Paul Glendinning

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Rotation numbers have played a central role in the study of (unforced) monotone circle maps. In such a case it is possible to obtain a priori bounds of the form » ¡ 1=n µ …1=n †…y n ¡ y 0 † µ » ‡ 1=n, where …1=n †…y n ¡ y 0 † is an estimate of the rotation number obtained from an orbit of length n with initial condition y 0 , and » is the true rotation(More)
Tennyson's 'Nature, red in tooth and claw' [10] paints a rather more vivid picture of the struggle for life than I encounter on the moors. Moorland deaths seem bloodless; the work of the weather, land and time rather than the result of battles between rival animals. The only regular wars I witness are the squabbles of finches and tits over the peanut(More)
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)
In this paper we show that the border collision normal form of continuous but non-differentiable discrete time maps is affected by a curse of dimensionality: it is impossible to reduce the study of the general case to low dimensions, since in every dimension the bifurcation produces fundamentally different attractors (contrary to the case of smooth(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)