arXiv subjects

David J. W. Simpson

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

At least 37 records · Page 2Linked to original sources

arXiv2021

Inclusion of higher-order terms in the border-collision normal form: persistence of chaos and applications to power converters

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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arXiv2021

Renormalisation of the two-dimensional border-collision normal form

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. Rev. Lett. 80(14):3049--3052, 1998]. We use renormalisation to partition this region by the number of connected components of a chaotic Milnor attractor. This reveals previously undescribed bifurcation structure in a succinct way.

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arXiv2021

Unfolding globally resonant homoclinic tangencies

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 tangency, in this paper we study one-parameter perturbations of typical globally resonant homoclinic tangencies. We assume the tangencies are formed by the stable and unstable manifolds of saddle fixed points of two-dimensional maps. We show the perturbations display two infinite sequences of bifurcations, one saddle-node the other period-doubling, between which single-round periodic solutions are asymptotically stable. Generically these scale like $|λ|^{2 k}$, as $k \to \infty$, where $-1 < λ< 1$ is the stable eigenvalue associated with the fixed point. If the perturbation is taken tangent to the surface of codimension-one homoclinic tangencies, they instead scale like $\frac{|λ|^k}{k}$. We also show slower scaling laws are possible if the perturbation admits further degeneracies.

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arXiv2021

On the stability of boundary equilibria in Filippov systems

The leading-order approximation to a Filippov system $f$ about a generic boundary equilibrium $x^*$ is a system $F$ that is affine one side of the boundary and constant on the other side. We prove $x^*$ is exponentially stable for $f$ if and only if it is exponentially stable for $F$ when the constant component of $F$ is not tangent to the boundary. We then show exponential stability and asymptotic stability are in fact equivalent for $F$. We also show exponential stability is preserved under small perturbations to the pieces of $F$. Such results are well known for homogeneous systems. To prove the results here additional techniques are required because the two components of $F$ have different degrees of homogeneity. The primary function of the results is to reduce the problem of the stability of $x^*$ from the general Filippov system $f$ to the simpler system $F$. Yet in general this problem remains difficult. We provide a four-dimensional example of $F$ for which orbits appear to converge to $x^*$ in a chaotic fashion. By utilising the presence of both homogeneity and sliding motion the dynamics of $F$ can in this case be reduced to the combination of a one-dimensional return map and a scalar function.

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arXiv2020

Detecting invariant expanding cones for generating word sets to identify chaos in piecewise-linear maps

We show how the existence of three objects, $Ω_{\rm trap}$, ${\bf W}$, and $C$, for a continuous piecewise-linear map $f$ on $\mathbb{R}^N$, implies that $f$ has a topological attractor with a positive Lyapunov exponent. First, $Ω_{\rm trap} \subset \mathbb{R}^N$ is trapping region for $f$. Second, ${\bf W}$ is a finite set of words that encodes the forward orbits of all points in $Ω_{\rm trap}$. Finally, $C \subset T \mathbb{R}^N$ is an invariant expanding cone for derivatives of compositions of $f$ formed by the words in ${\bf W}$. We develop an algorithm that identifies these objects for two-dimensional homeomorphisms comprised of two affine pieces. The main effort is in the explicit construction of $Ω_{\rm trap}$ and $C$. Their existence is equated to a set of computable conditions in a general way. This results in a computer-assisted proof of chaos throughout a relatively large regime of parameter space. We also observe how the failure of $C$ to be expanding can coincide with a bifurcation of $f$. Lyapunov exponents are evaluated using one-sided directional derivatives so that forward orbits that intersect a switching manifold (where $f$ is not differentiable) can be included in the analysis.

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arXiv2020

Numerical Bifurcation Analysis of Pacemaker Dynamics in a Model of Smooth Muscle Cells

Evidence from experimental studies shows that oscillations due to electro-mechanical coupling can be generated spontaneously in smooth muscle cells. Such cellular dynamics are known as \textit{pacemaker dynamics}. In this article we address pacemaker dynamics associated with the interaction of $\text{Ca}^{2+}$ and $\text{K}^+$ fluxes in the cell membrane of a smooth muscle cell. First we reduce a pacemaker model to a two-dimensional system equivalent to the reduced Morris-Lecar model and then perform a detailed numerical bifurcation analysis of the reduced model. Existing bifurcation analyses of the Morris-Lecar model concentrate on external applied current whereas we focus on parameters that model the response of the cell to changes in transmural pressure. We reveal a transition between Type I and Type II excitabilities with no external current required. We also compute a two-parameter bifurcation diagram and show how the transition is explained by the bifurcation structure.

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arXiv2019

Characterisation and classification of signatures of spanning trees of the $n$-cube

The signature of a spanning tree $T$ of the $n$-cube $Q_n$ is the $n$-tuple $\mathrm{sig}(T)=(a_1,a_2,\dots,a_n)$ such that $a_i$ is the number of edges of $T$ in the $i$th direction. We characterise the $n$-tuples that can occur as the signature of a spanning tree, and classify a signature $\mathcal{S}$ as reducible or irreducible according to whether or not there is a proper nonempty subset $R$ of $[n]$ such that restricting $\mathcal{S}$ to the indices in $R$ gives a signature of $Q_{|R|}$. If so, we say moreover that $\mathcal{S}$ and $T$ reduce over $R$. We show that reducibility places strict structural constraints on $T$. In particular, if $T$ reduces over a set of size $r$ then $T$ decomposes as a sum of $2^r$ spanning trees of $Q_{n-r}$, together with a spanning tree of a contraction of $Q_n$ with underlying simple graph $Q_r$. Moreover, this decomposition is realised by an isomorphism of edge slide graphs, where the edge slide graph of $Q_n$ is the graph $\mathcal{E}(Q_n)$ on the spanning trees of $Q_n$, with an edge between two trees if and only if they are related by an edge slide. An edge slide is an operation on spanning trees of the $n$-cube given by ``sliding'' an edge of a spanning tree across a $2$-dimensional face of the cube to get a second spanning tree. The signature of a spanning tree is invariant under edge slides, so the subgraph $\mathcal{E}(\mathcal{S})$ of $\mathcal{E}(Q_n)$ induced by the trees with signature $\mathcal{S}$ is a union of one or more connected components of $\mathcal{E}(Q_n)$. Reducible signatures may be further divided into strictly reducible and quasi-irreducible signatures, and as an application of our results we show that $\mathcal{E}(\mathcal{S})$ is disconnected if $\mathcal{S}$ is strictly reducible. We conjecture that the converse is also true.

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arXiv2019

Unfolding codimension-two subsumed homoclinic connections in two-dimensional piecewise-linear maps

For piecewise-linear maps, the phenomenon that a branch of a one-dimensional unstable manifold of a periodic solution is completely contained in its stable manifold is codimension-two. Unlike codimension-one homoclinic corners, such `subsumed' homoclinic connections can be associated with stable periodic solutions. The purpose of this paper is to determine the dynamics near a generic subsumed homoclinic connection in two dimensions. Assuming the eigenvalues associated with the periodic solution satisfy $0 < |λ| < 1 < σ< \frac{1}{|λ|}$, in a two-parameter unfolding there exists an infinite sequence of roughly triangular regions within which the map has a stable single-round periodic solution. The result applies to both discontinuous and continuous maps, although these cases admit different characterisations for the border-collision bifurcations that correspond to boundaries of the regions. The result is illustrated with a discontinuous map of Mira and the two-dimensional border-collision normal form.

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arXiv2019

Constructing robust chaos: invariant manifolds and expanding cones

Chaotic attractors in the two-dimensional border-collision normal form (a piecewise-linear map) can persist throughout open regions of parameter space. Such robust chaos has been established rigorously in some parameter regimes. Here we provide formal results for robust chaos in the original parameter regime of [S. Banerjee, J.A. Yorke, C. Grebogi, Robust Chaos, Phys. Rev. Lett. 80(14):3049--3052, 1998]. We first construct a trapping region in phase space to prove the existence of a topological attractor. We then construct an invariant expanding cone in tangent space to prove that tangent vectors expand and so no invariant set can have only negative Lyapunov exponents. Under additional assumptions we also characterise an attractor as the closure of the unstable manifold of a fixed point.

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arXiv2019

Robust Chaos and the Continuity of Attractors

As the parameters of a map are varied an attractor may vary continuously in the Hausdorff metric. The purpose of this paper is to explore the continuation of chaotic attractors. We argue that this is not a helpful concept for smooth unimodal maps for which periodic windows fill parameter space densely, but that for piecewise-smooth maps it provides a way to delineate structure within parameter regions of robust chaos and form a stronger notion of robustness. We obtain conditions for the continuity of an attractor and demonstrate the results with coupled skew tent maps, the Lozi map, and the border-collision normal form.

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arXiv2019

Twenty Hopf-like bifurcations in piecewise-smooth dynamical systems

For many physical systems the transition from a stationary solution to sustained small amplitude oscillations corresponds to a Hopf bifurcation. For systems involving impacts, thresholds, switches, or other abrupt events, however, this transition can be achieved in fundamentally different ways. This paper reviews 20 such `Hopf-like' bifurcations for two-dimensional ODE systems with state-dependent switching rules. The bifurcations include boundary equilibrium bifurcations, the collision or change of stability of equilibria or folds on switching manifolds, and limit cycle creation via hysteresis or time delay. In each case a stationary solution changes stability and possibly form, and emits one limit cycle. Each bifurcation is analysed quantitatively in a general setting: we identify quantities that govern the onset, criticality, and genericity of the bifurcation, and determine scaling laws for the period and amplitude of the resulting limit cycle. Complete derivations based on asymptotic expansions of Poincare maps are provided. Many of these are new, done previously only for piecewise-linear systems. The bifurcations are collated and compared so that dynamical observations can be matched to geometric mechanisms responsible for the creation of a limit cycle. The results are illustrated with impact oscillators, relay control, automated balancing control, predator-prey systems, ocean circulation, and the McKean and Wilson-Cowan neuron models.

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arXiv2018

Hopf-like boundary equilibrium bifurcations involving two foci in Filippov systems

This paper concerns two-dimensional Filippov systems --- ordinary differential equations that are discontinuous on one-dimensional switching manifolds. In the situation that a stable focus transitions to an unstable focus by colliding with a switching manifold as parameters are varied, a simple sufficient condition for a unique local limit cycle to be created is established. If this condition is violated, three nested limit cycles may be created simultaneously. The result is achieved by constructing a Poincaré map and generalising analytical arguments that have been employed for continuous systems. Necessary and sufficient conditions for the existence of pseudo-equilibria (equilibria of sliding motion on the switching manifold) are also determined. For simplicity only piecewise-linear systems are considered.

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arXiv2018

Dimension reduction for slow-fast, piecewise-smooth, continuous systems of ODEs

The limiting slow dynamics of slow-fast, piecewise-linear, continuous systems of ODEs occurs on critical manifolds that are piecewise-linear. At points of non-differentiability, such manifolds are not normally hyperbolic and so the fundamental results of geometric singular perturbation theory do not apply. In this paper it is shown that if the critical manifold is globally stable then the system is forward invariant in a neighbourhood of the critical manifold. It follows that in this neighbourhood the dynamics is given by a regular perturbation of the dynamics on the critical manifold and so dimension reduction can be achieved. If the attraction is instead non-global, additional dynamics involving canards may be generated. For boundary equilibrium bifurcations of piecewise-smooth, continuous systems, the results are used to establish a general methodology by which such bifurcations can be analysed. This approach is illustrated with a three-dimensional model of ocean circulation.

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arXiv2017

Grazing-sliding bifurcations creating infinitely many attractors

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

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arXiv2016

The structure of mode-locking regions of piecewise-linear continuous maps: II. Skew sawtooth maps

In two-parameter bifurcation diagrams of piecewise-linear continuous maps on $\mathbb{R}^N$, mode-locking regions typically have points of zero width known as shrinking points. Near any shrinking point, but outside the associated mode-locking region, a significant proportion of parameter space can be usefully partitioned into a two-dimensional array of nearly-hyperbolic annular sectors. The purpose of this paper is to show that in these sectors the dynamics is well-approximated by a three-parameter family of skew sawtooth circle maps, where the relationship between the skew sawtooth maps and the $N$-dimensional map is fixed within each sector. The skew sawtooth maps are continuous, degree-one, and piecewise-linear, with two different slopes. They approximate the stable dynamics of the $N$-dimensional map with an error that goes to zero with the distance from the shrinking point. The results explain the complicated radial pattern of periodic, quasi-periodic, and chaotic dynamics that occurs near shrinking points.

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arXiv2016

The stability of fixed points on switching manifolds of piecewise-smooth continuous maps

This paper concerns piecewise-smooth maps on $\mathbb{R}^d$ that are continuous but not differentiable on switching manifolds (where the functional form of the map changes). The stability of fixed points on switching manifolds is investigated in scenarios for which one-sided derivatives are locally bounded. The lack of differentiability allows fixed points to be Milnor attractors despite being unstable. For this reason a measure-theoretic notion of stability is considered in addition to standard notions of stability. Locally the map is well approximated by a piecewise-linear map that is linearly homogeneous when the fixed point is at the origin. For the class of continuous, linearly homogeneous maps, and $o({\bf x})$ perturbations of these maps, a sufficient condition for the exponential stability of the origin is obtained. It is shown how the stability of the origin can be determined by analysing invariant probability measures of a map on $\mathbb{S}^{d-1}$. The results are illustrated for the two-dimensional border-collision normal form. The fixed point may be asymptotically stable even if both smooth components of the map are area-expanding, and unstable even if it is the $ω$-limit set of almost all points in $\mathbb{R}^d$.

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arXiv2016

Subsumed homoclinic connections and infinitely many coexisting attractors in piecewise-linear maps

We establish an equivalence between infinitely many asymptotically stable periodic solutions and subsumed homoclinic connections for $N$-dimensional piecewise-linear continuous maps. These features arise as a codimension-three phenomenon. The periodic solutions are single-round: they each involve one excursion away from a central saddle-type periodic solution. The homoclinic connection is subsumed in the sense that one branch of the unstable manifold of the saddle solution is contained entirely within its stable manifold. The results are proved by using exact expressions for the periodic solutions and components of the stable and unstable manifolds which are available because the maps are piecewise-linear. We also describe a practical approach for finding this phenomenon in the parameter space of a map and illustrate the results with the three-dimensional border-collision normal form.

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arXiv2016

Jitter in Dynamical Systems with Intersecting Discontinuity Surfaces

Mathematical models involving switches --- in the form of differential equations with discontinuities --- can accomodate real-world non-idealities through perturbations by hysteresis, time-delay, discretization, and noise. These are used to model the processes associated with switching in electronic control, mechanical contact, predator-prey preferences, and genetic or cellular regulation. The effect of such perturbations on rapid switching dynamics about a single switch are somewhat benign: in the limit that the size of the perturbation goes to zero the dynamics is given by Filippov's sliding solution. When multiple switches are involved, however, perturbations can have complicated effects, as shown in this paper. In the zero-perturbation limit, hysteresis, time-delay, and discretization cause erratic variation or `jitter' for stable sliding motion, whilst noise generates a relatively regular sliding solution similar to the canopy solution (an extension of Filippov's solution to multiple switches). We illustrate the results with a model of a switched power circuit, and showcase a variety of complex phenomena that perturbations can generate, including chaotic dynamics, exit selection, and coexisting sliding solutions.

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