Category Archives: Research

Postdoc position now available

3 year postdoc position available for 1 June 2017 start!

We seek a highly motivated Research Associate who is interested in working as part of a team at the interface between Engineering and Applied Mathematics to investigate new methods for exploring the nonlinear behaviour of engineered systems and to develop numerical continuation techniques for physical experiments.

Modern test methods for investigating the dynamics of engineered structures are inadequate for dealing with the presence of significant nonlinearity since they have largely been developed under the assumption of linear behaviour. In contrast, control-based continuation (CBC), a versatile non-parametric identification method, has been developed with nonlinearity in mind from the beginning. It has been demonstrated on simple experiments but now advances in underlying methodology are required to apply CBC to real-world experiments which have higher levels of measurement noise and many degrees of freedom. The versatility of CBC is such that, with these advances, it will also become relevant for researchers studying nonlinear systems in both engineering and other fields, such as in the biological sciences.

We are seeking a Research Associate to drive this research forward alongside researchers working on closely related problems from the Departments of Engineering Mathematics, Mechanical Engineering and Aerospace Engineering. Support will be readily available from the investigators David Barton, Simon Neild and Djamel Rezgui. More widely, you will be part of the Dynamics and Control research group and the Applied Nonlinear Mathematics research group both of which carry out cutting-edge research in a wide range of application areas.

CBC presently draws on a wide range of underlying areas including, but not limited to, dynamical systems and bifurcation theory, system identification, control theory and machine learning. Applicants are expected to have experience in at least one of these areas in addition to a first degree and preferably a PhD in Applied Mathematics/Physics/Engineering (or a closely related discipline).

Possible initial avenues of research include

  • Improving the robustness of CBC in the presence of noise using surrogate models. Gaussian processes have previously been investigated and may be useful.
  • Investigating the scaling up of CBC to many degree-of-freedom systems. Ideas from numerical continuation of PDE systems could yield insights.
  • Implementation of CBC on existing aerospace experiments for dynamic testing and wind tunnel testing.

The post is available from 1 June 2017 with funding available for up to three years.

To apply visit

Please direct any questions to David Barton,

Studentships available for immediate start

We have some EPSRC funded studentships available for immediate start (competitively awarded) in my department/school. The studentships pay an annual stipend of ~£14k plus tuition fees, though you must meet EPSRC eligibility requirements (there are no awards available for overseas students).

The deadline is very tight though, you need to have an application completed for 10 October.

Applications coming in after this time will be considered for a September 2017 start; but please do still get in touch.

If you are interested in nonlinear dynamics (from a numerical/computational/experimental viewpoint), please get in touch.  I’m particularly interested in stochastic dynamics and/or links with machine learning.

Email me at with your current CV and a short description of your (academic) interests. General enquires along the lines of “I don’t really know what I want to do” are also welcome.

Nonlinear vibrations, localisation and energy transfer

I recently had the privilege of being invited to the 6th conference on Nonlinear Vibrations, Localization and Energy Transfer in Liege, 4-8 July 2016. The conference was superbly organised by Gaetan Kerschen and Jean-Philippe Noël in remarkable surroundings and with very good food (and beer — a trip to the Val-Dieu abbey to taste good Belgian beer was a highlight of the social part of the conference!).

The conference was a good mix of cutting-edge research combined with tutorial lectures for PhD students (and above!) on a range of topics. The lectures on nonlinear system identification were particularly useful for me — it’s an area that I’m keen to learn more about and import into my own work.

I had the pleasure of presenting on “Control-based Continuation – A new experimental approach for testing nonlinear dynamic systems” (with co-authors Ludovic Renson and Simon Neild); hopefully I was able to impart some of the enthusiasm we have for working on nonlinear experiments and understanding their nonlinear dynamics and bifurcations via numerical continuation!

Other talks of particular interest were by Simon Peter (Stuttgart) and Peter Bruns (Hannover) on using phase-locked loops to extract backbone (nonlinear normal modes) from experiments. Their approaches are highly complementary to my own. Cyril Touzé also gave a fascinating talk on turbulence in metallic plates, along with convincing experiments that show energy transfers from low-frequency modes to higher-frequency modes of vibration. I was also pleased to see that software developments are slowly propagating through into applied communities; Bruno Cochelin presented an interesting take on using Automatic Differentiation for high-order numerical continuation methods, I’m curious as to how far this can be applied.

Overall a thoroughly productive week!

PhD opportunity in Industrial Mathematics

I am seeking to recruit a student who is keen to work on industrial applications of nonlinear dynamics. They are expected to have a good degree (first or upper second) in a related subject, for example, applied mathematics, physics or engineering. The project is centred around investigating the nonlinear dynamics of physical experiments; in this case, the dynamics and bifurcations of rotating machinery. The project will combine ideas from nonlinear dynamics, control theory and numerical analysis to extract the required information from the physical experiment in real-time. Experience in one (or more) of these areas is ideal but not necessary as training will be given.

For start of this project, a specific engineering experiment has been constructed (a rotor rig) but the methodology being developed is applicable to a wide range of areas, from engineering to biological systems. Any controllable experiment that exhibits nonlinear behaviour could potentially benefit.

This project is funded by an EPSRC iCASE award meaning that the studentship pays the standard EPSRC rate of £14,296 plus an industrial top-up of around £5,000 (to be confirmed). The studentship is open to any UK/EU student who has been normally resident in the UK for 3 years (see the EPSRC eligibility requirements at ).

For more information please send a short email to Dr David Barton outlining your academic background (i.e., current/most recent degree).

PhD studentship opportunities

Breaking news: PhD studentships (scholarships) available for UK and EU* students in the Department of Engineering Mathematics at the University of Bristol for immediate start.

Due to a number of students withdrawing for personal reasons, there are now several PhD studentships available that must be taken up by September 2016. I am looking for motivated and able  students who are interested in doing research at the intersection of Applied Mathematics and Engineering/Science (my interests are quite broad!). In particular, I have three different topics that I’m actively pursuing at the moment all of which feature numerical computation/numerical analysis:

  • Nonlinear dynamics of stochastic differential equations — I’m interested in investigating how the tools and concepts of nonlinear dynamics and bifurcation theory can be applied to stochastic differential equations arising from various application areas (e.g., neuroscience or climate science).
  • Control-based continuation — numerical continuation is a very effective tool for investigating the nonlinear behaviour and bifurcations of mathematical models, and control-based continuation is a means for applying this tool to physical experiments (engineered systems or, hopefully, biological systems) without the need for a mathematical model. Research in this area requires a very interesting mix of numerical analysis, control theory, system identification and the theory of stochastic processes (I don’t expect students to have a background in all of these subjects!).
  • Equation-free methods and agent-based modelling — equation-free methods are a means for obtaining a macroscopic model from microscopic simulations. They have been used for many physical processes previously and I am interested in how they can be extended to agent-based models, such as models of Zebrafish locomotion to investigate the dynamics of shoaling (Zebrafish are just one example).

I have also worked extensively with delay differential equations and, though I don’t have any active work in this area at the moment, I’m happy to solicit project suggestions in this area. All of these projects are very open ended and I’m happy to work with students to tailor the projects to their own interests.

There is no deadline for these studentships, though obviously it’s better to apply sooner rather than later. If you are interested, get in touch for more information.

* EU students are eligible provided they have been resident in the UK for at least 3 years. See the EPSRC website for more details.

Data-driven stochastic modelling of Zebrafish locomotion

Ever wondered how you can write down equations that govern the behaviour of a moving creature? Let’s start simple and try something like a fish. (OK, a fish isn’t that simple…) It turns out that Zebrafish are quite a convenient animal to study and one of our collaborators (Maurizio Porfiri from New York University, USA) has great facilities for the fish to happily swim around in while being tracked by video cameras.

With the data from Maurizio’s lab, we have constructed a mathematical model of Zebrafish locomotion based on stochastic differential equations (SDEs), that is equations that govern the rates of change of speed and of turning speed (the differential bit) and are driven by random noise (the stochastic bit). The randomness is there to provide a mathematical description of all the seemingly erratic behaviours that the Zebrafish exhibit which we don’t really understand — we can match the statistics of the randomness to the statistics of the erratic behaviour but we can’t say anything about the higher level thought (if that is an appropriate description for a fish!) that is generating it.

The SDEs are relatively simple (if you are used to such things) and take the form
$$ \begin{gathered}
\mathrm{d}U = -\theta_u(U – \mu_u – U^*)\mathrm{d}t + \sigma_u\mathrm{d}W_1(t) \\
\mathrm{d}\Omega = -\theta_\omega(\Omega – \Omega^*)\mathrm{d}t + \sigma_\omega\mathrm{d}W_2(t)
\end{gathered} $$
where $U$ is the speed and $\Omega$ is the turning speed. This is an Ito stochastic differential equation (though perhaps it should be Stratonovich…). This is in effect a mean-reverting process or Ornstein-Uhlenbeck process. In short, the speed is tracking the value $\mu_u + U^*$ with a time scale $\theta_u$ where $\mu_u$ is the natural speed of the Zebrafish and $U^*$ is a modification to the natural speed due to external factors (e.g., getting close to a wall or other Zebrafish). The same description holds for the turning speed equation (except that the natural turning speed is zero and so not included). The terms $U^*$, $\Omega^*$ and $\sigma_u$ are not actually constants but functions of the fish position relative to other fish and any obstacles.

Take a read of the article and see what you think! (The article is open access to all.)

If you like this, a follow on post will provide a nice graphical interface for playing with these equations (written in Python).

Reference: Data-driven stochastic modelling of zebrafish locomotion Adam Zienkiewicz, David A.W. Barton, Maurizio Porfiri and Mario di Bernardo, Journal of Mathematical Biology 71(5) 2015 pp. 1081–1105. DOI: 10.1007/s00285-014-0843-2 (open access).

Systematic experimental exploration of bifurcations with noninvasive control

One of the things I’ll be doing on this new website is highlighting some of my papers, past and present. This paper is one I’m particularly pleased with as it showcases what you can do when you link numerical methods (computer algorithms) with a physical experiment in real-time.

Normally to investigate the behaviour of a system or structure (e.g., an aerofoil) you would build a mathematical model of the system, then validate it (hopefully! lots of people skip this step) and then investigate it using some of the sophisticated mathematical tools we have for analysis. This paper looks at how we can throw out the model building step and use those same sophisticated mathematical tools directly on the original physical experiment. OK, there are a few constraints (we must have quite a bit of control over the experiment) but it does mean that we can be sure that the behaviour we see is real and not just because the model is wrong.

This paper looks at a very simple physical example, namely an energy harvester but it demonstrates the principle nicely.

Reference: Systematic experimental exploration of bifurcations with noninvasive control, David A.W. Barton and Jan Sieber, Physical Review E 87 2013, 052916. DOI: 10.1103/PhysRevE.87.052916. Preprint (open access): arXiv 1209.3713.