1 July - 31 December 1995

**Organisers**: P Constantin (*Chicago*), JD Gibbon (*Imperial College, London*), JK Hale (*Georgia Tech*), C Sparrow (*Cambridge*). ASI Organiser: P Glendinning (*Cambridge*)

**Background**

This small workshop will take place as part of the Newton Institute programme, `From Finite to Infinite Dimensional Dynamical Systems'. Piecewise linear ODEs have been extensively studied in the last ten years: for example, in engineering situations with oscillations in competition with a clearance (impact oscillator theory), and in mathematical biology, as analytically tractable caricatures of classical models. These models exhibit a full range of nonlinear and chaotic behaviour. However, the existence of explicit solutions has admitted the possibility of direct calculation of orbit parameters, stability boundaries and heteroclinic bifurcations which are only possible numerically with nonlinear models. The resultant theory is remarkably detailed. This has yielded greater understanding of standard bifurcations, providing alternatives to Duffing's equation, say, as a model equation for chaos. In addition, new kinds of bifurcation can arise from the piecewise linearity, for example, the `grazing' bifurcations in impact oscillator theory (vibro-impact dynamics) which are associated with dramatic global bifurcations and other changes in phase space.

The aim of the workshop is to consider how this experience with ODEs is relevant to the study of infinite dimensional piecewise linear differential equations (PDEs or delay DEs). These piecewise-linear differential equations arise in many fields of engineering interest, for example, impacting beams, earthquakes, composite beams, bridge dynamics, and models of the deflection of railroad tracks and pipelines; as well as in mathematical biology, for instance, in models of excitable media. Surprisingly, however, there is relatively little general theory tailored to the analysis of such equations. In the presented talks, we hope that a range of techniques will become apparent for the successful analysis of such equations, and patterns emerge which can form the basis of a more general theory.

- C J Budd, Bath, UK
- Piecewise linear PDEs arising from quantum and other effects in nonlinear electrostatics
- G W Desch, Graz, Austria
- Coupled elastic and viscoelastic rods
- W A Green, Loughborough, UK
- Wave propagation in layered solids
- U an der Heiden, Witten/Herdecke, Germany
- Piecewise linear delay differential equations
- P J McKenna, Connecticut, USA
- Periodic and travelling waves in a nonlinearly suspended beam
- J A Sherratt, Warwick, UK
- Piecewise linear models in mathematical biology
- F Zanolin, Udine, Italy
- Periodic solutions of piecewise linear ODEs

Each talk is scheduled for one hour. It is anticipated that included in this will be some time for discussion and questions.

**THURSDAY 16th November**

*09.45* Welcome & Introduction: S J Hogan, S H Doole

*10.00* P J McKenna

*11.00* COFFEE

*11.30* F Zanolin

*12.30* BUFFET LUNCH

*14.00* J A Sherratt

*15.00* G W Desch

*16.00* TEA

*16.30* U an der Heiden

*1730* NEWTON INSTITUTE DRINKS

**FRIDAY 17th November**

*09.30* C J Budd

*10.30* COFFEE

*11.00* W A Green

*12.00* Closing Discussion: S J Hogan, S H Doole

*12.30* PUB LUNCH

**Workshop location, costs, registration**

The workshop will take place in the Newton Institute's purpose-designed building, in a pleasant area in the west of Cambridge, about one mile from the centre of the City. The Newton Institute can provide assistance with finding local accommodation - the cost of which is likely to be 25 - 40 pounds per day including breakfast. There will be a nominal charge for lunch. There may be some financial help available to support UK participants.

To register, please obtain and return a **Registration Form** from/to Mike Sekulla at

Isaac Newton Institute,

20 Clarkson Road,

Cambridge CB3 0EH.

**Tel** (44) 1223-335984

**Fax**(44) 1223-330508

Further information about the workshop programme may be obtained from Stuart Doole.

School of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.

*Piecewise linear PDEs arising from quantum and other effects in nonlinear electrostatics.*

My talk will concern problems with very high fields where you have a jump in the electron energy levels at critical fields causing piecewise linear effects. We will consider charge motion in a high electric field, as well as charge creation in ionised gases and hot gases. In particular, we can obtain travelling wavefronts and solitary waves, or even chains of solitary waves. I have experimental confirmation of the latter.

[TOP | SPEAKERS | ABSTRACTS | TIMETABLE]

Institute fuer Mathematik, Graz University, Heinrichstratle 36, A-8010 Graz, Austria.

*Elastic and viscoelastic rods*

(joint work with Prof Ronald Grimmer, Carbondale, USA)

We consider the effect of boundary conditions on simple elastic or viscoelastic systems with linear constitutive laws.

1) Two rods, one elastic, one viscoelastic, are joined face to face. A fractional derivative model is assumed for the viscoelastic rod. One may distinguish between strong and weak fractional derivative models, depending on the type of singularity the stress relaxation modulus exhibits at time 0. The stronger the singularity, the more drastic smoothing and damping effects are implied by the viscoelasticity of the material. How strongly does this damping affect the coupled system? A viscoelastic rod with a strong fractional derivative constitutive law coupled to an elastic rod yields a spectrum, which asymptotically decomposes into two parts: Asymptotically one part of the spectrum approaches the spectrum of the viscoelastic rod (uncoupled), the other part of the spectrum approaches the spectrum of the elastic rod. The physical interpretation may be that the interface between two materials so different is almost impenetrable for vibrations at high frequencies. A viscoelastic rod with a weak fractional derivative law, coupled to an elastic rod, again yields a spectrum decomposing into two parts. But the ``elastic'' part can be shifted to a line parallel to the imaginary axis in the left half-plane. Only if the parameters of both rods fit an equation which can be interpreted as impedance matching, the real parts of all poles go to minus infinity, as their imaginary parts increase. Only in this special case, the viscoelastic damping of one rod is able to damp the whole system more efficiently than frictional damping.

2) While the problem above is a linear one, which deals with linear coupling of two linear systems, we sketch also one that is piecewise linear in the sense that there are two different linear regimes and the system may switch between one and the other. We consider a rod or a beam whose face is subject to an unilateral constraint. If the system is simple enough to be formulated in one space dimension, we can prove existence, uniqueness and continuous dependence of solutions. Surprisingly, Lipschitz continuous dependence does not hold in the physically motivated energy space W(1,2), but only in W(1,1).

[TOP | SPEAKERS | ABSTRACTS | TIMETABLE]

Department of Mathematical Sciences,Loughborough University, Leicestershire, LE11 3TU, UK.

*Wave propagation in layered solids*

The study of wave propagation in layered media has long been of interest in seismology, with the early work dating back for almost a century. More recently, the topic has received a renewed impetus from two quarters. These are:

(1) the development of fibre composite materials and their fabrication into laminated structural components;

and

(2) the development of the technology for constructing artificial crystals or superlattices, consisting of alternating layers of two different materials.

The use of man-made laminated structures has led to the investigation of time harmonic pulse propagation and of transient disturbances travelling in bounded anisotropic layered media. Time harmonic pulse propagation is of interest in the non-destructive evaluation of the integrity of structures whilst the transmission of transient disturbances has application to the study of the impact response of the structure and to the technique of flaw location by acoustic emission. The advent of superlattices has brought about an increased activity in the study of time harmonic wave propagation in periodic media. Each of these areas give rise to piecewise linear systems of partial differential equations.

In this talk, the governing equations for multilayered elastic and viscoelastic solids are derived and reduced to a system of piecewise linear ODEs by the use of transform techniques. The homogeneous system corresponds to infinite trains of time harmonic progressing waves and inserting the boundary conditions into the solution leads to the dispersion equation relating wavelength to frequency. Inhomogeneous equations are associated with events such as a time dependent surface loading or an internal crack formation and their solution leads to the transient motion resulting from the forcing term. The talk will outline some methods for solution and will illustrate the results for both surface and internal impulsive events.

[TOP | SPEAKERS | ABSTRACTS | TIMETABLE]

Institute fuer Mathematik, Universitat Witten-Herdecke, Stockumerstrasse 10, D-58448 Witten, Germany.

*Piecewise linear delay differential equations*

We consider equations of the type

\sum_{i=0}^{n} a_i d^i/dt^i x(t) = f(x (t-1)), a_i \in R, t \in [-1, \infty), x(t) \in R, f: R -> R,

where f is a (nearly) piecewise constant function. We demonstrate analytical results concerning the rich dynamics of these equations including new kinds of bifurcation trees, periodic solutions of arbitrary large and small frequencies, homoclinic orbits, deterministic chaos, and mixing properties, at least for n <=2.

[TOP | SPEAKERS | ABSTRACTS | TIMETABLE]

Department of Mathematics, University of Connecticut, Storrs, CT 06269-0001, USA.

*Periodic and travelling waves in a nonlinearly suspended beam.*

We will survey the history of both types of motions, and describe recent mathematical progress on their understanding. We shall then focus on some unexplained phenomena which arise when travelling waves are allowed to interact nonlinearly with each other and `soliton'-like behaviour is observed.

[TOP | SPEAKERS | ABSTRACTS | TIMETABLE]

Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK.

*Piecewise linear models in mathematical biology*

Piecewise linear differential equations have been a highly successful method of gaining analytical insight into biological models. I will describe three examples of this technique arising from different areas of biology, starting with a developmental biology problem. Chemical prepatterns are an established mechanism for the formation of many aspects of embryonic structure, and their mathematical study dates back to the work of Turing in the 1950s. An important example to which this theory has been applied is the formation of skeletal structure in the chick limb. However, in the early 1990s, experiments at University College London generated limbs with six rather than the usual three digits, a result that is inconsistent with the prepattern theory. I will describe the way in which this apparent contradiction was resolved using models in which chemical diffusion coefficients had a piecewise constant variation across the limb, which is in keeping with experimental data.

Excitable systems have application in many areas of biology, including in particular the transmission of electrical signals along nerve axons, which occurs fast and reliably at a speed of about 45 miles per hour. Hodgkin and Huxley originally studied this phenomenon using a model which they solved numerically. I will describe more recent work in which a piecewise linear caricature of the Hodgkin-Huxley system has been studied using singular perturbation theory to obtain a detailed analytical description of the signalling pulse.

Excitable dynamics are also important in cellular slime moulds, in which excitably-generated chemical signalling waves propagate between cells, causing them to aggregate when food is scarce. Spiral signalling waves are established and gradually radial streams of cells form. The mechanism of this stream formation has been a long-standing puzzle which we have recently solved, again using piecewise linear equations. This method enables solution of the Floquet problem governing stability of the spiral wave structure, so that the conditions for destabilisation into streams can be predicted.

[TOP | SPEAKERS | ABSTRACTS | TIMETABLE]

Universita di Udine, Dipartimento di Matematica e Informatica, 33100 Udine, Italy.

*Periodic solutions of piecewise linear ODEs*

I consider nonlinear boundary value problems with asymmetric nonlinearities and thereby study periodic solutions to piecewise linear ODEs. I will review some recent results by myself and co-workers Fonda, Rebelo and Ortega concerning piecewise linear equations as well give some general ideas about the (geometric and topological) techniques we use. For example, we used have some recent versions of the Poincare - Birkhoff fixed point theorem which although a purely existence result can be used to produce multiplicity of solutions as well. In addition, recent `constructive' proofs of the P--B theorem have numerical applications.