David J. W. Simpson
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View article: From two-dimensional continuous maps to one-dimensional discontinuous maps: a novel reduction explaining complex bifurcation structures in piecewise-linear families of maps
From two-dimensional continuous maps to one-dimensional discontinuous maps: a novel reduction explaining complex bifurcation structures in piecewise-linear families of maps Open
Piecewise-linear maps describe dynamical phenomena that switch between distinct states and readily generate complex bifurcation structures due to their strong nonlinearity. We show that two-dimensional continuous piecewise-linear maps near…
View article: The VIVID function for numerically continuing periodic orbits arising from grazing bifurcations of hybrid dynamical systems
The VIVID function for numerically continuing periodic orbits arising from grazing bifurcations of hybrid dynamical systems Open
Periodic orbits of systems of ordinary differential equations can be found and continued numerically by following fixed points of Poincaré maps. However, this often fails near grazing bifurcations where a periodic orbit collides tangential…
View article: Exploring perspectives of knowledge users about reporting on health equity in observational studies: A qualitative study informing the development of the STROBE-Equity reporting guideline
Exploring perspectives of knowledge users about reporting on health equity in observational studies: A qualitative study informing the development of the STROBE-Equity reporting guideline Open
Background Health inequities arising from systemic factors and contextual conditions result in avoidable and unjust differences in health outcomes, with profound social and economic implications. Health inequities can frequently go unrepor…
View article: Three forms of dimension reduction for border-collision bifurcations
Three forms of dimension reduction for border-collision bifurcations Open
View article: Reliability of the Five Step Assessment and Its Coefficients of Impairment in Spastic Paresis
Reliability of the Five Step Assessment and Its Coefficients of Impairment in Spastic Paresis Open
The 5 parameters and 4 coefficients of impairment of the FSA have moderate-to-excellent intrarater and interrater reliability in chronic spastic paresis.
View article: The concept of controlled current output of ICCP anodes
The concept of controlled current output of ICCP anodes Open
A common issue with Impressed Current Cathodic Protection (ICCP) anodes is the lack of control of the current that each individual anode can deliver to the steel. Currently, the problem is faced by separating the structure into several ind…
View article: Robust chaos in $\mathbb{R}^n$
Robust chaos in $\mathbb{R}^n$ Open
We treat $n$-dimensional piecewise-linear continuous maps with two pieces, each of which has exactly one unstable direction, and identify an explicit set of sufficient conditions for the existence of a chaotic attractor. The conditions cor…
View article: Nonsmooth folds as tipping points
Nonsmooth folds as tipping points Open
A nonsmooth fold is where 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 tr…
View article: The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps
The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps Open
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 o…
View article: The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps
The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps Open
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 …
View article: Inclusion of higher-order terms in the border-collision normal form: Persistence of chaos and applications to power converters
Inclusion of higher-order terms in the border-collision normal form: Persistence of chaos and applications to power converters Open
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. F…
View article: The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps
The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps Open
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 o…
View article: The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps
The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps Open
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 …
View article: How to compute multi-dimensional stable and unstable manifolds of piecewise-linear maps
How to compute multi-dimensional stable and unstable manifolds of piecewise-linear maps Open
For piecewise-linear maps the stable and unstable manifolds of hyperbolic periodic solutions are themselves piecewise-linear. Hence compact subsets of these manifolds can be represented using polytopes (i.e. polygons, in the case of two-di…
View article: Robust chaos in orientation-reversing and non-invertible two-dimensional piecewise-linear maps
Robust chaos in orientation-reversing and non-invertible two-dimensional piecewise-linear maps Open
This paper concerns the two-dimensional border-collision normal form -- a four-parameter family of piecewise-linear maps generalising the Lozi family and relevant to diverse applications. The normal form was recently shown to exhibit a cha…
View article: Does a novel bridging collar in endoprosthetic replacement optimise the mechanical environment for osseointegration? A finite element study
Does a novel bridging collar in endoprosthetic replacement optimise the mechanical environment for osseointegration? A finite element study Open
Introduction: Limb-salvage surgery using endoprosthetic replacements (EPRs) is frequently used to reconstruct segmental bone defects, but the reconstruction longevity is still a major concern. In EPRs, the stem-collar junction is the most …
View article: Preface to VSI: Advances in nonsmooth dynamics
Preface to VSI: Advances in nonsmooth dynamics Open
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 epidem…
View article: Differentiable conjugacies for one-dimensional maps
Differentiable conjugacies for one-dimensional maps Open
Differentiable conjugacies link dynamical systems that share properties such as the stability multipliers of corresponding orbits. It provides a stronger classification than topological conjugacy, which only requires qualitative similarity…
View article: Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications
Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications Open
Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their s…
View article: Unstable dimension variability, heterodimensional cycles, and blenders in the border-collision normal form
Unstable dimension variability, heterodimensional cycles, and blenders in the border-collision normal form Open
Chaotic attractors commonly contain periodic solutions with unstable manifolds of different dimensions. This allows for a zoo of dynamical phenomena not possible for hyperbolic attractors. The purpose of this Letter is to demonstrate these…
View article: Normal forms for saddle-node bifurcations: Takens' coefficient and applications in climate models
Normal forms for saddle-node bifurcations: Takens' coefficient and applications in climate models Open
We show that a one-dimensional differential equation depending on a parameter $μ$ with a saddle-node bifurcation at $μ=0$ can be modelled by an extended normal form $\dot y = ν(μ)-y^2+a(μ)y^3$, where the functions $ν$ and $a$ are solutions…
View article: Border-collision bifurcations from stable fixed points to any number of coexisting chaotic attractors
Border-collision bifurcations from stable fixed points to any number of coexisting chaotic attractors Open
In diverse physical systems stable oscillatory solutions devolve into more complicated dynamical behaviour through border-collision bifurcations. Mathematically these occur when a stable fixed point of a piecewise-smooth map collides with …
View article: Pattern Formation in a Spatially Extended Model of Pacemaker Dynamics in Smooth Muscle Cells
Pattern Formation in a Spatially Extended Model of Pacemaker Dynamics in Smooth Muscle Cells Open
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 tha…
View article: Robust Devaney chaos in the two-dimensional border-collision normal form
Robust Devaney chaos in the two-dimensional border-collision normal form Open
The collection of all non-degenerate, continuous, two-piece, piecewise-linear maps on R2 can be reduced to a four-parameter family known as the two-dimensional border-collision normal form. We prove that throughout an open region of parame…
View article: Long term control of corrosion of steel reinforcement by a two-stage cathodic protection method
Long term control of corrosion of steel reinforcement by a two-stage cathodic protection method Open
It has been shown experimentally that corrosion of steel reinforcement can be arrested if sufficient cathodic charge at a current density higher than 20 mA/m 2 is applied over a period of weeks. Once corrosion is arrested, a cathodic preve…
View article: Unfolding globally resonant homoclinic tangencies
Unfolding globally resonant homoclinic tangencies Open
Global resonance is a mechanism by which a homoclinic tangency of a smooth map can have infinitely many asymptotically stable, single-round periodic solutions. To understand the bifurcation structure one would expect to see near such a tan…
View article: Inclusion of higher-order terms in the border-collision normal form: persistence of chaos and applications to power converters
Inclusion of higher-order terms in the border-collision normal form: persistence of chaos and applications to power converters Open
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. F…
View article: Pattern Formation in a Spatially-Extended Model of Pacemaker Dynamics in Smooth Muscle Cells
Pattern Formation in a Spatially-Extended Model of Pacemaker Dynamics in Smooth Muscle Cells Open
Spatiotemporal patterns are common in biological systems. For electrically-coupled cells previous studies of pattern formation have mainly used external forcing as the main bifurcation parameter. The purpose of this paper is to show that s…
View article: Renormalisation of the two-dimensional border-collision normal form
Renormalisation of the two-dimensional border-collision normal form Open
We study the two-dimensional border-collision normal form (a four-parameter family of continuous, piecewise-linear maps on $\mathbb{R}^2$) in the robust chaos parameter region of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Re…
View article: Unfolding globally resonant homoclinic tangencies
Unfolding globally resonant homoclinic tangencies Open
Global resonance is a mechanism by which a homoclinic tangency of a smooth map can have infinitely many asymptotically stable, single-round periodic solutions. To understand the bifurcation structure one would expect to see near such a tan…