arXiv subjects

D. J. W. Simpson

Publications and source records attributed to D. J. W. Simpson.

At least 19 recordsLinked to original sources

arXiv2026

Refracting Filippov systems: sliding dynamics without sliding regions

This paper develops fundamental mathematical theory for refracting Filippov systems. These are discontinuous ordinary differential equations with solutions defined in the sense of Filippov, and whose first Lie derivatives vary continuously across discontinuity surfaces. Unlike generic Filippov systems, discontinuity surfaces consist only of crossing regions and their boundaries where both adjacent vector fields are tangent to the discontinuity surface. Crossing orbits spiral around invisible-invisible tangency surfaces, and we derive a formula for the attractive or repulsive strength of these surfaces. We prove crossing orbits cannot converge to tangency surfaces in finite time (no Zeno), and that the limiting dynamics consists of Filippov solutions on the tangency surfaces (second-order sliding motion). We derive a vector field that governs this motion, and characterise the stability of equilibria on tangency surfaces. The methodology is applied to a model of a mechanical oscillator with compliant impacts, and a model of ant colony migration. We also relate refracting Filippov systems to second-order sliding mode control, and show that for two-dimensional systems the results reduce to known theory.

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arXiv2025

From two-dimensional continuous maps to one-dimensional discontinuous maps: a novel reduction explaining complex bifurcation structures in piecewise-linear families of maps

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 certain codimension-two homoclinic bifurcations are well approximated by a three-parameter family of one-dimensional maps. Each member of the one-dimensional family is discontinuous, because the family is constructed from the first return of iterates to a subset of phase space, and comprised of infinitely many linear pieces, where each piece corresponds to a fixed number of iterations near the saddle associated with the homoclinic bifurcation. The one-dimensional family exhibits period-incrementing, period-adding, bandcount-incrementing, and bandcount-adding structures (all typical for two-piece maps), as well as unique features caused by orbits repeatedly visiting more than two pieces of the map. These structures carry through to the two-dimensional maps with only minor differences in the arrangement of the bifurcations developing with the distance from the codimension-two bifurcations. This leads to a novel and vivid elucidation of the dynamics of the two-dimensional border-collision normal form.

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arXiv2025

Nonsmooth folds as tipping points

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 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 ODEs, 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 the bifurcation structures of the truncated systems.

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arXiv2024

Extended normal forms for one-dimensional border-collision bifurcations

The border-collision normal form describes the local dynamics in continuous systems with switches when a fixed point intersects a switching surface. For one-dimensional cases where the bifurcation creates or destroys only fixed points and period-two orbits, we show that the standard local equivalence of normal forms, topological conjugacy, can be replaced by differentiable conjugacy provided an extra term is added to the normal form. In these cases topological conjugacy is so weak that a range of values can be used for the coefficients in the normal form. The extension to differentiable conjugacy explains why the usual choice of slopes in the standard normal form is privileged. This highlights the importance of differentiable conjugacies and the need for extended normal forms.

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arXiv2023

How to compute multi-dimensional stable and unstable manifolds of piecewise-linear maps

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-dimensional manifolds). Such representations are efficient and exact so for computational purposes are superior to representations that use a large number of points on some mesh (as is usually done in the smooth setting). We introduce a method for computing convex polytope representations of stable and unstable manifolds. For an unstable manifold we iterate a suitably small subset of the local unstable manifold and prior to each iteration subdivide polytopes where they intersect the switching manifold of the map. We prove the output converges to the (entire) unstable manifold and use it to visualise attractors and bifurcations of the three-dimensional border-collision normal form: we identify a heterodimensional-cycle, a two-dimensional unstable manifold whose closure appears to be a unique attractor, and a piecewise-linear analogue of a first homoclinic tangency where an attractor appears to be destroyed.

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arXiv2023

Differentiable conjugacies for one-dimensional maps

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. We describe some of the techniques and recent results that allow differentiable conjugacies to be defined for standard bifurcations, and explain how this leads to a new class of normal forms. Closed-form expressions for differentiable conjugacies exist between some chaotic maps, and we describe some of the constraints that make it possible to recognise when such conjugacies arise. This paper focuses on the consequences of the existence of differentiable conjugacies rather than the conjugacy classes themselves.

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arXiv2022

Unstable dimension variability, heterodimensional cycles, and blenders in the border-collision normal form

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 phenomena in the border-collision normal form. This is a continuous, piecewise-linear family of maps that is physically relevant as it captures the dynamics created in border-collision bifurcations in diverse applications. Since the maps are piecewise-linear they are relatively amenable to an exact analysis and we are able to explicitly identify parameter values for heterodimensional cycles and blenders. For a one-parameter subfamily we identify bifurcations involved in a transition through unstable dimension variability. This is facilitated by being able to compute periodic solutions quickly and accurately, and the piecewise-linear form should provide a useful test-bed for further study.

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arXiv2022

Normal forms for saddle-node bifurcations: Takens' coefficient and applications in climate models

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 to equations that can be written down explicitly. The equivalence to the original equations is a local differentiable conjugacy on the basins of attraction and repulsion of stationary points in the parameter region for which these exist, and is a differentiable conjugacy on the whole local interval otherwise. (Recall that in standard approaches local equivalence is topological rather than differentiable.) The value $a(0)$ is Takens' coefficient from normal form theory. The results explain the sense in which normal forms extend away from the bifurcation point and provide a new and more detailed characterisation of the saddle-node bifurcation. The one-dimensional system can be derived from higher dimensional equations using centre manifold theory. We illustrate this using two examples from climate science and show how the functions $ν$ and $a$ can be determined analytically in some settings and numerically in others.

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arXiv2021

Pattern Formation in a Spatially-Extended Model of Pacemaker Dynamics in Smooth Muscle Cells

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 spatiotemporal patterns in electrically-coupled smooth muscle cells occur even in the absence of forcing. We study a reaction-diffusion system with 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 use travelling wave coordinates to analyse travelling waves.

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arXiv2021

Chaos in the border-collision normal form: A computer-assisted proof using induced maps and invariant expanding cones

In some maps the existence of an attractor with a positive Lyapunov exponent can be proved by constructing a trapping region in phase space and an invariant expanding cone in tangent space. If this approach fails it may be possible to adapt the strategy by considering an induced map (a first return map for a well-chosen subset of phase space). In this paper we show that such a construction can be applied to the two-dimensional border-collision normal form (a continuous piecewise-linear map) if a certain set of conditions are satisfied and develop an algorithm for checking these conditions. The algorithm requires relatively few computations, so it is a more efficient method than, for example, estimating the Lyapunov exponent from a single orbit in terms of speed, numerical accuracy, and rigor. The algorithm is used to prove the existence of an attractor with a positive Lyapunov exponent numerically in an area of parameter space where the map has strong rotational characteristics and the consideration of an induced map is critical for the proof of robust chaos.

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arXiv2020

Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions

We consider a homoclinic orbit to a saddle fixed point of an arbitrary $C^\infty$ map $f$ on $\mathbb{R}^2$ and study the phenomenon that $f$ has an infinite family of asymptotically stable, single-round periodic solutions. From classical theory, this requires $f$ to have a homoclinic tangency. We show it also necessary for $f$ to satisfy a `global resonance' condition and for the eigenvalues associated with the fixed point, $λ$ and $σ$, to satisfy $|λσ| = 1$. The phenomenon is codimension-three in the case $λσ= -1$, but codimension-four in the case $λσ= 1$ because here the coefficients of the leading-order resonance terms associated with $f$ at the fixed point must add to zero. We also identify conditions sufficient for the phenomenon to occur, illustrate the results for an abstract family of maps, and show numerically computed basins of attraction.

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arXiv2019

Chaotic attractors from border-collision bifurcations: stable border fixed points and determinant-based Lyapunov exponent bounds

The collision of a fixed point with a switching manifold (or border) in a piecewise-smooth map can create many different types of invariant sets. This paper explores two techniques that, combined, establish a chaotic attractor is created in a border-collision bifurcation in $\mathbb{R}^d$ $(d \ge 1)$. First, asymptotic stability of the fixed point at the bifurcation is characterised and shown to imply a local attractor is created. Second, a lower bound on the maximal Lyapunov exponent is obtained from the determinants of the one-sided Jacobian matrices associated with the fixed point. Special care is taken to accommodate points whose forward orbits intersect the switching manifold as such intersections can have a stabilising effect. The results are applied to the two-dimensional border-collision normal form focusing on parameter values for which the map is piecewise area-expanding.

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arXiv2018

A Compendium of Hopf-Like Bifurcations in Piecewise-Smooth Dynamical Systems

This Letter outlines 20 geometric mechanisms by which limit cycles are created locally in two-dimensional piecewise-smooth systems of ODEs. These include boundary equilibrium bifurcations of hybrid systems, Filippov systems, and continuous systems, and limit cycles created from folds and by the addition of hysteresis or time-delay. Scaling laws for the amplitude and period of the limit cycles are compared to (classical) Hopf bifurcations.

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arXiv2018

A general framework for boundary equilibrium bifurcations of Filippov systems

As parameters are varied a boundary equilibrium bifurcation (BEB) occurs when an equilibrium collides with a discontinuity surface in a piecewise-smooth system of ODEs. Under certain genericity conditions, at a BEB the equilibrium either transitions to a pseudo-equilibrium (on the discontinuity surface) or collides and annihilates with a coexisting pseudo-equilibrium. These two scenarios are distinguished by the sign of a certain inner product. Here it is shown that this sign can be determined from the number of unstable directions associated with the two equilibria by using techniques developed by Feigin. A new normal form is proposed for BEBs in systems of any number of dimensions. The normal form involves a companion matrix, as does the leading order sliding dynamics, and so the connection to the stability of the equilibria is explicit. In two dimensions the parameters of the normal form distinguish, in a simple way, the eight topologically distinct cases for the generic local dynamics at a BEB. A numerical exploration in three dimensions reveals that BEBs can create multiple attractors and chaotic attractors, and that the equilibrium at the BEB can be unstable even if both equilibria are stable. The developments presented here stem from seemingly unutilised similarities between BEBs in discontinuous systems (specifically Filippov systems as studied here) and BEBs in continuous systems for which analogous results are, to date, more advanced.

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arXiv2016

The instantaneous local transition of a stable equilibrium to a chaotic attractor in piecewise-smooth systems of differential equations

An attractor of a piecewise-smooth continuous system of differential equations can bifurcate from a stable equilibrium to a more complicated invariant set when it collides with a switching manifold under parameter variation. Here numerical evidence is provided to show that this invariant set can be chaotic. The transition occurs locally (in a neighbourhood of a point) and instantaneously (for a single critical parameter value). This phenomenon is illustrated for the normal form of a boundary equilibrium bifurcation in three dimensions using parameter values adapted from of a piecewise-linear model of a chaotic electrical circuit. The variation of a secondary parameter reveals a period-doubling cascade to chaos with windows of periodicity. The dynamics is well approximated by a one-dimensional unimodal map which explains this bifurcation structure. The robustness of the attractor is also investigated by studying the influence of nonlinear terms.

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arXiv2015

Fast phase randomisation via two-folds

A two-fold is a singular point on the discontinuity surface of a piecewise-smooth vector field, at which the vector field is tangent to the discontinuity surface on both sides. If an orbit passes through an invisible two-fold (also known as a Teixeira singularity) before settling to regular periodic motion, then the phase of that motion cannot be determined from initial conditions, and in the presence of small noise the asymptotic phase of a large number of sample solutions is highly random. In this paper we show how the probability distribution of the asymptotic phase depends on the global nonlinear dynamics. We also show how the phase of a smooth oscillator can be randomised by applying a simple discontinuous control law that generates an invisible two-fold. We propose that such a control law can be used to desynchronise a collection of oscillators, and that this manner of phase randomisation is fast compared to existing methods (which use fixed points as phase singularities) because there is no slowing of the dynamics near a two-fold.

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arXiv2010

Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems

Systems that are not smooth can undergo bifurcations that are forbidden in smooth systems. We review some of the phenomena that can occur for piecewise-smooth, continuous maps and flows when a fixed point or an equilibrium collides with a surface on which the system is not smooth. Much of our understanding of these cases relies on a reduction to piecewise linearity near the border-collision. We also review a number of codimension-two bifurcations in which nonlinearity is important.

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arXiv2010

Resonance near Border-Collision Bifurcations in Piecewise-Smooth, Continuous Maps

Mode-locking regions (resonance tongues) formed by border-collision bifurcations of piecewise-smooth, continuous maps commonly exhibit a distinctive sausage-like geometry with pinch points called "shrinking points". In this paper we extend our unfolding of the piecewise-linear case [{\em Nonlinearity}, 22(5):1123-1144, 2009] to show how shrinking points are destroyed by nonlinearity. We obtain a codimension-three unfolding of this shrinking point bifurcation for $N$-dimensional maps. We show that the destruction of the shrinking points generically occurs by the creation of a curve of saddle-node bifurcations that smooth one boundary of the sausage, leaving a kink in the other boundary.

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