Dr Philip J. Aston

Reader in Mathematics

Research Interests

My recent research work has been in the following areas:

Symmetry-Breaking Bifurcation Theory
Of particular interest are mode interactions which occur when two bifurcations coalesce as a second parameter is varied. When the second parameter is perturbed from its critical value, the two bifurcations separate and secondary bifurcations appear.

Numerical methods for dealing with steady state/steady state mode interactions were considered with A. Ganesh Sittampalam where it was shown that the mode interaction corresponds to a bifurcation in two extended systems. Numerical methods for dealing with steady state/Hopf mode interactions were considered with a former PhD student, Dr F. Amdjadi. It was shown that such a mode interaction corresponds to a symmetry-breaking bifurcation of a Hopf extended system and also to a Hopf bifurcation of a symmetry-breaking extended system.

Numerical methods for the continuation of a path of Hopf bifurcation points arising from a double singular point have been considered in collaboration with Prof A. Spence (Bath) and Dr W. Wu (Jilin, China).

Numerical methods for dealing with Hopf bifurcations in equations with O(2) symmetry have been considered with Dr F. Amdjadi and Dr P. Plechac. Numerical results were obtained for multiple Hopf bifurcation points in the Kuramoto-Sivashinsky equation.

Work in progress with Dr J. Furter (Brunel) is concerned with obtaining a classification of mode interactions. This is achieved by considering the reduced problem on a two-dimensional space.

Symmetric Chaos
Transitions which occur in chaotic systems with symmetry as a parameter is varied have been considered in two different cases:

• iterated maps with O(2) symmetry can have chaotic motion in a fixed point space which is fixed by a reflection. This attractor can lose stability to an attractor which drifts round the group orbit and has full O(2) symmetry on average. It can be shown that for the attractor in the fixed point space, there are Lyapunov exponents associated with perturbations normal to the space and one of these is always zero. If another of these Lyapunov exponent changes sign from negative to positive, then this attractor loses stability and the motion begins to drift around the group orbit (Int. J. Bif. Chaos 5, 701-724, 1995).
  
  
• work done with Michael Dellnitz considered chaotic motion of systems of coupled oscillators which have symmetry due to the coupling arrangement. Lyapunov exponents were classified according to their symmetry type and this lead to efficient numerical methods for the computation of the dominant Lyapunov exponent of each symmetry type, which determine the bifurcations of attractors. It was also shown that problems with continuous Lie symmetries give rise to attractors with several zero Lyapunov exponents. Results on both the pointwise and the average symmetry of attractors after a bifurcation were obtained. The theoretical results were applied to coupled Lorenz systems and coupled Duffing oscillators (Int. J. Bif. Chaos 5, 1643-1676, 1995).

Turbulence in the CGL Equation
This is a project funded by EPSRC under the Applied Nonlinear Mathematics Initiative and involves myself, Dr Michele Bartuccelli and Dr Peter Voke (Department of Mechanical Engineering). Eddie Wilson is working on the project.

The aim of the project is to investigate numerically the transition from soft to hard turbulence which was predicted analytically by Bartuccelli, Constantin, Doering, Gibbon and Gisselfalt (Physica D 44, 421-444, 1990). In this paper, upper bounds on the solutions of the CGL equation in various norms were obtained and so the numerical work aims to investigate how tight these bounds are and to see whether the predicted transition from soft to hard turbulence occurs in practice.

Control of Chaos
In 1990, Ott, Grebogi and Yorke proposed a new method for controlling chaos by using small parameter perturbations to stabilise a fixed point contained in the attractor (Phys. Rev. Letts 64, 1196-1199, 1990). A PhD student, Carl Bird, has been working in this area. The main topics we have considered are

• the rate of convergence of the OGY method and new methods for improving the rate of convergence (Int. J. Bif. Chaos 5, 1157-1165, 1995);
• extending the basin of attraction in order to reduce the chaotic transient (Chaos, Sol. & Fract. 8 , 1413-1429, 1997);
• improving the approximation to a fixed/periodic point which has been stabilised using the OGY method;
• targeting methods in the presence of noise;
• achieving synchronisation of coupled systems using small parameter perturbations.

Integration Methods for Computing the Dominant Lyapunov Exponent
Lyapunov exponents are usually computed using time averaging over a long orbit. However, there are a number of pitfalls of this approach. An alternative approach is to use integration with respect to the invariant measure which overcomes all the pitfalls of the time averaging approach.

Michael Dellnitz and I have developed methods for using a spatial average to compute the dominant Lyapunov exponent.

For a one-dimensional map, the Ergodic Theorem can be used to replace the time average with a spatial average, integrating with respect to the invariant measure. However, for maps of higher dimension, this cannot be used. Instead, a sequence of integrals can be found which converges to the dominant Lyapunov exponent. In practice, convergence of this sequence is found to be very slow. We prove that the dominant term in the asymptotic expansion of the sequence is proportional to 1/n. This allows us to use two successive terms in the sequence to eliminate this slowly decaying term and then much faster convergence is obtained. Good numerical results confirm the theory. This approach is particularly efficient for determining blowout bifurcations of coupled oscillators. (Comp. Meth. Appl. Mech. Engng 170, 223-237, 1999; Proc. Roy. Soc. 2003 in press. download here )

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