About the centre

The Bristol Centre for Applied Nonlinear Mathematics is a £1M EPSRC funded research programme to address both the mathematical themes of the Bristol Laboratory for Advanced Dynamic Engineering (BLADE), and the grand engineering challenge of real-time dynamic substructuring. Building on the internationally leading interdiscipinary base of the Applied Nonlinear Mathematics group, and together with the Laboratory for Advanced Computation and the Department of Mathematics, the centre will address these aims with

• Five postdoctoral research assistants, appointed on a rolling programme over five years, to work on specific projects within each theme.
• A series of workshops and extensive visitor programme centered on the themes.

The mathematical themes the programme will address are:

	A: Dynamical systems with delays
	B: Dynamics of piecewise smooth systems
	C: Qualitative numerical analysis of high-dimensional nonlinear systems
	D: Modelling spatially extended systems; continuous vs discrete

In addition, the permanent academic members of the centre have a wide range of research interests, including

• analysis and control of piecewise smooth systems
• analysis of local and global bifurcations
• numerical continuation of global bifurcations and solitary waves
• computation of invariant manifolds
• control and synchronisation of chaotic systems
• nonlinear dynamics of laser systems
• liquid crystal dynamics
• electrical control of porous media flows
• non-integrable nonlinear optics
• localised elastic buckling in long structures
• nonlinear dynamics of power electronics systems
• fractal analysis of medical images

Workshop and visitor programme | Back to top

A crucial part of this multidisciplinary research is a programme that allows us to attract the world's leading experts for research visits on a continuous rolling programme. Visitors will be able to conduct collaborative research towards each theme's objectives, and make full use of the world-class facilities of BLADE. Follow-up visits are also encouraged, to continue collaboration. Each visitor will be encouraged to visit other UK institutions, and also to present a set of tutorial lectures, which will take place at Bristol, but be advertised nationally.

We plan to hold five week-long workshops, one in each of the four themes, plus a closing meeting. In addition to long-term visitors, each of the other workshops will bring together further international experts in the relevant mathematical areas, as well as practitioners from the engineering and other applied disciplines. Each workshop will be held in Bristol, making full use of BLADE facilities, with with selected tutorial lectures & accompanying material being made available on the web.

• The first workshop, Delay Equations and their Applications took place 8-12 September 2003.
• The second workshop, Piecewise smooth dynamical systems: analysis, numerics and applications took place 13-16 September 2004.
• The third workshop, Qualitative Numerical Analysis of High-dimensional Nonlinear Systems took place 21-24 March 2005.
• The fourth workshop, Successes and Failures of Continuous Models for Discrete Systems took place 5-8 September 2005.

Theme A: Methodological advances in differential delay equations | Back to top

The overall aim of this RA project is to develop practical methodologies for understanding the global dynamics of differential equations with delay, in particular the onset of high-dimensional dynamics. This is motivated by the real-time dynamic subtructuring element of the BLADE project (RA Z), where any generic control scheme introduces a delay. The project will focus on a hierarchy of models for the archetypal system of an inverted pendulum supported on a motorised cart.

• Understand the dynamics and (global) bifurcations of a two-degree of freedom model of a controlled inverted pendulum with one fixed delay.
• In order to allow comparison with physical experiments, develop and analyse more realistic higher dimensional models which incorporate angular momentum of the supporting cart and compliance of its drive line. Investigate the combined effect of delay and backlash arising from the cart's transmission (in co-operation with RA B)
• Classify new types of dynamical behaviour and bifurcations leading to high-dimensional attractors in DDEs, including those describing low frequency fluctuations in lasers with optical feedback.

Theme B: Analysis and computation of bifurcations in piecewise smooth dynamical systems | Back to top

The overall aims of this RA project are to develop analytical and numerical methods for the study of bifurcations unique to piecewise smooth (PWS) dynamical systems and to apply them to investigate the dynamics of a high dimensional PWS model of an aeroelastic wing system with backlash.

• Analyse codimension-one and codimension-two C-bifurcations in mechanical systems with friction and backlash.
• Develop numerical algorithms for the continuation of periodic solutions in PWS systems and the detection and unfolding/branch switching of codimension-one and codimension-two C-bifurcations.
• Use these models to analyse the limit cycle behaviour and the onset of high-frequency motion of an aeroelastic wing system with a trailing edge which has structural backlash.

BCANM is also a node of SICONOS, a FP5 European Project on Nonsmooth Systems, with which the research of theme B is strongly connected. In particular, Alan Champneys is the leader of Workpackage 4 - Bifurcation Analysis on which Petri Piiroinen is working.

An informal meeting on codimension-two bifurcations in nonsmooth dynamical systems was held at the University of Bristol, 25-26 July 2003.

The Theme B workshop: Piecewise smooth dynamical systems: analysis, numerics and applications, took place from 13-16 September 2004.

Theme C: Overcoming instabilities in helicopter dynamics | Back to top

The overall aim of this RA project is to understand the onset and subsequent large-amplitude motion of an oscillatory instability in a typical helicopter teetering tail rotor. This subject is of practical interest to Westland Helicopters Ltd. The project entails the development of specialised numerical methods for the investigation of the onset of quasi-periodic motion and subsequent breakdown of invariant tori in high-dimensional dynamical systems.

• In a simplified model of the teetering tail rotor, accurately predict and perform parameter continuation of secondary Hopf bifurcations, subsequent invariant tori and their breakdown.
• Develop and assess complementary continuation methods for quasi-periodic motion and invariant tori of high-dimensional systems.
• Apply these methods to analyse the oscillatory instabilities in models of the teetering rotor that couple to the whole helicopter motion including unsteady aerodynamic effects.

Theme D: Models of shear band formation in soil mechanics | Back to top

This modelling project seeks to understand a particular localization an pattern formation phenomenon observed in soil under load. Our analysis will be based on a combination of discrete (at the level of individual grains) and continuum models, in order to gain qualitative insight into the wavelength selection process. The long term aim is to predict the macroscopic behaviour (including failure) of soils under load, which is a problem of much interest to structural engineers.

• Develop, simulate and analyse the solutions of simple discrete models of touching discs or spheres. Pass to a continuum limit.
• Take various simplified continuum models of the statics of sand and perform numerical bifurcation analysis of their spatially-periodic solutions, both stripes and criss-crosses.
• Use the `Maxwell load' criterion to predict wavelengths of periodic patterns produced with both modelling approaches and compare with laboratory tests and computations in realistic geometries.

Theme Z: Understanding the theoretical basis behind making real-time substructuring work | Back to top

The overall aim of this part of the programme is that we will have mathematical confidence in real-time dynamic substructuring. The theoretical study will be carried out in parallel with experimental substructuring tests within the BLADE facility. Whilst we have a clear idea of the first year, the precise course that this part of the programme will take depends on the results from all the other four themes, including workshops and associated visitor programmes. Therefore, the programme Scientific Steering Committee will select specific objectives from the three main challenges below.

• Delay
• Backlash in the real substructure
• Computational and modelling issues in the virtual substructure

We will apply our results to two test-bed problems: an auto-parametric resonant system, and cable-deck interaction in cable-stayed bridges.