UK Nonlinear News, August 2003


Geometry and dynamics in Hamiltonian lattices, with application to the Fermi-Pasta-Ulam problem

Bob Rink, Mathematisch Instituut, University of Utrecht

Thesis adviser: F. Verhulst, 2d adviser J.J. Duistermaat

Thesis to be defended September 17, 2003

E. Fermi, J. Pasta and S. Ulam introduced the Fermi-Pasta-Ulam lattice in the 1950s as a classical mechanical model for a mono-atomic crystal or a one-dimensional continuum. The model consisted of a discrete number of equal point masses that interact with their nearest neighbours only. On the basis of statistical mechanics, they expected that when the interparticle forces were anharmonic, the lattice would reach a thermal equilibrium. This means that averaged over time, its initial energy would be equipartitioned among all its Fourier modes. The famous computer experiment that they performed in Los Alamos in 1955, was intended to investigate in what manner and at what time-scale the equipartitioning would take place. The result was astonishing: the lattice did not come close to thermal equilibrium, but behaved more or less quasi-periodically. Only when the initial energy was larger than a certain threshold, did the lattice seem to `thermalise'. This paradox is nowadays known as the `Fermi-Pasta-Ulam problem'.

The observations of Fermi, Pasta and Ulam were a great impulse for nonlinear dynamics. One possible explanation for the quasi-periodic behaviour of the FPU system, is based on the Kolmogorov-Arnol'd-Moser theorem, which states that most of the invariant Lagrangean tori of a Liouville integrable Hamiltonian system survive under small perturbations. It is required though for the KAM theorem that the integrable system satisfies a nondegeneracy condition. Unfortunately, it has for a long time been unclear how the Fermi-Pasta-Ulam lattice can be viewed as a perturbation of a nondegenerate integrable system.

Nishida in 1971 and Sanders in 1977, investigated a Birkhoff normal form for the FPU lattice. Under the assumption of a rather strong nonresonance condition on the linear frequencies ωk = 2 Sin(kπ/n)  of the lattice, they showed that this Birkhoff normal form is integrable and nondegenerate. This means that the KAM theorem can indeed be applied. The problem is of course that their required nonresonance condition is usually not met. This leaves a large gap in the proofs. The new idea in this thesis is to incorporate the discrete symmetries of the lattice in the argument. It turns out that this enables us to show that the nonresonance condition of Nishida and Sanders is not needed: every resonance is overruled by a symmetry. Hence the Birkhoff normal form is integrable and this proves the applicability of the KAM theorem.

Moreover, much attention is paid to the analysis of exact and approximate solutions of the lattice equations. New exact solutions and invariant manifolds are found in the fixed point sets of the symmetries of the lattice. Also, the Birkhoff normal form reveals approximate integrals and solutions in the low energy domain of the phase space. The analysis makes use of invariant theory and singular reduction. One of the conclusions is that the lattice with an even number of particles contains travelling wave solutions that change their direction. Moreover the integrable normal form contains nontrivial monodromy, meaning that it does not allow global action-angle variables.


Nonlinear Dynamics of Self-excitation in Autoparametric Systems

Abadi, Mathematisch Instituut, University of Utrecht

Thesis adviser: F. Verhulst

Thesis to be defended October 6, 2003

Various types of self-excited oscillators are implemented into an autoparametric system [4], and the study of the solutions, stabilities and bifurcations, shows very di.erent results. This thesis describes recent results on this topic by Abadi [1, 2], Verhulst and Abadi [8], and the modi.cation of the results by Tondl [5, 6]. For more details about the background, see the references therein. 

First, we implement the Rayleigh type oscillator into a suitable autoparametric system. The bifurcation analysis of the solution gives the result that there exists a stable robust heteroclinic cycle as nontrivial solution of the system. Taking the detuning σ = 0 (near resonance) results in symmetry breaking of the heteroclinic cycle; a long periodic solution occurs. 

Second, we replace the Rayleigh oscillator of the autoparametric system with a dry friction oscillator characterised by a small parameter. We can determine the boundary value of parameters of the system for having a nonsmooth periodic solution. By using SlideCont [3], sliding bifurcations of the nonsmooth periodic solution can be studied. A numerical simulation shows that a 3-dimensional nonsmooth invariant manifold existing in the (full) system can be obtained. 

In this thesis we also implement a relaxation oscillator of van der Pol type into an autoparametric system. This extends the result by Verhulst [7]. The possibility of destabilising the undesirable vibrations due to the stable normal mode of the system is studied by choosing a suitable tuning and appropriate deformation of the slow manifold. In the case of normal mode vibration derived from a relaxation oscillation, we need low-frequency tuning of the attached oscillator. In this way it is possible to make the quenching e.ective. 

Conditions for suppression of self-excitation of a 3 d.o.f system is also studied. We consider vibrating systems containing self-excitation and parametric excitation. In the analysis we show that full and partly suppression of the excitation are possible to achieve when some conditions are met. Surprisingly, an ÷autoparametric÷ phenomenon takes place in the normal form of a three-mass system in 1 : 2 : 3-resonance; partial decoupling of the system occurs. We conclude that nonlinear dynamics obtained from embedding a self-excited oscillator in a higher dimensional system is of practical interest and at the same time it is a rich source of interesting phenomena.


  1. Abadi, On Self-Excited Autoparametric System, Nonlinear Dynamics, 24, 2001, 147ű166. 
  2. Abadi, A Self-excited Autoparametric System with Dry Friction, ZAMM, [submitted], 2003.
  3. Dercole, F., and Kuznetsov, Yu. A., SlideCont: An AUTO97 Driver for Sliding Bifurcation Analysis, Preprint 1261, Universiteit Utrecht, 2002.
  4. Tondl, A., Ruijgrok, M., Verhulst, F., and Nabergoj, R., Autoparametric Resonance in Mechanical Systems, Cambridge University Press, New York, 2000. 
  5. Tondl, A., Suppressing Self-excited Vibration by Means of Parametric Excitation, Proc. Colloquium Dynamics of Machines 2000, 225, Institute of Thermomechanics ASCR, Prague, 2000. 
  6. Tondl, A. and Nabergoj, R., The E.ect of Parametric Excitation on a A Self-excited Three-mass System, Pre-print, 2002.
  7. Verhulst, F., Normal Mode Instability and Quenching, Proc. 6th Int. Conf. Vibrations Problems, Liberec, 2003.
  8. Verhulst F. and Abadi, Autoparametric Resonance of Relaxation Oscillations, [to be submitted], 2003.

PhD Studentships

Mathematical Biology and Biophysical Chemistry Molecular Organisation and Assembly in Cells

University of Warwick

Eight multi-disciplinary PhD studentships (stipend: ˙11,000) will be available for UK students in the new multi-disciplinary doctoral training centre funded by the Life Sciences Interface of the Engineering and Physical Sciences Research Council in October 2003. Graduates in mathematical and physical sciences who wish to develop multidisciplinary research programmes in the areas of mathematical-biology and biophysical chemistry are invited to apply for this four year doctoral training programme, the first year of which will lead to an MSc in Molecular Organisation and Assembly in Cells.

Further details of the MOAC programme can be obtained from the director:

Dr. Alison Rodger
Department of Chemistry
University of Warwick
Coventry, CV4 7AL, UK
Phone:024 7652 3234

Source: David Rand

Dynamics/Numerical Analysis

University of Bristol

See advertisement for same in Postdoctoral Positions.

2 EPSRC PhD Studentships in Dynamics/Solid Mechanics

University of Aberdeen

Applications are invited for a three year PhD studentship available NOW for a home student to work in the areas of dynamics/ solid mechanics. We seek suitably-qualified students with an engineering and mathematical background who are willing to undertake an analytical and experimental work in one of the areas listed below:

For further information and application forms please contact:

Prof. M. Wiercigroch
Telephone 01124 272509

Dr. E. Pavlovskaia 
Telephone 01124 272782 

Source: Marian Wiercigroch


Postdoctoral Position

Mathematical Chemistry

University of Lethbridge, Canada

A postdoctoral research position is available in the Department of Chemistry and Biochemistry at the University of Lethbridge in the area of mathematical chemistry, with an emphasis on problems of biological relevance. Current interests of the group include invariant manifold methods and their application to model reduction, pattern formation in the context of developmental biology, and the dynamics of systems subjected to on-off switching. Both the development of fundamental theory and applied mathematical modelling projects are pursued. Accordingly, this position affords candidates a variety of avenues for growth and exploration.

The successful applicant will have a strong background in nonlinear dynamics, normally obtained through appropriate thesis work in the natural or mathematical sciences. Experience in the analysis of partial or stochastic differential equations is particularly desirable. A strong interest in biological systems is a definite asset.

The initial appointment would be for a period of one year, with renewal contingent on performance. The start date is flexible. Salary is negotiable and will depend on the candidate's experience. A relocation allowance will also be available.

For further information or to apply, contact

Dr Marc R. Roussel
Department of Chemistry and Biochemistry
University of Lethbridge
Lethbridge, Alberta
T1K 3M4

phone: +1 403 329 2326
fax: +1 403 329 2057

Applications should include a cover letter outlining the candidate's qualifications, a CV, and the names, phone numbers and email addresses of at least three referees. For applications sent by email, the following file formats, listed in order of preference, are acceptable: Adobe Acrobat (pdf), PostScript, LaTeX, Rich Text Format, Microsoft Word. Applications will be accepted until October 19, 2003.

Mathematical Architecture of Biological Regulation

University of Warwick

Applications are invited for a number of 3-year postdoctoral research fellowships in the newly created Interdisciplinary programme for Cellular Regulation (IPCR). These fellows will be part of a major new multidisciplinary research programme funded by the EPSRC and BBSRC applying mathematics and statistics to the understanding of cellular regulation. This team of investigators leading the projects is a multidisciplinary one involving mathematical and physical scientists and as well as biologists and medics. The research programme is organised under two themes divided into seven projects:

Theme 1: Networks & data: Mathematical analysis and statistical inference
Project 1: Mathematical analysis of network architecture: function, dynamics and noise.
Project 2: Coordinating and controlling rhythms throughout the cell.
Project 3: Regulatory networks: statistical links between models and data.
Project 4: Integration of bioinformatics data in the construction of Bayesian networks for gene regulation.
Theme 2: Spatial processes and regulation.
Project 5: Modelling transport and protein sorting across cell membranes and compartments.
Project 6: The dynamic mechanical properties of biopolymer networks.
Project 7: TCR signalling and raft dependent spatial clustering.

Each fellow will be attached to one of the projects but will also be expected to interact with some others. There will be a organised training programme that will both cover both the relevant biology and the underlying mathematical and statistical technologies and fellows will be encouraged to form strong links with relevant experimental groups.

Applicants should possess strong mathematical and/or statistical skills. Our expectation is that they will possess a PhD in Mathematics, Statistics, Computer Science or Physics although we are happy to consider other areas. For some of the posts we are particularly interested in candidates with a strong background in either probability theory, statistics, bioinformatics, nonlinear dynamics or numerical methods/scientific computing. Previous experience with biology would be an advantage but is not necessary. However, a willingness to engage with biological ideas is vital.

The salary scale for these posts is ˙18.265 - ˙27.339.00.

Further details can be obtained from the IPCR's website Candidates wishing to discuss these posts further should contact one of the programme managers: Dr Nigel Burroughs or Professor Andrew Millar.

Application packs are available from the Personnel Office on 024 7652 3685 (24 hour answerphone) or by email to An application form MUST be completed if you wish to be considered for this post.

Application Form Please quote reference 42/4R/02. Closing date for applications is 27 June 2003 but late applications will be considered.

Source David Rand

(Professor Rand asked UK Nonlinear News to carry this announcement even though the deadline has passed. He encourages interested applicants to contact Dr Rodger).

Dynamics/Numerical Analysis

University of Bristol

A 3-year EPSRC-funded RA position starting Nov 2003 is available on

"Pseudospectra; Uncertainty, Vulnerability and Bifurcation in Structural Mechanics"

The work will be based the interdisciplinary centre BLADE (Bristol Laboratory for Advanced Dynamic Engineering) and specifically involves the Engineering Mathematics, Aerospace and Civil Engineering Departments. Candidates should hold a PhD or equivalent in mathematics, physics, engineering or related disciplines. Expertise in one of numerical linear algebra, dynamical systems or structural mechanics would be highly beneficial. So would the ability to work as part of a interdisciplinary team. Funding for salary is available up to spine point 6 on the RA scale (20,311 pounds per annum), and will be incremented annually subject to satisfactory progress.

Further details and application procedure may be found online at quoting job reference number 9488

There is also support for a PhD studentship position attached to this grant. Any interested applicants for that are encouraged to respond directly to one of us.

Alan Champneys
Mike Friswell
Jintendra Agarwal
Nick Lieven
John Hogan

Source: Alan Champneys

Characeterisation of Chaos and Synchronisation in Biomedical Time Series with Novel Surrogate Techniques

Hong Kong Polytechnic University

A vacancy exists for a Postdoctoral Fellow (12 months) in the Department of Electronic and Information Engineering at The Hong Kong Polytechnic University. Applicants must have a doctorate degree in mathematics, physics or engineering and normally no more than three years of postdoctoral experience. A good research record in nonlinear time series analysis and an interest in biomedical applications are also required. Further information about the department and the relevant research areas may be found on the following pages

Salary is competitive and will be commensurate with experience.

For further information and to apply please contact:

Dr Michael Small 
Electronic and Information Engineering 
Hong Kong Polytechnic University 
Tel.: +852 2766 4744 

providing a brief CV and a description of your research experience.

Source: Michael Small

Interdisciplinary Research in DNA Microarray Data Analysis & Modelling

Department of Mathematics, UMIST

Postdoctoral Research Associate (up to four years)

DNA Microarrays are a revolutionary technology allowing biologists to measure the levels of expression of thousands of genes at a stroke. The methodology is new - less than a decade old - and interesting mathematical and statistical problems arise at every stage of the experiments, from design and basic data analysis and through to interpretation and modelling of gene control networks. The UMIST component of this interdisciplinary research programme will concentrate on techniques for modelling and statistical analysis. We will be particularly interested in time-series data from experiments that, for example, take snapshots of gene expression throughout a cycle of cell replication.

This project is part of a consortium coordinated by Prof. Olaf Wolkenhauer (Systems Biology Research & Data Engineering, University of Rostock, Germany) and visiting Reader in UMIST's Department of Mathematics. His main UMIST collaborators are Prof. David Broomhead and Dr. Mark Muldoon, both also in the Department of Mathematics.

The consortium also includes two experimental groups: Dr. Paul Kellam's, at UCL, who study host-virus interactions and Prof. Colin Smith's, at Surrey, who study Streptomyces Coelicolor. Other mathematically-inclined collaborators include the Prof. David Lowe and Dr. Ian Nabney of the Neural Computing Research Group at Aston, Prof. Xiaohui LiuĂs Intelligent Data Analysis group at Brunel and Dr. Ernst Wit's Bioinformatics group in the Statistics Department at Glasgow.

Applications are invited from a broad range of post-doctoral researchers in the physical and mathematical sciences. Through close collaborations within the consortium and with the BBSRC life science community this position provides an ideal opportunity for interdisciplinary research and training. Good written and spoken English, research experience and an eagerness to communicate with other scientists are essential. You should expect to start work in the Autumn of 2003.

For more details, please contact:

Dr. Mark Muldoon (
Department of Mathematics, 
P.O. Box 88 
Manchester M60 1QD

Source: Mark Muldoon

Theoretical Biophysics

University of Surrey

(Reference: 3934)

Salary: up to ˙22,191 per annum

We are seeking a postdoctoral research fellow to work on a 2-year EPSRC-funded project to model the mechanics of myosin V, a processive molecular motor, using analytical and numerical techniques. Recent experiments have provided new images of myosin stepping; we aim to capture this new information in a model of the myosin gait.

The successful candidate will be working under the supervision of Dr Rebecca Hoyle in the Department of Mathematics and Statistics at the University of Surrey, where the post will be based, and Dr Matthew Turner of the Department of Physics at the University of Warwick.

Applicants should have a PhD in applied mathematics, physics or a related discipline. An aptitude for mathematical modelling is essential and a background in biophysics is desirable.

For further details see

Informal enquiries may be made to Dr Rebecca Hoyle (+44 (0)1483 682638 or email or Dr Matthew Turner (+44(0)24 765 2257 or email 

For an application pack and details of how to apply please contact:

Miss Julie Boatfield, 
School Human Resources Assistant, 
School of Electronics and Physical Sciences (SEPS), 
University of Surrey, 
Surrey, GU2 7XH. 
Tel:+44 (0) 1483 686125 
answer phone/fax or 689135 during office hours. 

or download application documents from: under "Employment Opportunities". Please quote Post Reference Number 3934, supply your postal address and where you saw this advertisement. 

Closing date for applications is 21st August 2003, with Interviews taking place the week of 14th September.

The University is committed to an Equal Opportunities Policy

Source: Rebecca Hoyle

T.H. Hildebrandt Research Assistant Professorships and VIGRE Assistant Professorships in Applied & Interdisciplinary Mathematics

University of Michigan at Ann Arbor

These positions for up to three years are designed to provide applied mathematical scientists with favourable circumstances for academic career development in both research and teaching. T.H. Hildebrandt and VIGRE positions have a teaching responsibility of one course per semester. These positions may be combined with other postdoctoral fellowships giving additional reductions in teaching responsibility. 

Preference is given to candidates who received the Ph.D. degree in 2002 or later and who submit a completed application by December 13, 2003. Salary is competitive and there are opportunities for supplemental summer salary. (VIGRE appointments have some citizenship/residency restrictions.)

Application forms and further important information are available at,  by Email at  or by mail from:

Hiring Committee 
Department of Mathematics, 
2074 East Hall 
University of Michigan 
525 E. University Avenue 
Ann Arbor, 
MI 48109-1109 

Source: Charlie Doering

Research Fellow in Dynamics/Solid Mechanics

University of Aberdeen

The Engineering and Physical Science Research Council has funded a multidisciplinary project on "Nonlinear Dynamics and Rock Contact Fracture Mechanics in Modelling of Vibration Enhanced Drilling". Duration of the post is 25 months commencing on 1 October 2003.

The project intends to develop and analyse new mathematical models of dynamic interactions between the drill-bit and the drilled formation occurring during rotary vibro-impact (percussive) downhole drilling. This gives rise to three principal objectives:

Applicants must have a strong background in experimental and theoretical dynamics and solid mechanics. The concept of percussive drilling and impacting oscillators will be explored. The experimental part of the project is crucial and therefore candidates should have a relevant experimental experience.

Applicants should have a PhD degree in applied mechanics or mechanical engineering. A degree in mathematics/ computer science would also be advantageous. Deadline for submitting application is 1 August 2003.

Informal enquiries should be directed to :

Prof. M. Wiercigroch 

Source: Marian Wiercigroch


5-year Lectureship in Mathematical Biology

Department of Mathematics and MOAC Doctoral Training Centre, University of Warwick

Applications are invited for a five-year lectureship to be held in the Department of Mathematics and the new EPSRC-funded Doctoral Training Centre MOAC (Molecular Organisation and Assembly in Cells). The lecturer is expected to contribute to the teaching and training activities of MOAC. The Lecturer will be expected to communicate strongly across the biology-mathematics interface and/or the chemistry-mathematics interface and contribute to the research programmes of MOAC, the recently created Interdisciplinary Programme for Cellular Regulation (IPCR) and the Mathematics Department. Candidates with related interests in Scientific Computing are also encouraged to apply.

Web sites:

Mathematics Department:  

Informal enquiries can be directed to Professor D A Rand, Chair of the Department of Mathematics or Dr Alison Roger the Director of MOAC.:

To receive a hard copy application pack, please contact the Personnel Office on 024 7652 3685 (24 hour answerphone) or by email to Or web site address: An application form MUST be completed if you wish to be considered for this post. Please note the hard copy and on-line application pack are the same.

Please quote reference 52/A/02. Closing date for applications is 1 October 2003.

Source: David Rand

Lecturer in Computational Applied Mathematics

School of Mathematical Sciences, University of Nottingham

Applications are invited for a Lectureship in Computational Applied Mathematics, to be held in the Division of Applied Mathematics at the University of Nottingham. The Unit of Assessment for Applied Mathematics was graded 5 in the 2001 Research Assessment Exercise and the successful candidate will be expected to contribute strongly to maintaining and enhancing our research record.

Candidates should have achieved research distinction or have strong research potential in a branch of Computational Applied Mathematics and should be committed to high quality teaching. Applications are encouraged from candidates whose research activity complements current activity in the Division, particularly in modelling applications such as fluid mechanics, materials, complex systems, and medicine and biology.

Salary will be within the range 22,191-33,679 pounds per annum, depending on qualifications and experience.

Informal enquiries may be addressed to:

 Professor O E Jensen, 
telephone: 0115 951 3866

Information about the Division of Applied Mathematics is available on the WWW at

Further details and application forms are available at 

Closing date: 12 September 2003

Source: Oliver Jensen

Tenure Track Position in Stochastic Analysis and PDE

Department of Mathematics, University of Wyoming

Applications are invited for an assistant professor tenure-track position starting August 2004. The minimum qualifications are an earned Ph.D., significant record of accomplishments in research, and evidence of a strong commitment to teaching. Candidates with research emphasis in stochastic analysis and partial differential equations related to fluid dynamics, electromagnetic and acoustic fields in random media, and geophysical phenomena are preferred. The position requires the ability and interest to supervise masters and doctoral students, to collaborate with colleagues in the math department and faculty in related disciplines, and to develop a competitive, externally funded, research program. Review of completed applications will begin December 15, 2003. A complete application will consist of a letter of application, a complete CV, a statement of research interests and accomplishments, and a statement of teaching philosophy. Please forward applications to: The Analysis Search Committee, Department of Mathematics, P.O. Box 3036, University of Wyoming, Laramie, WY 82071-3036. Please have three letters of recommendations, one of which should address the candidate's teaching, sent directly to the search committee. For further information please refer to: or contact S.S. Sritharan. UW is an EO/AA employer.

Source: S.S. Sritharan.

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