arXiv subjects

David J. W. Simpson

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

At least 19 recordsLinked to original sources

arXiv2026

Hopf-like bifurcations induced by hysteresis and time-delay near monodromic tangential singularities

We investigate planar piecewise-analytic vector fields, focusing on the merged focus and other monodromic tangential singularities when hysteresis or time-delay is incorporated into the switching condition. If the singularity is an asymptotically stable solution of the system with an instantaneous switch, then the introduction of hysteresis or time-delay causes an attracting limit cycle to be formed locally. We derive asymptotic expressions for the size and period of the limit cycle, allowing any degrees of tangency and any order for the smallest non-zero Lyapunov coefficient. We find that the growth rate of the limit cycle differs for hysteresis and time-delay, and differs to that of the related pseudo-Hopf bifurcation.

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arXiv2026

Grazing bifurcations of linear impact oscillators in the zero damping limit

We consider a harmonically forced linear impact oscillator, where impact events are instantaneous with energy loss. We study the dynamics at the grazing bifurcation of the non-impacting periodic solution in the limit that the damping coefficient of the oscillator is zero. Through numerical computations we show that a recurring sequence of bifurcations exists between points of resonance. Specifically, resonance creates a stable periodic solution that subsequently loses stability in a secondary grazing bifurcation, then regains stability in a saddle-node bifurcation, then transitions to a chaotic attractor through a period-doubling cascade. The dynamics persist under mild parameter variation, so apply to weakly-damped impact oscillators near grazing.

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arXiv2026

One-dimensional first return maps for the two-dimensional border-collision normal form with a zero determinant

The two-dimensional border-collision normal form is a four-parameter family of continuous, piecewise-linear maps. When this form has a zero determinant, all of its nonlinear dynamics are captured a one-dimensional first return map. The first return map is discontinuous and piecewise-linear, where each piece of the map corresponds to a constant return time. We show that when the normal form has a repelling focus fixed point, the configuration of the first return map is dictated by a rational rotation number whereby the set of return times and ordering of the pieces of the map are given by the denominators of the left and right sequences of Farey parents of this number. The result is proved by characterising polygons formed from preimages of the switching manifold, and employing an inductive argument on the Farey web. Several surfaces in parameter space where the configuration changes are bifurcations from chaotic to quasiperiodic or mode-locked dynamics.

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arXiv2026

Boundary Hopf bifurcations in three-dimensional Filippov systems

For piecewise-smooth ordinary differential equations, the occurrence of a Hopf bifurcation on a switching surface is known as a boundary Hopf bifurcation. Boundary Hopf bifurcations are codimension-two, so occur at points in two-parameter bifurcation diagrams. From any such point there issues a curve of grazing bifurcations, where the limit cycle born in the Hopf bifurcation hits the switching surface. For Filippov systems, these are usually grazing-sliding bifurcations whose local dynamics are dictated by piecewise-linear maps. In general, these maps have many independent parameters and extraordinarily rich dynamical behaviour. We show that for three-dimensional Filippov systems only a two-parameter family of piecewise-linear maps is relevant, because sliding motion induces a loss of dimension, and the stability of the limit cycle is degenerate at the Hopf bifurcation. We derive explicit formulas for the two parameters in terms of quantities associated with the boundary Hopf bifurcation, and perform a comprehensive numerical analysis to characterise the attractor of the family, which may be chaotic. The results are illustrated with a pedagogical example, a pest control model, and a model of a food chain with threshold-based harvesting. To evaluate the parameters, we use a formula for the linear term of the discontinuity map associated with grazing-sliding bifurcations. In this paper we present a new, simpler derivation of this formula for $n$-dimensional systems based on displacements from a virtual counterpart.

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arXiv2026

Resonant grazing bifurcations revisited

In vibro-impact mechanics, the division between an impact and a near miss is a zero-velocity grazing event. Grazing bifurcations of stable periodic motions often produce complicated attractors when grazing generates a square-root term in the Poincaré map. This paper concerns codimension-two scenarios for which the square-root term vanishes in some iterate of the Poincaré map. For forced one-degree-of-freedom oscillators, this occurs when the forcing frequency is a certain rational multiple of the damped natural frequency, i.e., the system is in resonance. In two-parameter bifurcation diagrams, curves of saddle-node and period-doubling bifurcations of single-impact periodic motions emanate from the codimension-two points. In this paper we prove these curves are quadratically tangent to the curve of grazing bifurcations, and derive explicit formulas for their quadratic coefficients. This is achieved by modifying the Poincaré map in a way that circumvents the square-root singularity, enabling us to use the implicit function theorem to demonstrate smoothness and perform asymptotic calculations of the saddle-node and period-doubling bifurcation curves. In doing so we resolve a long-standing conjecture on the admissibility of single-impact periodic motions by supplementing raw asymptotic computations with geometric and topological arguments. We illustrate the results with a linear impact oscillator model, matching the theoretical unfolding to numerically computed bifurcation curves. The results explain why previously reported physical experiments reveal an absence of chaos shortly past the grazing bifurcation.

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arXiv2026

The stability of boundary equilibria of three-dimensional Filippov systems

For three-dimensional piecewise-smooth systems of ordinary differential equations, this paper characterises the stability of points that belong to a switching surface and are equilibria of exactly one of the two neighbouring pieces of the system. Stability is challenging to characterise when nearby orbits repeatedly switch between regular motion on one side of the switching surface, and sliding motion on the switching surface, as defined via Filippov's convention. We prove that in this case stability is governed by the behaviour of a global reinjection mechanism of a four-parameter family of piecewise-linear hybrid systems, and perform a detailed numerical study of this family.

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arXiv2025

The VIVID function for numerically continuing periodic orbits arising from grazing bifurcations of hybrid dynamical systems

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 tangentially with a boundary of phase space. Failure occurs when the map contains a square-root singularity and the root-finding algorithm searches beyond the domain of viable values. We show that by instead following the zeros of a function that maps Velocity Into Variation In Displacement (VIVID) this issue is circumvented and there is no such failure. We illustrate this with a prototypical one-degree-of-freedom impact oscillator model by applying Newton's method to the VIVID function to follow periodic orbits collapsing into grazing bifurcations. We also follow curves of saddle-node and period-doubling bifurcations of periodic orbits that issue from a codimension-two resonant grazing bifurcation. The VIVID function provides a simple alternative to the more sophisticated collocation method and enables periodic orbits and their bifurcations to be resolved easily and accurately near grazing bifurcations.

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arXiv2024

Three forms of dimension reduction for border-collision bifurcations

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.

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arXiv2024

A piecewise-linear fixed point theorem

We prove that if a continuous piecewise-smooth map on $\mathbb{R}^n$ is comprised of two linear functions, has a bounded orbit, and satisfies a certain non-degeneracy condition, then it has a fixed point. The result has important consequences to the bifurcation theory of nonsmooth dynamical systems, yet the proof requires only elementary linear algebra.

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arXiv2024

Robust chaos in $\mathbb{R}^n$

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 correspond to an open set within the space of all such maps, allow all $n \ge 2$, and allow all possible values for the unstable eigenvalues in the limit that all stable eigenvalues tend to zero. To prove an attractor exists we use the stable manifold of a fixed point to construct a trapping region; to prove the attractor is chaotic we use the unstable directions to construct an invariant expanding cone for the derivatives of the pieces of the map. We also show the chaotic attractor is persistent under nonlinear perturbations, thus when such an attractor is created locally in a border-collision bifurcation of a general piecewise-smooth system, it persists and is chaotic for an interval of parameter values beyond the bifurcation.

nlin.CD↗

arXiv2024

The two-dimensional border-collision normal form with a zero determinant

The border-collision normal form is a piecewise-linear family of continuous maps that describe the dynamics near border-collision bifurcations. Most prior studies assume each piece of the normal form is invertible, as is generic from an abstract viewpoint, but in applied problems one piece of the map often has degenerate range, corresponding to a zero determinant. This provides simplification, yet even in two dimensions the dynamics can be incredibly rich. The purpose of this paper is to determine broadly how the dynamics of the two-dimensional border-collision normal form with a zero determinant differs for different values of its parameters. We identify parameter regions of period-adding, period-incrementing, mode-locking, and component doubling of chaotic attractors, and characterise the dominant bifurcation boundaries. The intention is for the results to enable border-collision bifurcations in mathematical models to be analysed more easily and effectively, and we illustrate this with a flu epidemic model and two stick-slip friction oscillator models. We also describe three novel bifurcation structures that remain to be explored.

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arXiv2024

The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps

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.

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arXiv2023

The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps

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: {\em persistence}, where the symbolic itinerary of a periodic solution changes by one symbol, and a {\em 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 $\mathbb{R}^n$ 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.

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arXiv2023

Robust chaos in orientation-reversing and non-invertible two-dimensional piecewise-linear maps

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 chaotic attractor throughout an open region of parameter space. This was achieved by constructing a trapping region in phase space and an invariant expanding cone in tangent space, but only allowed parameter combinations for which the normal form is invertible and orientation-preserving. This paper generalises the construction to include the non-invertible and orientation-reversing cases. This provides a more complete and unified picture of robust chaos by revealing its presence to be disassociated from the global topological properties of the map. We identify a region of parameter space in which the map exhibits robust chaos, and show that part of the boundary of this region consists of bifurcation points at which the chaotic attractor is destroyed.

nlin.CD↗

arXiv2022

A synopsis of the non-invertible, two-dimensional, border-collision normal form with applications to power converters

The border-collision normal form is a canonical form for two-dimensional, continuous maps comprised of two affine pieces. In this paper we provide a guide to the dynamics of this family of maps in the non-invertible case where the two pieces fold onto the same half-plane. We identify parameter regimes for the occurrence of key bifurcation structures, such as period-incrementing, period-adding, and robust chaos. We then apply the results to a classic model of a boost converter for adjusting the voltage of direct current. It is known that for one combination of circuit parameters the model exhibits a border-collision bifurcation that mimics supercritical period-doubling and is non-invertible due to the switching mechanism of the converter. We find that over a wide range of parameter values, even though the dynamics created in border-collision bifurcations is in general extremely diverse, the bifurcation in the boost converter can only mimic period-doubling, although it can be subcritical.

nlin.CD↗

arXiv2022

Border-collision bifurcations from stable fixed points to any number of coexisting chaotic attractors

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 a switching manifold as parameters are varied. The purpose of this paper is to highlight the extreme complexity possible in the subsequent dynamics. We perturb instances of the border-collision normal form in $n \ge 2$ dimensions for which the $n^{\rm th}$ iterate is a direct product of identical skew tent maps that have chaotic attractors comprised of $k \ge 2$ disjoint intervals. The resulting maps have coexisting attractors and we use Burnside's lemma to count the number of mutually disjoint trapping regions produced by taking unions of Cartesian products of slight enlargements of the disjoint intervals. The attractors are shown to be chaotic by demonstrating that some iterate of the map is piecewise-expanding. The resulting transition from a stable fixed point to many coexisting chaotic attractors is shown to occur throughout open subsets of parameter space and not destroyed by adding higher order terms to the normal form, hence can be expected to arise generically in mathematical models.

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arXiv2022

Normal forms, differentiable conjugacies and elementary bifurcations of maps

We strengthen the standard bifurcation theorems for saddle-node, transcritical, pitchfork, and period-doubling bifurcations of maps. Our new formulation involves adding one or two extra terms to the standard truncated normal forms with coefficients determined by algebraic equations. These extended normal forms are differentiably conjugate to the original maps on basins of attraction and repulsion of fixed points or periodic orbits. This reflects common assumptions about the additional information in normal forms despite standard bifurcation theorems being formulated only in terms of topological equivalence.

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arXiv2021

Robust Devaney chaos in the two-dimensional border-collision normal form

The collection of all non-degenerate, continuous, two-piece, piecewise-linear maps on $\mathbb{R}^2$ 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 parameter space this family has an attractor satisfying Devaney's definition of chaos. This strengthens existing results on the robustness of chaos in piecewise-linear maps. We further show that the stable manifold of a saddle fixed point, despite being a one-dimensional object, densely fills an open region containing the attractor. Finally we identify a heteroclinic bifurcation, not described previously, at which the attractor undergoes a crisis and may be destroyed.

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