysis very close to P = 2. In particular, when P approaches the Maxwell load P M , the localized solutions approach a large–amplitude heteroclinic connection between an unbuckled solution and a periodic solution. An asymptotic description of P M in terms of the system parameters is given. The agreement between the numerical calculations and the asymptotic approximations is striking.

In this paper we consider symmetry-preserving difference schemes for the nonlinear Schrodinger equation where n is the number of space dimensions. This equation describes one-dimensional waves in n space dimensions in many physical situations, including phenomena in plasma physics and nonlinear optics. We will consider the nonintegrable case n≥2 for which the equation admits solutions that blow up in a finite time, and construct discretizations based upon moving mesh schemes that have the same Lie group properties and Lagrangian structures as the continuous counterpart.

This paper analyses a so-called corner-collision bifurcation in piecewise-smooth systems of ordinary differential equations (ODEs), for which a periodic solution grazes with a corner of the discontinuity set. It is shown under quite general circumstances that this leads to a normal form that is to lowest order a piecewise-linear map. This is the first generic derivation from ODE theory of the so-called C -bifurcation (or border collision) for piecewise-linear maps. The result contrasts with the equivalent results when a periodic orbit grazes with a smooth discontinuity set, which has recently been shown to lead to maps that have continuous first derivatives but not second. Moreover, it is shown how to calculate the piecewise-linear map for arbitrary dimensional systems, using only properties of the single periodic trajectory undergoing corner collision. The calculation is worked out for two examples, including a model for a commonly used power electronic converter where complex dynamics associated with corner collision was previously found numerically, but is explained analytically here for the first time.

This review paper examines a synthesis of adaptive mesh methods with the use of symmetry to solve ordinary and partial differential equations. It looks at the effectiveness of numerical methods in preserving geometric structures of the underlying equations such as scaling invariance, conservation laws and solution orderings. Studies are made of a series of examples including the porous medium equation and the nonlinear Schrödinger equation.

In this paper, we investigate the numerical solution of a model equation uxxe \frac{1}{\epsilon^2}eexp(−\frac{x}{\epsilon}) (and several slightly more general problems) when eL1 using the standard central difference scheme on nonuniform grids. In particular, we are interested in the error behaviour in two limiting cases: (i) the total mesh point number N is fixed when the regularization parameter e→0, and (ii) e is fixed when N→∞. Using a formal analysis, we show that a generalized version of a special piecewise uniform mesh 12 and an adaptive grid based on the equidistribution principle share some common features. And the “optimal” meshes give rates of convergence bounded by vlog(e)v as e→0 and N is given, which are shown to be sharp by numerical tests.

We propose a model for the folding of rock under the compression of tectonic plates. This models an elastic rock layer imbedded in a viscous foundation by a fourth-order parabolic equation with a nonlinear constraint. The large-time behavior of solutions of this problem is examined and found to be of approximately self-similar form.

A long structural system with an unstable (subcritical)post-buckling response that subsequently restabilizes typically deformsin a cellular manner, with localized buckles first forming and thenlocking up in sequence. As buckling continues over a growing number ofcells, the response can be described by a set of lengthening homoclinicconnections from the fundamental equilibrium state to itself. In thelimit, this leads to a heteroclinic connection from the fundamentalunbuckled state to a post-buckled state that is periodic. Under suchprogressive displacement the load tends to oscillate between twodistinct values. The paper is both a review and a pointer tofuture research. The response is described via a typical system, asimple but ubiquitous model of a strut on a foundation which includesinitially-destabilizing and finally-restabilizing nonlinear terms. Anumber of different structural forms, including the axially-compressedcylindrical shell, a typical sandwich structure, a model of geologicalfolding and a simple link model are shown to display such behaviour. Amathematical variational argument is outlined for determining the globalminimum postbuckling state under controlled end displacement (rigidloading). Finally, the paper stresses the practical significance of aMaxwell-load instability criterion for such systems. This criterion,defined under dead loading to be where the pre-buckled and post-buckledstate have the same energy, is shown to have significance in the presentsetting under rigid loading also. Specifically, the Maxwell load isargued to be the limit of minimum energy localized solutions asend-shortening tends to infinity.

We study the blow-up self-similar solutions of the radially symmetric nonlinear Schrodinger equation (NLS) given by iut+urr+d?1rur+u|u|2, with dimension d>2. These solutions become infinite in a finite time T. By a series of careful numerical computations, partly supported by analytic results, we demonstrate that there is a countably infinite set of blow-up self-similar solutions which satisfy a second order complex ordinary differential equation with an integral constraint. These solutions are characterised by the number of oscillations in their amplitude when d is close to 2. The solutions are computed as functions of d and their behaviour in the critical limit as d?2 is investigated. The stability of these solutions is then studied by solving the NLS by using an adaptive numerical method. This method uses moving mesh partial differential equations and exploits the scaling invariance properties of the underlying equation. We demonstrate that the single-humped self-similar solution is globally stable whereas the multi-humped solutions all appear to be unstable.

This paper examines a synthesis of adaptive mesh methods with the use of symmetry to study a partial differential equation. In particular, it considers methods which admit discrete self–similar solutions, examining the convergence of these to the true self–similar solution as well as their stability. Special attention is given to the nonlinear diffusion equation describing flow in a porous medium.

Since their introduction by Sir Isaac Newton, diffierential equations have played a decisive role in the mathematical study of natural phenomena. An important and widely acknowledged lesson of the last three centuries is that critical information about the qualitative nature of solutions of diffierential equations can be determined by studying their geometry. Perhaps the most important example of this approach was the formulation of the laws of mechanics by Alexander Rowan Hamilton, which allowed deep geometric tools to be used in understanding the dynamics of complex systems such as rigid bodies and the Solar System. Conserved quantities of a Hamiltonian system, such as energy, linear and angular momentum, could be understood in terms of the symmetries of the underlying Hamiltonian function, its ergodic properties determined from the underlying symplectic nature of the formulation and constraints on the system could be incorporated in a natural manner. The Hamiltonian geometric formulation of many other problems in science modelled by ordinary and partial diffierential equations, such as ocean dynamics, nonlinear optics and elastic deformations, continues to play a vital role in our qualitative understanding of these systems. An equally important geometric approach to the study of diffierential equations is the application of symmetry–based methods pioneered by Sophus Lie. Exploiting underlying symmetries of a partial or ordinary difierential equation, it can be often greatly simplified and sometimes solved altogether in closed form. Such methods, which lie at the heart of the construction of self–similar solutions of diffierential equations and the symmetry reduction of complex systems, have become increasingly popular with the development of symbolic algebra packages. It is no coincidence that the most important equations of mathematical physics are precisely those for which geometric and symmetry–based methods are most effiective. Arguably, these equations are really a shorthand for the deep underlying symmetries in nature that they encapsulate.

We consider the finite element solution of the parameterized semilinear elliptic equation $\Delta u + \lambda u + u^{5} = 0, u > 0$, where $u$ is defined in the unit cube and is 0 on the boundary of the cube. This equation is important in analysis, and it is known that there is a value $\lambda_{0} > 0$ such that no solutions exist for $\lambda \lambda_{0}$ and for the form of the spurious numerical solutions which are known to exist when $\lambda < \lambda_{0}$. These estimates are then used to post-process the numerical results to obtain a sharp estimate for $\lambda_{0}$ which agrees with the conjectured value.

We study a nonlinear diffusion equation which is invariant under a stretching group of transformations and which reduces to a linear diffusion equation in the limit of σ → 0. We show that if σ > 0 then the equation has solutions which form singularities which have a self–similar profile. By considering a nonlinear eigenvalue problem the existence and stability of the self–similar profiles is discussed. The existence of approximately self–similar behaviour is also considered for evolution profiles with several maxima and minima. This behaviour is compared to similar behaviour for the linear diffusion problem. The paper uses a combination of techniques from analysis and the theory of dynamical systems.

This paper provides an analytical insight into the observed nonlinear behaviour of a simple widely used power electronic circuit (the buck converter) and draws parallels with a wider class of piecewise-smooth systems. After introducing the buck converter model and background, the most fascinating features of its dynamical behaviour are reviewed. So-called grazing and sliding solutions are discussed and their role in determining many of the buck converter's dynamical oddities is demonstrated. In particular, a local map is studied which explains how grazing bifurcations cause sharp turning points in the bifurcation diagram of periodic orbits. Moreover, these orbits are shown to accumulate onto a sliding trajectory through a `spiralling' impact adding scenario. The structure of such a diagram is derived analytically and is shown to be closely related to the analysis of homoclinic bifurcations. The results are shown to match perfectly with numerical simulations. The sudden jump to large-scale chaos and the fingered structure of the resulting attractor are also explained.

Copyright (c) 1997 Elsevier Science B.V. All rights reserved. We consider the Cauchy problem for the nonlinear diffusion equation, u t =(u m u x ) x which is posed on an infinite domain. The PDE and a conservation law are invariant to a Lie group of stretchings which is used to construct the invariant quantities, xt −1/(2+m) and ut 1/(2+m) . Using these invariants as similarity variables the problem is reduced to a second order ODE and then integrated to give the well known similarity solutions. The problem is semi-discretized using the method of lines. The mesh movement is governed by the conservation of mass law, so that the computational domain expands as the solution diffuses. The resulting semi-discretization is a system of ODEs which is invariant to the same Lie group as the PDE and so the mesh X i (t) behaves like the discretized invariant Y i t 1/(2+m) and the solution U i (t) like V i t −1/(2+m) . Furthermore, it is shown that, using these invariants, the same reduction and integration is possible in the semi-discrete case as in the continuous case.

We study the behaviour of the non-negative blowing up solutions to the quasilinear parabolic equation with a typical reaction–diffusion right-hand side and with a singularity in the space variable which takes the formwhere m ≧ 1, p > 1 are arbitrary constants, in the critical exponent case q = (p–1)/m > 0. We impose zero Dirichlet boundary conditions at the singular point x = 0 and at x = 1, and take large initial data. For a class of ‘concave’ initial functions, we prove focusing at the origin of the solutions as t approaches the blow-up time T in the sense that x = 0 belongs to the blow-up set. The proof is based on an application of the intersection comparison method with an explicit ‘separable’ solution which has the same blow-up time as u. The method has a natural generalisation to the case of more general nonlinearities in the equation. A description of different fine structures of blow-up patterns in the semilinear case m = 1 and in the quasilinear one m > 1 is also presented. A numerical study of the semilinear equation is also made using an adaptive collocation method. This is shown to give very close agreement with the fine structure predicted and allows us to make some conjectures about the general behaviour.

We consider the effectiveness of adaptive finite element methods for finding the finite element solutions of the parametrised semi-linear elliptic equation Δ u + λ u + u 5 = 0, u > 0, where u ϵ C 2 (Ω), for a domain Ω ⊂ R and u = 0 on the boundary of Ω. This equation is important in analysis and it is known that there is a value λ 0 > 0 such that no solutions exist for λ 0 and a singularity forms as λ → λ 0 . Furthermore the linear operator L defined by L φ = Δφ + λφ + 5 u 4 φ has a singular inverse in this limit. We demonstrate that conventional adaptive methods (using both static and dynamic regridding) based on usual error estimates fail to give accurate solutions and indeed admit spurious solutions of the differential equation when λ 0 . This is directly due to the lack of invertibility of the operator L . In contrast we show that error estimates which take this into account can give answers to any prescribed tolerance.

We consider a periodically forced oscillator which impacts against a rigid constraint. The forcing is both sinusoidal and non-sinusoidal and is made up of a combination of the first and second harmonics of a Fourier series. A simple restitution law is used to describe the impact. This paper centres around the analysis of double-impact periodic solutions, that is motions which repeat after every second impact. Solutions for these double-impact orbits are found, including non-physical orbits and unstable orbits. Behaviour typical of nonlinear dynamical systems, such as period-doubling bifurcations are detected and grazing bifurcations, which are unique to discontinuous dynamical systems are also observed. A combination of analytical and numerical techniques are used to uncover the behaviour of the system at odd multiples of the natural frequency, where numerical experiments tell us that stable double-impact periodic solutions exist. The persistence of these stable double-impact orbits are also examined as parameters are varied. We conclude by presenting some numerical calculations of two more general cases of impacting systems.

In this paper we consider the numerical solution of PDEs with blow-up for which scaling invariance plays a natural role in describing the underlying solution structures. It is a challenging numerical problem to capture the qualitative behaviour in the blow-up region, and the use of nonuniform meshes is essential. We consider moving mesh methods for which the mesh is determined using so-called moving mesh partial differential equations (MMPDEs). Specifically, the underlying PDE and the MMPDE are solved for the blow-up solution and the computational mesh simultaneously. Motivated by the desire for the MMPDE to preserve the scaling invariance of the underlying problem, we study the effect of different choices of MMPDEs and monitor functions. It is shown that for suitable ones the MMPDE solution evolves towards a (moving) mesh which close to the blow-up point automatically places the mesh points in such a manner that the ignition kernel, which is well known to be a natural coordinate in describing the behaviour of blow-up, approaches a constant as $t\to T$ (the blow-up time). Several numerical examples are given to verify the theory for these MMPDE methods and to illustrate their efficacy.

We study the self-similar solutions of the quasilinear parabolic equationWe show that there is an exponentsuch that if σ> then the equation admits a countable set {uk(x, t)} of self-similar blow-up solutions. These solutions have the formwhere T> 0 is a finite blow-up time, θ(ξ) solves a nonlinear ODE and each function uk(x, t) is nonconstant in a neighbourhood of the origin and has exactly k maxima and minima for x ≧ 0. There is a further critical exponent σ = ф such that if σ > ф there is a second set of self-similar solutions which are constant (in x) in a neighbourhood of the origin. We conjecture (and provide formal arguments and numerical evidence for) the existence of an infinite sequence σk→σ∞ of critical values, such that σ1 = 0 and uk exists only in the range σ>σk (when σ> 0 the equation has no nontrivial self-similar solutions). The proof of existence when σ>σ∞(σ>ф) is obtained by a combination of comparison and dynamical systems arguments and relates the existence of the self-similar solutions to a homoclinic bifurcation in an appropriate phase-space.