Nonlinear Workshops

Default Time: Mondays 1:00-1:55 pm (unless indicated otherwise)
Location: VC's Room, Queen's building
Bring your own lunch!

Autumn 1999

Here is a List of Seminar Series (with links) in Engineering and other Faculties.

Previously Seminars:

Back to Engineering Mathematics

This page maintained by Alan Champneys
E-mail: a.r.champneys@bristol

Lust Abstract


There is a growing interest in the study of periodic phenomena in large-scale nonlinear dynamical systems. Often the high-dimensional system has only low-dimensional dynamics, e.g., many reaction-diffusion systems or Navier-Stokes flows at low Reynolds number. We present an approach that exploits this property in order to compute branches of periodic solutions of the large system of ordinary differential equations (ODEs) obtained after a space discretisation of the PDE. We call our approach the Newton-Picard method. Our method is based on the recursive projection method (RPM) of Shroff and Keller but extends this method in many different ways. Out technique tries to combine the performance of straightforward time integration with the advantages of solving a nonlinear boundary value problem using Newton's method and a direct solver. Time integration works well for very stable limit cycles. Solving a boundary value problem is expensive, but works also for unstable limit cycles.

We will present some background material on RPM. Next we will explain the basic features of the Newton-Picard method for single shooting. The linearised system is solved by a combination of direct and iterative techniques. First, we isolate the low-dimensional subspace of unstable and weakly stable modes (using orthogonal subspace iteration) and project the linearised system on this subspace and on its (high-dimensional) orthogonal complement. In the high-dimensional subspace we use iterative techniques such as Picard iteration or GMRES. In the low-dimensional (but "hard") subspace, direct methods such as Gaussian elimination or a least-squares are used. While computing the projectors, we also obtain good estimates for the dominant, stability-determining Floquet multipliers. We will present a framework which allows us to monitor and steer the convergence behaviour of the method.

RPM and the Newton-Picard technique have been developed for PDEs which reduce to large systems of ODEs after space discretisation. In fact, both methods can be applied to any large system of ODEs. We will indicate how these methods can be applied to the discretisation of the Navier-Stokes equations for incompressible flow (which reduce to an index-2 system of differential-algebraic equations after space discretisation when written in terms of velocity and pressure.)

The Newton-Picard method has already been extended to the computation of bifurcation points on paths of periodic solutions and to multiple shooting. Extension to certain collocation and finite difference techniques is also possible.

Fraser abstract


It is well known that if a column exceeds a certain critical length it will, when placed upright, buckle under its own weight. In a recent experiment Acheson and Mullin (1998) have demonstrated that a column that is longer than its critical length can be stabilized by subjecting its bottom support point to a vertical oscillation of appropriate amplitude and frequency.

In this Talk I shall derive the nonlinear equations that govern the stability of such a column. I will then discuss a weakly nonlinear asymptotic analysis of these equations for a column that is slightly longer than its critical length. It will be shown that such a `slightly too long column' can be stabilized by oscillating its support point, but that it may be difficult to obtain a sharp experimental determination of the stability boundary.

Kuske abstract


Localization occurs in systems of coupled oscillators when the oscillations of one component are small compared to the oscillations of other components. This phenomenon can be observed in coupled laser arrays, biological and chemical oscillations, and repetitive engineering structures. Symmetry-breaking, initial conditions, and resonance effects can all contribute to localization. Phase equations for coupled oscillators can not capture this phenomenon, and a new method for predicting localization will be described for several applications.