Browsing by Author "Simpson DJW"

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Inclusion of higher-order terms in the border-collision normal form: Persistence of chaos and applications to power converters
(Elsevier BV, 2024-06) Simpson DJW; Glendinning PA
The dynamics near a border-collision bifurcation are approximated to leading order by a continuous, piecewise-linear map. The purpose of this paper is to consider the higher-order terms that are neglected when forming this approximation. For two-dimensional maps we establish conditions under which a chaotic attractor created in a border-collision bifurcation persists for an open interval of parameters beyond the bifurcation. We apply the results to a prototypical power converter model to prove the model exhibits robust chaos.
Nonsmooth folds as tipping points.
(AIP Publishing LLC, 2025-02-05) Simpson DJW
A nonsmooth fold occurs when an equilibrium or limit cycle of a nonsmooth dynamical system hits a switching manifold and collides and annihilates with another solution of the same type. We show that beyond the bifurcation, the leading-order truncation to the system, in general, has no bounded invariant set. This is proved for boundary equilibrium bifurcations of Filippov systems, hybrid systems, and continuous piecewise-smooth ordinary differential equations, and grazing-type events for which the truncated form is a continuous piecewise-linear map. The omitted higher-order terms are expected to be incapable of altering the local dynamics qualitatively, implying the system has no local invariant set on one side of a nonsmooth fold, and we demonstrate this with an example. Thus, if the equilibrium or limit cycle is attracting, the bifurcation causes the local attractor of the system to tip to a new state. The results also help explain global aspects of bifurcation structures of the truncated systems.
Pattern Formation in a Spatially Extended Model of Pacemaker Dynamics in Smooth Muscle Cells.
(Springer Nature Switzerland AG on behalf of the Society for Mathematical Biology, 2022-07-08) Fatoyinbo HO; Brown RG; Simpson DJW; van Brunt B
Spatiotemporal patterns are common in biological systems. For electrically coupled cells, previous studies of pattern formation have mainly used applied current as the primary bifurcation parameter. The purpose of this paper is to show that applied current is not needed to generate spatiotemporal patterns for smooth muscle cells. The patterns can be generated solely by external mechanical stimulation (transmural pressure). To do this we study a reaction-diffusion system involving the Morris-Lecar equations and observe a wide range of spatiotemporal patterns for different values of the model parameters. Some aspects of these patterns are explained via a bifurcation analysis of the system without coupling - in particular Type I and Type II excitability both occur. We show the patterns are not due to a Turing instability and that the spatially extended model exhibits spatiotemporal chaos. We also use travelling wave coordinates to analyse travelling waves.
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.
The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps
(Elsevier Ltd, 2024-07) Ghosh I; McLachlan RI; Simpson DJW
We study two-dimensional, two-piece, piecewise-linear maps having two saddle fixed points. Such maps reduce to a four-parameter family and are well known to have a chaotic attractor throughout open regions of parameter space. The purpose of this paper is to determine where and how this attractor undergoes bifurcations. We explore the bifurcation structure numerically by using Eckstein's greatest common divisor algorithm to estimate from sample orbits the number of connected components in the attractor. Where the map is orientation-preserving the numerical results agree with formal results obtained previously through renormalisation. Where the map is orientation-reversing or non-invertible the same renormalisation scheme appears to generate the bifurcation boundaries, but here we need to account for the possibility of some stable low-period solutions. Also the attractor can be destroyed in novel heteroclinic bifurcations (boundary crises) that do not correspond to simple algebraic constraints on the parameters. Overall the results reveal a broadly similar component-doubling bifurcation structure in the orientation-reversing and non-invertible settings, but with some additional complexities.
The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps
(Elsevier BV, 2024-06) Simpson DJW
Piecewise-smooth maps are used as discrete-time models of dynamical systems whose evolution is governed by different equations under different conditions (e.g. switched control systems). By assigning a symbol to each region of phase space where the map is smooth, any period-p solution of the map can be associated to an itinerary of p symbols. As parameters of the map are varied, changes to this itinerary occur at border-collision bifurcations (BCBs) where one point of the periodic solution collides with a region boundary. It is well known that BCBs conform broadly to two cases: persistence, where the symbolic itinerary of a periodic solution changes by one symbol, and a nonsmooth-fold, where two solutions differing by one symbol collide and annihilate. This paper derives new properties of periodic solutions of piecewise-linear continuous maps on Rn to show that under mild conditions BCBs of mode-locked solutions on invariant circles must be nonsmooth-folds. This explains why Arnold tongues of piecewise-linear maps exhibit a sausage-string structure whereby changes to symbolic itineraries occur at codimension-two pinch points instead of codimension-one persistence-type BCBs. But the main result is based on the combinatorical properties of the itineraries, so the impossibility of persistence-type BCBs also holds when the periodic solution is unstable or there is no invariant circle.
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.