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University of Bristol
Engineering Mathematics
BCAMN
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
Submitted
(BCANM preprint)
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.
Stability calculations for piecewise-smooth delay equations
David A.W. Barton
Accepted for publication in the International Journal of Bifurcation and Chaos
(BCANM preprint)
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.
Equilibrium configurations for an equilibrium model
Ronald Votel, David A.W. Barton, and Jeff Moehlis
Submitted
(BCANM preprint)
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.
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
(International
Journal of Non-linear Mechanics, BCANM preprint)
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.
Nonlinear dynamics of torsional waves in a drill-string model with spatial extent
David A.W. Barton, Bernd Krauskopf, and R. Eddie Wilson
Accepted for publication in the Journal of Vibration and Control
(BCANM preprint)
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
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
(Nonlinearity,
BCANM preprint)
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.
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
(Journal of Difference Equations and
Applications, BCANM preprint)
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.
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
(Dynamical Systems, BCANM preprint)
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.
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
(Nonlinearity, BCANM preprint)
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.