Dr. David A.W. Barton (david.barton at bristol.ac.uk) Great Western Research Fellow Department of Engineering Mathematics University of Bristol Queen's Building, University Walk Bristol, BS8 1TR, United Kingdom Tel: +44 (0)117 3315613 Fax: +44 (0)117 3315606 Office: 2.43

Research Interests

I have a range of research interests in the areas of applied mathematics:

• Mathematical modelling (in general, but particularly engineering related)
• Nonlinear systems and their dynamics (particularly nonlinear vibrations)
• Delay differential equations (DDEs) and their applications
• Non-smooth/piecewise-smooth differential equations and their applications
• Numerical methods for dynamical systems/bifurcation analysis, particularly numerical continuation
• Stochastic (noisy) systems

• Energy harvesting/power scavenging
• Micro-electromechanical devices (MEMS)

Previous Positions

• October 2006 to November 2007: Lloyds Tercentenary Foundation Fellow (
  details
  )
  • At the University of Bristol
  • Funded by the Lloyds Tercentenary Foundation
  • To investigate The Mathematical Challenges of Hybrid Test Engineering
• October 2003 to September 2006: PhD student (
  details
  )
  • At the University of Bristol
  • Supervised by Prof Bernd Krauskopf and Dr R Eddie Wilson
  • Thesis was on the Dynamics and bifurcations of non-smooth delay equations

PhD Students

My current PhD students are:

• Andrea Cammarano: co-supervised with Dr Steve Burrow on Energy harvesting: vibration powered generators with non-linear compliance
• Matthew Dawson: on Energy harvesting with multi-degree-of-freedom devices

Grants

• EP/H018107/1: Modelling and testing of nonlinear energy harvesters (PI)

Publications

Journal papers:

1. On the global dynamics of chatter in the orthogonal cutting model
   Zoltán Dombóvári, David A.W. Barton, R. Eddie Wilson and Gabor Stépán
   Submitted (
   abstract
   )

   Abstract: The large-amplitude motions of a one degree-of-freedom model of orthogonal cutting are analysed. The model takes the form of a delay differential equation which is non-smooth at the instant at which the tool loses contact with the workpiece, and which is coupled to an algebraic equation that stores the profile of the cut surface whilst the tool is not in contact. This system is approximated by a smooth delay differential equation without algebraic effects which is analysed with numerical continuation software. The grazing bifurcation that defines the onset of chattering motion is thus analysed as are secondary (period-doubling etc.) bifurcations of chattering orbits, and convergence of the bifurcation diagrams is established in the vanishing limit of the smoothing parameters. The bifurcation diagrams of the smoothed system are then compared with initial value simulations of the full non-smooth delay differential algebraic equation. These simulations mostly validate the smoothing technique and show in detail how chaotic chattering dynamics emerge from the non-smooth bifurcations of periodic orbits.

2. Tuning a resonant energy harvester using a generalised electrical load
   Andrea Cammarano, Stephen G. Burrow, David A.W. Barton, Alex Carrella and Lindsay R. Clare
   Submitted (
   abstract
   )

   Abstract: A fundamental drawback of vibration-based energy harvesters is that they typically feature a resonant mass/spring mechanical system to amplify the small source vibrations; the limited bandwidth of the mechanical amplifier restricts the effectiveness of the energy harvester considerably. By extending the range of input frequencies over which a vibration energy harvester can generate useful power, e.g. through adaptive tuning, it is not only possible to open up a wider range of applications, such as those where the source frequency changes over time, but it is also possible to relax the requirements for precision manufacture or the need for mechanical adjustment in~situ. In this paper, a vibration-based energy harvester connected to a generalised electrical load (containing both real and reactive impedance) is presented and it is demonstrated that reactive component of the electrical load can be used to electrically tune the harvester system to significantly increase output power away from the resonant peak of the device. An analytical model of the system is developed, which includes non-ideal components arising from the physical implementation, and the results are confirmed by experiment. The -3dB (half-power) bandwidth of the prototype energy harvester is shown to be over three times greater when presented with an optimised generalised load impedance compared to the same harvester presented with an optimised resistive-only load.

3. Numerical continuation in a physical experiment: investigation of a nonlinear energy harvester
   David A.W. Barton and Stephen G. Burrow
   Submitted (
   abstract
   )

   Abstract: In this paper we demonstrate the use of numerical continuation within a physical experiment: a nonlinear energy harvester, which is used to convert vibrational energy into usable electrical energy. To continue a branch of periodic orbits through a saddle-node bifurcation and along the associated branch of unstable periodic orbits, a modified time-delay controller is used. At each step in the continuation the pseudo-arclength equation is appended to a set of equations that ensure that the controller is non-invasive. The resulting nonlinear system is solved using a quasi-Newton iteration, where each evaluation of the nonlinear system requires changing the excitation parameters of the experiment and measuring the response. We present the continuation results for the energy harvester in a number of different configurations.

4. Energy harvesting from vibrations with a nonlinear oscillator
   David A.W. Barton, Stephen G. Burrow, and Lindsay R. Clare
   In press, ASME Journal of Vibrations and Acoustics (
   abstract
   )

   Abstract: In this paper we present a nonlinear electromagnetic energy harvesting device that has a broadly resonant response. The nonlinearity is generated by a particular arrangement of magnets in conjunction with an iron-cored stator. We show the resonant response of the system to both pure-tone excitation and narrow-band random excitation. In addition to the primary resonance, the super-harmonic resonances of the harvester are also investigated and we show that the corresponding mechanical up-conversion of the excitation frequency may be useful for energy harvesting. The harvester is modeled using a Duffing-type equation and the results compared to the experimental data.

5. Nonlinear dynamics of torsional waves in a drill-string model with spatial extent
   David A.W. Barton, Bernd Krauskopf, and R. Eddie Wilson
   In press, Journal of Vibration and Control (BCANM preprint,
   abstract
   )

   Abstract: In this paper we investigate the dynamics and bifurcations of an oil-well drill-string model that takes the form of a neutral delay differential equation (NDDE). We consider the torsional mode of the drill-string and investigate the associated stick-slip motion. To analyse the model we develop numerical continuation routines based on Fourier-methods since existing routines based on polynomial approximations are unable to cope with the presence of arbitrarily weakly damped modes. We find `resonance peaks' in the dynamics where a high-frequency mode is superimposed on the underlying periodic behaviour causing large torsional waves in the drill-string. We show that the resonance peaks are robust to small perturbations in the friction parameters but disappear if static friction forces are neglected completely

6. How to turn a genetic circuit into a synthetic tunable oscillator, or a bistable switch
   Lucia Marucci, David A.W. Barton, Irene Cantone, Maria Aurelia Ricci, Maria Pia Cosma, Stefania Santini, Diego di Bernardo, and Mario di Bernardo
   PLoS ONE 4(12) 2009 e8083 (full article,
   abstract
   )

   Abstract: Systems and Synthetic Biology use computational models of biological pathways in order to study in silico the behaviour of biological pathways. Mathematical models allow to verify biological hypotheses and to predict new possible dynamical behaviours. Here we use the tools of non-linear analysis to understand how to change the dynamics of the genes composing a novel synthetic network recently constructed in the yeast Saccharomyces Cerevisiae for In-vivo Reverse-engineering and Modelling Assessment (IRMA). Guided by previous theoretical results that link the dynamics of a biological network to its topological properties, through the use of simulation and continuation techniques, we found that the network can be easily turned into a robust and tunable synthetic oscillator, or a bistable switch. Our results provide guidelines to properly re-engineering in vivo natural and synthetic networks in order to tune their dynamics.

7. Transitions to spike-wave oscillations and epileptic dynamics in a human cortico-thalamic mean-field model
   Serafim Rodrigues, David A.W. Barton, Robert Szalai, Oscar J. Benjamin, Mark P. Richardson, John R. Terry
   Journal of Computational Neuroscience 27(3) 2009 pgs. 507-526 (full article, BCANM preprint,
   abstract
   )

   Abstract: In this paper we present a detailed theoretical analysis of the onset of spike-wave activity in a model of human EEG activity, relating this to clinical recordings from patients with absence seizures. We present a complete explanation of the transition from healthy activity to spike and wave using a combination of bifurcation theory, numerical continuation and techniques for detecting the occurrence of inflection points in systems of DDEs. We observe the the initial transition to oscillatory behaviour occurs as a result of a Hopf bifurcation, whereas the addition of spikes arises as a result of an inflection point of the vector field. Strikingly these findings are consistent with EEG data recorded from patients with absence seizures and we present a discussion of the clinical significance of these results, suggesting potential new techniques for detection and anticipation of seizures.

8. Equilibrium configurations for a territorial model
   Ronald Votel, David A.W. Barton, Takahide Gotou, Takeshi Hatanaka, Masayuki Fujita and Jeff Moehlis
   SIAM Journal on Applied Dynamical Systems 8(3) 2009 pgs. 1234-1260 (full article, BCANM preprint,
   abstract
   )

   Abstract: We consider a territorial model based on Voronoi tessellations. Such tessellations form a partitioning of a planar region by enclosing each agent in a polygon such that every point within the polygon is closest to that agent instead of any other. For rectangular domains and for small population sizes, we show that there can be distinct coexisting stable equilibrium configurations, including the possibility of stable equilibria that are not related by symmetry. By considering randomly distributed initial positions, we give a statistical characterization of the basins of attraction for these equilibria in the case of a square domain. Furthermore, we show that the final territory that an agent occupies can have a wide range of sizes, which suggests that an individual can obtain a competitive advantage or disadvantage due entirely to its initial position. Finally, by treating the ratio of the length of the shorter side to the length of the longer side of the rectangle as a bifurcation parameter, we numerically explore how stable and unstable equilibrium configurations are related to each other.

9. Stability calculations for piecewise-smooth delay equations
   David A.W. Barton
   International Journal of Bifurcation and Chaos 19 (2) 2009 pgs. 639-650 (full article, BCANM preprint,
   abstract
   )

   Abstract: This paper describes a new method for computing the stability of nonsmooth periodic orbits of piecewise-smooth dynamical systems with delay. Stability computations for piecewise-smooth dynamical systems without delay have previously been performed using discontinuity mappings to `correct' the linearized period map. However, this approach is less convenient for systems with delays due to the infinite dimensional nature of the problem. Additional problems arise due to the discontinuity propagation properties of delay differential equations. The method proposed is based around a multi-point boundary value solver, which allows the correct linearized period map to be constructed directly. We present numerical examples showing the rapid convergence of the method and also illustrate its use as part of a numerical bifurcation study.

10. Criticality of Hopf bifurcation in state-dependent delay model of turning processes
    Tamás Insperger, David A.W. Barton, and Gabor Stépán
    International Journal of Non-linear Mechanics 43(2) 2008 pgs. 140-149 (full article, BCANM preprint,
    abstract
    )

    Abstract: In this paper the nonlinear dynamics of a state-dependent delay model of the turning process is analyzed. The size of the regenerative delay is determined not only by the rotation of the workpiece, but also by the vibrations of the tool. A numerical continuation technique is developed that can be used to follow the periodic orbits of a system with implicitly defined state-dependent delays. The numerical analysis of the model reveals that the criticality of the Hopf bifurcation depends on the feed rate. This is in contrast to simpler constant delay models where the criticality does not change. For small feed rates, subcritical Hopf bifurcations are found, similar to the constant delay models. In this case, periodic orbits coexist with the stable stationary cutting state and so there is the potential for large amplitude chatter and bistability. For large feed rates, the Hopf bifurcation becomes supercritical for a range of spindle speeds. In this case, stable periodic orbits instead coexist with the unstable stationary cutting state, removing the possibility of large amplitude chatter. Thus, the state-dependent delay in the model has a kind of stabilizing effect, since the supercritical case is more favorable from a practical viewpoint than the subcritical one.

11. Homoclinic bifurcations in a neutral delay model of a transmission line oscillator
    David A.W. Barton, Bernd Krauskopf, and R. Eddie Wilson
    Nonlinearity 20(4) 2007 pgs. 809-829 (full article, BCANM preprint,
    abstract
    )
    Selected as one of Nonlinearity's 'High-Profile Articles' of 2007!

    Abstract: In a transmission line oscillator (TLO) a linear wave travels along a piece of cable, the transmission line, and interacts with terminating electrical components. A fixed time delay arises due to the transmission time through the transmission line. Recent experiments on a TLO driven by a negative resistor demonstrated rich delay-induced dynamics and high-frequency chaotic behaviour. Furthermore, good agreement was found with a neutral delay differential equation (NDDE) model.

    In this paper we perform a numerical bifurcation analysis of the NDDE model of the TLO. Our main focus is on homoclinic orbits, which give rise to complicated dynamics and bifurcations. For small time delay there is a homoclinic orbit to a steady-state. However, past a codimension-two Shil'nikov-Hopf bifurcation the homoclinic orbit connects to a saddle-type periodic solution, which exists in a region bounded by homoclinic tangencies. Both types of homoclinic bifurcations are associated with accumulating branches of periodic solutions. We summarise our results in a two-parameter bifurcation diagram in the plane of resistance against time delay.

    Our study demonstrates that the theory of homoclinic bifurcations in ordinary differential equations largely carries over to NDDEs. However, we find that the neutral delay nature of the problem influences some bifurcations, especially convergence rates of folds associated with homoclinic tangencies.

12. Collocation schemes for periodic solutions of neutral delay differential equations
    David A.W. Barton, Bernd Krauskopf, and R. Eddie Wilson
    Journal of Difference Equations and Applications 12(11) 2006 pgs. 1087-1101 (full article, BCANM preprint,
    abstract
    )

    Abstract: We introduce two collocation schemes for the computation of periodic solutions of neutral delay differential equations (NDDEs): one based on a direct discretisation of the underlying NDDE, and one based on a discretisation of a related delay differential difference equation (i.e., a delay differential equation coupled with a difference equation). Numerical examples are used to demonstrate these schemes and their respective orders of convergence. Both collocation schemes are implemented in DDE-BIFTOOL, a numerical continuation tool for delay equations. Their use in a continuation setting is shown with one- and two-parameter bifurcation studies of a transmission line model.

13. Periodic solutions and their bifurcations in a non-smooth second-order delay differential equation
    David A.W. Barton, Bernd Krauskopf, and R. Eddie Wilson
    Dynamical Systems 21(3) 2006 pgs. 289-311 (full article, BCANM preprint,
    abstract
    )

    Abstract: We consider a non-smooth second order delay differential equation (DDE) that was previously studied as a model of the pupil light reflex. It can also be viewed as a prototype model for a system operated under delayed relay control.

    We use the explicit construction of solutions of the non-smooth DDE hand-in-hand with a numerical continuation study of a related smoothed system. This allows us to produce a comprehensive global picture of the dynamics and bifurcations, which extends and completes previous results. Specifically, we find a rich combinatorial structure consisting of solution branches connected at resonance points. All new solutions of the smoothed system were subsequently constructed as solutions of the non-smooth system. Furthermore, we show an example of the unfolding in the smoothed system of a non-smooth bifurcation point, from which infinitely many solution branches emanate. This shows that smoothing of the DDE may provide insight even into bifurcations that can only occur in non-smooth systems.

14. Explicit periodic solutions in a model of a relay controller with delay and forcing
    David A.W. Barton, Bernd Krauskopf, and R. Eddie Wilson
    Nonlinearity 18(6) 2005 pgs. 2637-2656 (full article, BCANM preprint,
    abstract
    )

    Abstract: In this paper we use a combination of numerical and analytical methods to find and construct solutions of a cameo model of relay control, formulated as a piecewise-constant delay differential equation (DDE). Numerical solutions of a related equation, where the discontinuities of the original DDE are smoothed out, are used to guide the construction of explicit solutions of the original DDE. On the other hand, the construction of explicit solutions provides starting data for numerical continuation of the smoothed equation. The stability of the explicit solutions can also be inferred from the numerical approach.

Refereed conference proceedings papers:

1. How to turn a genetic circuit into a synthetic tunable oscillator, or a bistable switch
   Lucia Marucci, David A.W. Barton, Irene Cantone, Maria Aurelia Ricci, Maria Pia Cosma, Stefania Santini, Diego di Bernardo, and Mario di Bernardo
   Accepted for RECOMB Systems Biology 2009, MIT, Boston, 2nd-6th December 2009
2. Numerical continuation in a physical experiment: investigation of a nonlinear energy harvester
   David A.W. Barton, Stephen G. Burrow
   Proceedings of ASME 2009 International Design Engineering Technical Conferences, San Diego, USA, 30th August-2nd September 2009
3. Energy harvesting from vibrations with a nonlinear oscillator
   David A.W. Barton, Stephen G. Burrow, and Lindsay R. Clare
   Proceedings of ASME 2009 International Design Engineering Technical Conferences, San Diego, USA, 30th August-2nd September 2009
4. Supercritical bifurcations in the state-dependent delay model of turning process
   Tamás Insperger, David A.W. Barton, and Gabor Stépán
   Proceedings of the International Congress of Theoretical and Applied Mechanics (ICTAM2008), Adelaide, Australia, 24-29th August 2008
5. Vibration energy harvesters with non-linear compliance
   Stephen G. Burrow, Lindsay R. Clare, Alex Carrella and David A.W. Barton
   Proceedings of SPIE vol. 6928, 692807, Apr. 4, 2008
6. Bifurcation analysis tools for neutral delay equations: a case study
   David A.W. Barton, Bernd Krauskopf, and R. Eddie Wilson
   Proceedings of the 6th IFAC conference on Time-Delay Systems 2006, L'Aquila, Italy, 10-12th July 2006 (BCANM preprint)
7. Analytical construction techniques for some solutions of forced piecewise constant delay equations
   David A.W. Barton and R. Eddie Wilson
   Proceedings of the 5th IFAC conference on Time-Delay Systems 2004, Leuven, Belgium, 8-10th September 2004 (BCANM preprint)

Conferences, Workshops & Invited Talks

• 1st February 2010 invited talk at the Dynamics Interest Group, University of Illinois at Urbana-Champaign, IL, USA
  Talk title: Control-based continuation for investigating nonlinear experiments
• 1st December 2009 invited talk at the Center for Nonlinear and Complex Systems, Duke University, NC, USA
  Talk title: Numerical continuation for nonlinear systems: from model to experiment
• 12-16th October 2009 State-Dependent Delay Equations at the Max Planck Institute for Physics of Complex Systems, Dresden, Germany
  Talk title: State-dependent delay equations arising in engineering applications
• 30th August-2nd September 2009 ASME 2009 International Design Engineering Technical Conferences, San Diego, USA
  Talk title #1: Numerical continuation in a physical experiment: investigation of a nonlinear energy harvester
  Talk title #2: Energy harvesting from vibrations with a nonlinear oscillator
• 7-9th April 2009 British Applied Mathematics Colloquium 2009 at the University of Nottingham, UK
  Talk title: Hybrid experimental/numerical investigation of a nonlinear oscillator
• 30th March - 3rd April 2009 68th European Study Group with Industry at the University of Southampton, UK
  Talk/report title: Modelling and simulation of drill-string dynamics
• 30th June - 4th July 2008 8th World Congress on Computational Mechanics/5th European Conference on Computational Methods in Applied Sciences and Engineering, Venice, Italy
  Talk title: Numerical continuation methods for models with nonsmooth dynamics (and delay)
• 1st April 2008 British Applied Mathematics Colloquium 2008 at the University of Manchester, UK
  Talk title: Nonlinear energy harvesting from vibrations
• 7th December 2007 invited talk at the University of Surrey, UK
  Talk title: Delay and nonsmoothness in machine tool models
• 13th November 2007 invited talk at the University of Bath, UK
  Talk title: Different flavours of delay differential equations and their (engineering) applications
• 4-7th September 2007 postgraduate workshop on Computational Methods for Dynamical Systems in Bristol, UK
  Co-organiser of the workshop
  Talk title: Dynamical systems with delay
• 16-20th July 2007 ICIAM in Zurich, Switzerland
  Organiser of mini-symposium: Dynamics of cutting processes
  Talk title: Numerical tools for understanding chattering behaviour in non-smooth DDE models
• 2nd-7th July 2007 Advanced Algorithms and Numerical Software for the Bifurcation Analysis of Dynamical Systems in Montreal, Canada
  Talk title: Numerical continuation for delay equations - beyond DDE-BIFTOOL
• 28th May-1st June 2007 SIAM Dynamical Systems (Snowbird) in Snowbird, Utah, USA
  Organiser of mini-symposium: Delay effects in engineering and nature
  Talk title: Numerical continuation for neutral delay systems: application to an electronic oscillator
  Poster title: Global dynamics of a non-smooth model of metal cutting
• 17th April 2007 British Applied Mathematics Colloquium 2008 at the University of Bristol, UK
  Talk title: Numerics for non-smoothness: understanding machine tool chatter
• 5-16th February 2007 CRM workshop on Complex Non-Smooth Dynamical Systems in Barcelona, Spain
  Talk title: Numerical continuation for piecewise-smooth delay differential equations
• 21st November 2006 invited talk at the University of Bath, UK
  Talk title: Numerical methods for nonlinear systems with delay: a case study
• 10th-12th July 2006 6th IFAC conference on Time-Delay Systems in L'Aquila, Italy
  Talk title: Bifurcation analysis tools for neutral delay equations: a case study
• 22nd-23rd June 2006 Advanced numerical methods for mathematical modelling at Ghent University, Belgium
  Talk title: Continuation for neutral delay differential equations - a case study of a transmission line oscillator
• 9th May 2006 invited talk at the University of Exeter, UK
  Talk title: Delay induced dynamics of a nonlinear transmission line oscillator model
• 24th-27th April 2006 British Applied Mathematics Colloquium 2006 at Keele University, UK
  Talk title: Homoclinic orbits in a neutral delay model of a transmission line oscillator
• 3rd-7th April 2006 56th European Study Group with Industry at the University of Bath, UK
  Talk/report title: Shimmy in aircraft landing gears
• 4th-9th September 2005 Mathematics for Biomedical Engineering at the University of Warwick, UK
  Talk title: Compartmental modelling of the gut
• 22nd-26th May 2005 SIAM conference on Applications of Dynamical Systems in Snowbird, Utah, USA
  Talk title: Explicit periodic solutions in a model of a relay controller with delay and forcing
• 4th-7th April 2005 Mathematics 2005 at the University of Liverpool, UK
  Talk title: Explicit periodic solutions in a model of a relay controller with delay and forcing
• 8th-10th September 2004 5th IFAC conference on Time-Delay Systems at K.U. Leuven, Belgium
  Talk title: Analytical construction techniques for some solutions of forced piecewise constant delay equations
• 13th-22nd August 2004 ECMI Modelling week at Lappeenranta University of Technology, Finland
  Talk/report title: Modelling the crystallisation of polymers
• 10th-11th November 2003 SICONOS meeting at the Technical University of Catalonia, Barcelona, Spain
  Poster title: Periodic solutions of piecewise linear DDEs