Paul Glendinning

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It is well-known that the dynamics of the Arnol ′ d circle map is phase-locked in regions of the parameter space called Arnol ′ d tongues. If the map is invertible, the only possible dynamics is either quasiperiodic motion, or phase-locked behavior with a unique attracting periodic orbit. Under the influence of quasiperiodic forcing the dynamics of the map(More)
We investigate the dynamic mechanisms underlying intermittent state transitions in a recently proposed neural mass model of epilepsy. A low dimensional model is constructed, which preserves two key features of the neural mass model, namely (i) coupling between oscillators and (ii) heterogeneous proximity of these oscillators to a bifurcation between(More)
Homoclinic orbits to bifocus-type stationary points have been studied theoretically by a number of authors, but up until now, only one analytic example has been found. In this paper we summarise and extend the known theory regarding bifocal homoclinic bifurcations and present numerical verification of some of the more interesting theoretical predictions(More)
(2012). Grazing-sliding bifurcations, the border collision normal form, and the curse of dimensionality for nonsmooth bifurcation theory. Nonlinearity. General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: Explore Bristol(More)
Homoclinic bifurcations in autonomous ordinaty differential equations provide useful organizing centres for the analysis of examples. There are four generic types of homoclinic bifurcation, depending on the dominant eigenvalues of the Jacobian matrix of the flow near a stationary point. A family of differential equations is presented which, for suitable(More)
Infinite cascades of periodicity hubs were predicted and very recently observed experimentally to organize stable oscillations of some dissipative flows. Here we describe the global mechanism underlying the genesis and organization of networks of periodicity hubs in control parameter space of a simple prototypical flow, namely a Rössler's oscillator. We(More)
The deterministic border collision normal form describes the bifurcations of a discrete time dynamical system as a fixed point moves across the switching surface with changing parameter. If the position of the switching surface varies randomly, but within some bounded region, we give conditions which imply that the attractor close to the bifurcation point(More)
We show that the classic examples of quasi-periodically forced maps with strange nonchaotic attractors described by Grebogi et al and Herman in the mid-1980s have some chaotic properties. More precisely, we show that these systems exhibit sensitive dependence on initial conditions, both on the whole phase space and restricted to the attractor. The results(More)