Journal Articles

Permanent URI for this collectionhttps://mro.massey.ac.nz/handle/10179/7915

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Subject: search.filters.subject.Bifurcation

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    Three forms of dimension reduction for border-collision bifurcations
    (Elsevier B V, 2025-08-05) Simpson DJW
    For dynamical systems that switch between different modes of operation, parameter variation can cause periodic solutions to lose or acquire new switching events. When this causes the eigenvalues (stability multipliers) associated with the solution to change discontinuously, we show that if one eigenvalue remains continuous then all local invariant sets of the leading-order approximation to the system occur on a lower dimensional manifold. This allows us to analyse the dynamics with fewer variables, which is particularly helpful when the dynamics is chaotic. We compare this to two other codimension-two scenarios for which dimension reduction can be achieved.
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    Preface to VSI: Advances in nonsmooth dynamics
    (Elsevier B V, 2023-11) Jeffrey MR; Piiroinen PT; Simpson DJW
    This Special Issue on nonsmooth dynamics brings together recent developments in nonsmooth dynamics, from applications in control engineering and mechanics, economics, climate modelling, physiological modelling, medicine, ecology and epidemiology, and others, to theory of novel forms of unpredictability and nonlinearity, chaos and bifurcations, and the study of higher dimensions.
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    Numerical bifurcation theory for high-dimensional neural models
    (BioMed Central, 25/07/2014) Laing CR
    Numerical bifurcation theory involves finding and then following certain types of solutions of differential equations as parameters are varied, and determining whether they undergo any bifurcations (qualitative changes in behaviour). The primary technique for doing this is numerical continuation, where the solution of interest satisfies a parametrised set of algebraic equations, and branches of solutions are followed as the parameter is varied. An effective way to do this is with pseudo-arclength continuation. We give an introduction to pseudo-arclength continuation and then demonstrate its use in investigating the behaviour of a number of models from the field of computational neuroscience. The models we consider are high dimensional, as they result from the discretisation of neural field models—nonlocal differential equations used to model macroscopic pattern formation in the cortex. We consider both stationary and moving patterns in one spatial dimension, and then translating patterns in two spatial dimensions. A variety of results from the literature are discussed, and a number of extensions of the technique are given.