Journal Articles

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

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Author: search.filters.author.Simpson DJWSubject: search.filters.subject.Robust chaos

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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.
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    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.