Chris Budd

Bifurcations in Nonsmooth Dynamical Systems

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.

Moving Mesh Methods for Problems with Blow-up

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

Cellular buckling in long structures

t\to T

A Model of the Evolution of Duopoly: Does the Asymmetry between Firms Tend to Increase or Decrease?

(the blow-up time). Several numerical examples are given to verify the theory for these MMPDE methods and to illustrate their efficacy.

Normal form maps for grazing bifurcations in n -dimensional piecewise-smooth dynamical systems

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.

Adaptivity with moving grids

This paper is an attempt to identify some of the factors that affect the evolution of market structure in a model of dynamic competition between two firms. The stochastic evolution of the state of competition depends on the respective effort rates of the firms. The question is whether the current leader works harder than the laggard—does the ‘gap’ between firms tend to increase or decrease? We show that several effects are at work. The state tends to evolve in the direction where joint payoffs are greater. Since joint payoffs are related to joint product-market profits less joint effort costs, there are two classes of effect: the joint-profit effect and various joint-cost effects. The latter result in part from the pattern of profits, and in part from endpoint effects that give relief from efforts. Asymptotic expansions illuminate these influences. Moreover, we show by numerical simulation that there is another kind of joint-cost effect. The pattern of joint effort costs can influence the pattern of evolution of market structure, and the evolution of the pattern of market structure can influence the pattern of efforts, in a mutually self-reinforcing manner. In particular, there may be equilibria in which this last effect means that the laggard works harder than the leader even though all the other effects work in favour of the leader.

Chattering and related behaviour in impact oscillators

Abstract This paper presents a unified framework for performing local analysis of grazing bifurcations in n-dimensional piecewise-smooth systems of ODEs. These occur when a periodic orbit has a point of tangency with a smooth (n−1)-dimensional boundary dividing distinct regions in phase space where the vector field is smooth. It is shown under quite general circumstances that this leads to a normal-form map that contains to lowest order either a square-root or a (3/2)-type singularity according to whether the vector field is discontinuous or not at the grazing point. In particular, contrary to what has been reported in the literature, piecewise-linear local maps do not occur generically. First, the concept of a grazing bifurcation is carefully defined using appropriate non-degeneracy conditions. Next, complete expressions are derived for calculating the leading-order term in the normal form Poincare map at a grazing bifurcation point in arbitrary systems, using the concept of a discontinuity mapping. Finally, the theory is compared with numerical examples including bilinear oscillators, a relay feedback controller and general third-order systems.

Geometric integration: numerical solution of differential equations on manifolds

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.

Corner collision implies border-collision bifurcation

One of the most interesting properties of an impacting system is the possibility of an infinite number of impacts occurring in a finite time (such as a ball bouncing to rest on a table). Such behaviour is usually called chatter. In this paper we make a systematic study of chattering behaviour for a periodically forced, single-degree-of-freedom impact oscillator with a restitution law for each impact. We show that chatter can occur for such systems and we compute the sets of initial data which always lead to chatter. We then show how these sets determine the intricate form of the domains of attraction for various types of asymptotic periodic motion. Finally, we deduce the existence of periodic motion which includes repeated chattering behaviour and show how this motion is related to certain types of chaotic behaviour.

Self-Similar Blow-Up in Higher-Order Semilinear Parabolic Equations

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.