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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.
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.