STATEMENT OF RESEARCH INTERESTS
Member of the London Mathematical Society.
Referee for the Institute of Physics, the London Mathematical Society and Mathematical Reviews.
As an author, I am supplied with free copies of Maple, Mathematica and MATLAB, and all their add-ons.
The research material has informed and supported both final year undergraduate and MSc level units at
Manchester Metropolitan University and Southampton University. I have had over twenty refereed
journal articles published, two books and one invited book chapter. Two more books are due for publication
in 2007. My research can be divided into four areas:
1. Nonlinear Ordinary Differential Equations
Collaborators:
(i) Dr Colin Christopher, Department of Mathematics and Statistics, University of Plymouth, UK.
(ii) Dr Lida Nejad, Department of Computing and Mathematics, Manchester Metropolitan University, UK.
(iii) Professor Yu Pei, University of Western Ontario, CANADA.
(iv) Professor Han Maoan, Shanghai Normal University, CHINA.
(v) Professor Armengol Gasull, Departament de Matematiques, Edifici C, Universitat Autonoma de Barcelona
08193 Bellaterra, Barcelona, SPAIN.
One of the most intriguing problems in the theory of planar systems is the famous second part of David
Hilbert’s sixteenth problem: estimate the maximal number and relative positions of limit cycles of poly-
nomial vector fields of degree n. Although easily stated, this problem has proved to be one of the most
difficult to solve from his list of twenty three problems presented to theWorld Congress of Mathematicians
in 1900. One particular sub-class of the investigation into Hilbert’s problem is the Liénard equation:
x´´+f(x)x´+g(x)=0.
This system has proved remarkably useful in the investigation of limit cycles in the plane and appears
in many applications. Our work has focused on local bifurcations; that is, bifurcation of limit cycles in
a small neighbourhood of a critical point. This work has led to a number of publications which give the
maximum possible number of small-amplitude limit cycles for Liénard systems of varying degrees.
We are currently investigating simultaneous bifurcations of limit cycles in symmetric Liénard
systems. In 2004, I was invited to write a book chapter by Dongming Wang and Zhiming Zheng. The
book entitled ”Differential Equations with Symbolic Computation” was published in 2005.
Future work:
Work will continue investigating simultaneous bifurcations from multiple critical points. Applications of
this theory will be investigated in wing rock in modern aircraft, surge in jet engines, and periodic
behaviour in artificial neural networks, for example. Lida Nejad and I will be applying for a PhD grant.
The bid will be entitled ”Instabilities, bistability and feedback in astrochemical cloud models”. I will be
visiting Professor Armengol Gasull in Barcelona next year to discuss future collaborations. I have been
invited to China by Professor Jibin Li and Professor Han Maoan. I will be seeking funding and hope to
visit them in summer 2007.
Some publications:
1. Yu P., Han M., Christopher C.J. & Lynch S., On limit cycles of the Liénard equations with Z2 sym-
metry, in preparation (2006).
2. Jiang J., Han M., Yu P. & Lynch S., Limit cycles in two types of symmetric Liénard systems, Int. J.
of Bifurcation and Chaos, (in press) (2006).
3. Lynch S., Chapter 1: Symbolic computation of Lyapunov quantities and the second part of Hilbert’s
sixteenth problem, Differential Equations with Symbolic Computations, Wang, Dongming; Zheng, Zhim-
ing (Eds.), Series: Trends in Mathematics, 1-26 (2005). ISBN: 3-7643-7368-7.
4. Christopher C.J. & Lynch S., Small-amplitude limit cycles of Liénard equations with either quadratic
or cubic damping or restoring coefficients, Nonlinearity, 12(4), 1099-1112 (1999).
5. Lynch S. & Christopher C.J., Limit cycles in highly nonlinear differential equations, J. Sound and
Vibration, 224(3), 505-517 (1999).
6. Lloyd N.G. & Lynch S., Small amplitude limit cycles of certain Liénard systems, Proc. Roy. Soc.
Lond. Ser. A, 418, 199-208 (1988).
2. Applications of Multifractal Analysis
Collaborators:
(i) Dr Graham Lees and Dr Chris Liauw, Department of Materials Science, Manchester Metropolitan
University, UK.
(ii) Dr Steve Mills, BICC Cables, Wrexham, UK.
(iii) Professor Jo Verran, Department of Biology, Manchester Metropolitan University, UK.
(iv) Professor Peter Kelly, Materials Science, MMU.
When plotting the solutions of chaotic systems one nearly always sees fractal like patterns. A true fractal
is a geometrical figure consisting of an identical motif repeated on an ever reduced scale. Most of the
fractals appearing in books or on posters are early generation fractals, the real fractals generated after
an infinite number of generations are theoretical figures which we can only picture in our minds.
Unfortunately, in the real world, fractals are not homogeneous, there is rarely an identical motif
repeated on all scales; instead the fractals are heterogeneous and there is always some kind of scaling
restriction. The generalized fractal dimension curves may be used as a means of measuring dispersion.
One useful tool used in nonlinear dynamics is the multifractal analysis of density distributions. In this
particular case, the analysis is applied to elemental dot maps produced by scanning electron microscopy
coupled with energy dispersive X-ray spectroscopy. This is being applied to study distribution patterns
of particulate fillers in the production of polymers and the effects on the physical properties of plastics
used in coating.
Future work:
In 2006, Professor Jo Verran (Biology), Professor Peter Kelly (Materials Science) and I are applying for
an EPSRC grant (£120,000), the bid is entitled ”TiN/Ag Nanocomposite Coatings for Novel Hygienic Surfaces”.
Some publications:
1. Mills S.L., Lees G.C., Liauw C.M., Rothon R.N. & Lynch S., Prediction of physical properties following
the dispersion assessment of flame retardant filler/polymer composites based on the multifractal analysis
of SEM images, J. Macromolecular Sci. B-Physics, 44(6), 1137-1151 (2005).
2. Mills S.L., Lees G., Liauw C. & Lynch S., An improved method for the dispersion assessment of flame
retardent filler/polymer systems based on the multifractal analysis of SEM images, Macromolecular Materials
and Engineering, 289(10), 864-871 (2004).
3. Mills S.L., Lees G., Liauw C. & Lynch S., Dispersion assessment of flame retardent filler/polymer sys-
tems using a combination of X-ray mapping and multifractal analysis, Polymer Testing, 21(8), 941-948
(2002).
3. Nonlinear Optics
Collaborators:
(i) Dr Alan Steele, Department of Electrical Engineering, Carleton University, Ottawa, CANADA.
The development of optical fibre technology has continued to expand dramatically over recent years with
the underlying aim to increase the rate of data transmission. The nonlinearity of the optical fibre is
usually included as a refractive index that is dependent on the intensity of light propagating through the
fibre. Optical bistability using nonlinear optical fibre has become an area of growing interest in optical logic
and all-optical signal processors. Investigations have focused upon three different types of resonator;
the ring cavity, the simple fibre resonator and the nonlinear optical loop mirror with feedback. All
three devices can exhibit bistable behaviour, but instabilities can encroach on these regions. A range
of different techniques have been applied including a linear stability analysis, graphical methods and
iterative methods which may be used to produce bifurcation diagrams. It is possible to choose parameter
values such that the bistable regions are isolated from any instabilities. Current work is concentrating on
the shape of the pulse entering the bistable device. The shape of the pulse can lead to a ’ringing’ on the
hysteresis loop, we are attempting to minimise this effect. We are also investigating the stability of other
nonlinear optical devices, for example, the double ring and double coupler resonators.
A grant was awarded in Canada for ”Facilities for Research in Design Automation Algorithms in
Optoelectronic and High-speed Systems”. The grant holders are, Pavan Gunupudi (Project Leader),
Alan Steele & Stephen Lynch. Project number 9602. Sum 537,102 CAD over 4 years from 2004. For
large memory computer servers, computer workstations and associated infrastructure. Note: Total sum
comes from three sources, Canada Foundation for Innovation, the Ontario Innovation Trust and in-kind
contributions from equipment suppliers. Award is only for equipment and infrastructure not students or
other personnel.
Future work:
There are a number of papers in preparation and we are going to apply for joint funding in the near future.
Some publications:
1. Lynch S. & Steele A.L., Analysis of a dual nonlinear optical fibre loop mirror resonator, in preparation
(2006).
2. Steele A.L. & Lynch S., Characteristics of auxiliary synchronization in optical fibre ring resonators, in
preparation, (2006).
3. Steele A.L. & Lynch S, Chaos synchronization of a passive fibre resonator using the auxiliary system
and applications to chaos masking, Nonlinear Guided Waves and Their Applications (Topical Meeting)
on CD-ROM (The Optical Society of America, Washington, DC, 2004), MC15.
4. Lynch S. & Steele A.L., Controlling chaos in nonlinear bistable optical resonators, Chaos, Solitons
and Fractals, 11(5), 721-728 (2000).
5. Steele A.L., Lynch S. & Hoad J.E., Analysis of optical instabilities and bistability in a nonlinear optical
fibre loop mirror with feedback, Optics Comm., 137(1-3), 136-142 (1997).
4. Neural Networks
Collaborators:
(i) Dr Jon Borresen, Department of Computing and Mathematics, Manchester Metropolitan University,
UK.
(ii) Dr Zuhair Bandar, Department of Computing and Mathematics, Manchester Metropolitan University,
UK.
(iii) Dr Prasad Ponnapalli, Department of Engineering, Manchester Metropolitan University, UK.
It is now understood that chaos, oscillations, synchronization effects, wave patterns, and feedback are
present in higher-level brain functions and on different levels of signal processing. In recent years, the
disciplines of neuroscience and nonlinear dynamics have increasingly coalesced, leading to a new branch
of science called neurodynamics. Early work has concentrated on feedback in neural networks.
Future work:
Adaptive and intelligent control using artificial neural networks, with Dr Prasad Ponnappalli
in the Engineering department at MMU. This work will be concerned with artificial neural networks which work
through feedback. Investigating small- and large-amplitude limit cycle bifurcations in artificial neural
networks.
Some publications:
1. Borresen J. & Lynch S., Limit cycles in coupled Hodgkin-Huxley neuromodules, in preparation (2006).
2. Lynch S. & Bandar Z., Analysis of a bistable neuromodule, in preparation, (2006).
3. Lynch S. & Bandar Z., Bistable neuromodules, Nonlinear Anal. Theory, Meth. & Appl., 63(5-7),
669-677 (2005).