This is a list of my journal and conference publications with download links as appropriate. Please note that access to the download articles may be restricted depending on where you are connecting from (and whether you have a subscription to the journal or not). If you have any problems getting any of these, just drop me an email.

1. A pseudospectral method for free-surface flow with varying bathymetry. S.G. Parkinson, A.R. Champneys, D.A.W. Barton, A.J. Hogg, M. Thomson and S. J. Hogan.
   • Abstract The primary interest of this study is to recreate the basic physics of the accelerational and decelerational effects in subcritical free-surface flow caused by bathymetry variation. A pseudospectral matrix method is applied to the quasi-steady, turbulent, two-dimensional shallow water equations using a Chebyshev polynomial basis set. Turbulence closure is achieved using an eddy-viscosity model. Upon application of the pseudospectral method the equations reduce to a system of coupled quasi-linear ordinary differential equations which may be solved as an initial-value problem in the streamwise direction given some arbitrary bathymetry zb (x). The model has only a few tunable parameters and the solution method is computationally efficient such that flow details may be resolved quickly and accurately. This has potential application for the optimal siting of tidal energy converters.
     
     Convergence of the method is tested by varying the number of modes and the mixing parameter in the eddy-viscosity model. In addition, close agreement is found with simulations using oceanographic computational fluid dynamics and with previously published experimental data. For sufficiently shallow bathymetry gradients the model performs well. The validity of the turbulence closure model is discussed in light of integration failure during simulations where steep gradients occur, especially in the lee of a ridge.
2. Uncertainty in performance for linear and nonlinear energy harvesting strategies. B.P. Mann, D.A.W. Barton and B.A.M. Owens. Accepted by Journal of Intelligent Material Systems and Structures.
   • Abstract Vibrational energy harvesters are often linear mass-spring-damper-type devices which have their resonant frequency tuned to the dominant vibration frequency of their host environment. As such, they can be highly sensitive to uncertainties which may arise from the imprecise characterization of the host environment or, alternatively, from manufacturing defects and tolerances. It has previously been claimed that the use of nonlinear energy harvesters may be one way to alleviate the problems of these uncertainties.
     
     This paper presents a systematic uncertainty propagation study of a proto-typical electromagnetic energy harvester. More specifically, the response of a linear harvester in the presence of parametric uncertainty is compared to the response of harvesters containing some common forms of nonlinearity, i.e. hardening, softening, or bistability. Analytical solutions are used in combination with presumed levels of parametric uncertainty to quantify the resulting uncertainty in the power output. Consequently, these studies can determine the regions in the parameter space where a nonlinear strategy may outperform a more traditional linear approach.
3. Control-based continuation for investigating nonlinear experiments. David A.W. Barton, Brian P. Mann and Stephen G. Burrow. Accepted by Journal of Vibration and Control.
   • Abstract This paper presents a systematic experimental study of two one-degree-of-freedom nonlinear devices using the newly introduced control-based continuation method by Sieber and Krauskopf [Nonlinear Dynamics 2008]. By considering hardening, softening and bistable spring characteristics we demonstrate the versitility and power of the control-based continuation method for investigating nonlinear experiments. We show that, using this method, it is possible to track the stable orbits of the devices through a saddle-node bifurcation (fold) where they lose stability and continue them up to the resonance peak where they undergo a second saddle-node bifurcation. For the bistable case, a bifurcation diagram is produced that is strongly reminiscent of the bifurcation diagram produced using the classical harmonic balance solution. A detailed introduction to general continuation methods is included to enable implementation by other experimentalists.

1. Periodic solutions of nonlinear delay differential equations using spectral element method. Firas A. Khasawneh, David A.W. Barton and Brian P. Mann. Nonlinear Dynamics 67(1), 641-658, 2012.
   • Abstract We extend the temporal spectral element method further to study the periodic orbits of general autonomous nonlinear delay differential equations (DDEs) with one constant delay. Although we describe the approach for one delay to keep the presentation clear, the extension to multiple delays is straightforward. We also show the underlying similarities between this method and the method of collocation. The spectral element method that we present here can be used to find both the periodic orbit and its stability. This is demonstrated with a variety of different examples, namely, the delayed versions of Mackey-Glass equation, Van der Pol equation, and Duffing equation. For each example, we show the method's convergence behavior using both p and h refinement and we provide comparisons between equal size meshes that have different distributions.
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2. Modelling and experimental characterization of an energy harvester with bistable compliance characteristics. Andrea Cammarano, Stephen G. Burrow and David A.W. Barton. Journal of Systems and Control Engineering 225, 475-484, 2011.
   • Abstract This paper presents a novel design for a vibrational energy harvester. The design uses high permeability magnetic materials which brings about two key advantages. Firstly, it gives strong coupling between the mechanical and electrical domains, thus enabling effective energy conversion. Secondly, it gives the device a bi-stable compliance characteristic, which gives the harvester a broad-band frequency response. An explicit analytical model is developed using a combination of experimental data and finite element modelling in order to accurately incorporate the magnetic forces. The model is then validated using dynamic tests of the experimental rig. The main features of the dynamic response of the bi-stable oscillator are highlighted and benefits discussed in the context of energy harvesting. Finally, comments are made on the relationship between the complicated behaviour resulting from the bi-stable compliance characteristic and the benefits of increased electrical coupling.
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3. On the global dynamics of chatter in the orthogonal cutting model. Zoltan Dombovari, David A.W. Barton, R. Eddie Wilson and Gabor Stepan. International Journal of Non-linear Mechanics 46(1), 330-338, 2011.
   • 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.
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4. Numerical continuation in a physical experiment: investigation of a nonlinear energy harvester. David A.W. Barton and Stephen G. Burrow. ASME Journal of Computational and Nonlinear Dynamics 6(1), 011010, 2011.
   • 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.
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5. A method for detecting false bifurcations in dynamical systems: application to neural-field models. Serafim Rodrigues, David A.W. Barton, Frank Marten, Moses Kibuuka, Gonzalo Alarcon, Mark P. Richardson and John R. Terry. Biological Cybernetics 102(2), 145-154, 2010.
   • Abstract In this article, we present a method for tracking changes in curvature of limit cycle solutions that arise due to inflection points. In keeping with previous literature, we term these changes false bifurcations, as they appear to be bifurcations when considering a Poincaré section that is tangent to the solution, but in actual fact the deformation of the solution occurs smoothly as a parameter is varied. These types of solutions arise commonly in electroencephalogram models of absence seizures and correspond to the formation of spikes in these models. Tracking these transitions in parameter space allows regions to be defined corresponding to different types of spike and wave dynamics, that may be of use in clinical neuroscience as a means to classify different subtypes of the more general syndrome.
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6. Nonlinear dynamics of torsional waves in a drill-string model with spatial extent. David A.W. Barton, Bernd Krauskopf and R. Eddie Wilson. Journal of Vibration and Control 16(7-8), 1049-1065, 2010.
   • 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.
   • Full text, BCANM preprint
7. 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. Smart Materials and Structures 19(5), 055003, 2010.
   • 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.
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8. Energy harvesting from vibrations with a nonlinear oscillator. David A.W. Barton, Stephen G. Burrow and Lindsay R. Clare. ASME Journal of Vibrations and Acoustics 132(2), 021009, 2010.
   • 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.
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9. 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), e8083, 2009.
   • 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.
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10. 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 and John R. Terry. Journal of Computational Neuroscience 27(3), 507-526, 2009.
    • 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.
    • Full text, BCANM preprint
11. 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), 1234-1260, 2009.
    • 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.
    • Full text, BCANM preprint
12. Stability calculations for piecewise-smooth delay equations. David A.W. Barton. International Journal of Bifurcation and Chaos 19 (2), 639-650, 2009.
    • 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.
    • Full text, BCANM preprint
13. 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), 140-149, 2008.
    • 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.
    • Full text, BCANM preprint
14. 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), 809-829, 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.
    • Full text, BCANM preprint
    • Selected as one of Nonlinearity's 'High-Profile Articles' of 2007!
15. 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), 1087-1101, 2006.
    • 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.
    • Full text, BCANM preprint
16. 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), 289-311, 2006.
    • 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.
    • Full text, BCANM preprint
17. 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), 2637-2656, 2005.
    • 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.
    • Full text, BCANM preprint

1. Experimental evaluation of vibration energy harvesting with non-linear oscillators. Andrea Cammarano, Stephen G. Burrow and David A.W. Barton. Proceedings of ISMA 2010 International Conference on Noise and Vibration Engineering, Leuven, Belgium, 20th-22nd September 2010.
2. Vibrational energy harvesting using a multi-degree-of-freedom device. Matthew D. Dawson and David A.W. Barton. Proceedings of ISMA 2010 International Conference on Noise and Vibration Engineering, Leuven, Belgium, 20th-22nd September 2010.
3. An energy harvester with bistable compliance characteristics. Andrea Cammarano, Stephen G. Burrow and David A.W. Barton. Proceedings of ASME 2010 International Design Engineering Technical Conferences, Montreal, Canada, 15-18th August 2010.
4. The influence of power harvesting circuits on energy harvesting performance. D. Dane Quinn, Stephen G. Burrow and David A.W. Barton. Proceedings of Active and Passive Smart Structures and Integrated Systems 2010, San Diego, CA, USA, 7-11th March 2010.
5. 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. Proceedings of RECOMB Systems Biology 2009, MIT, Boston, USA, 2nd-6th December 2009.
6. 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.
7. 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.
8. 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.
9. 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.
10. 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.
11. 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.