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The geometry of generic sliding bifurcations
 SIAM Review
, 2011
"... Abstract. Using the singularity theory of scalar functions, we derive a classification of sliding bifurcations in piecewisesmooth flows. These are global bifurcations which occur when distinguished orbits become tangent to surfaces of discontinuity, called switching manifolds. The key idea of the p ..."
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Cited by 15 (11 self)
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Abstract. Using the singularity theory of scalar functions, we derive a classification of sliding bifurcations in piecewisesmooth flows. These are global bifurcations which occur when distinguished orbits become tangent to surfaces of discontinuity, called switching manifolds. The key idea of the paper is to attribute sliding bifurcations to singularities in the manifold’s projection along the flow, namely to points where the projection contains folds, cusps, and twofolds (saddles and bowls). From the possible local configurations of orbits we obtain sliding bifurcations. In this way we derive a complete classification of generic oneparameter sliding bifurcations at a smooth codimension one switching manifold in ndimensions for n ≥ 3. We uncover previously unknown sliding bifurcations, all of which are catastrophic in nature. We also describe how the method can be extended to sliding bifurcations of codimension two or higher.
Bifurcations of piecewise smooth flows: perspectives, methodologies and open problems
, 2011
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Canards and curvature: nonsmooth approximation by pinching
, 2011
"... In multiple timescale (singularly perturbed) dynamical systems, canards are counterintuitive solutions that evolve along both attracting and repelling invariant manifolds. In two dimensions, canards result in periodic oscillations whose amplitude and period grow in a highly nonlinear way: they ar ..."
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Cited by 11 (9 self)
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In multiple timescale (singularly perturbed) dynamical systems, canards are counterintuitive solutions that evolve along both attracting and repelling invariant manifolds. In two dimensions, canards result in periodic oscillations whose amplitude and period grow in a highly nonlinear way: they are slowly varying with respect to a control parameter, except for an exponentially small range of values where they grow extremely rapidly. This sudden growth, called a canard explosion, has been encountered in many applications ranging from chemistry to neuronal dynamics, aerospace engineering and ecology. Canards were initially studied using nonstandard analysis, and later the same results were proved by standard techniques such as matched asymptotics, invariant manifold theory and parameter blowup. More recently, canardlike behaviour has been linked to surfaces of discontinuity in piecewisesmooth dynamical systems. This paper provides a new perspective on the canard phenomenon by showing that the nonstandard analysis of canard explosions can be recast into the framework of piecewisesmooth dynamical systems. An exponential coordinate scaling is applied to a singularly perturbed system of ordinary differential equations. The scaling acts as a lens that resolves dynamics across all timescales. The changes of local curvature that are responsible for canard explosions are then analyzed. Regions where different timescales dominate are separated by hypersurfaces, and these are pinched together to obtain a piecewisesmooth system, in which curvature changes manifest as discontinuityinduced bifurcations. The method is used to classify canards in arbitrary dimensions, and to derive the parameter values over which canards form either small cycles (canards without head) or large cycles (canards with head).
Teixeira singularities in 3D switched feedback control systems.
, 2010
"... This paper is concerned with the analysis of a singularity that can occur in threedimensional discontinuous feedback control systems. The singularity is the twofold – a tangency of orbits to both sides of a switching manifold. Particular attention is placed on the Teixeira singularity, which previ ..."
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Cited by 9 (6 self)
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This paper is concerned with the analysis of a singularity that can occur in threedimensional discontinuous feedback control systems. The singularity is the twofold – a tangency of orbits to both sides of a switching manifold. Particular attention is placed on the Teixeira singularity, which previous literature suggests as a mechanism for dynamical transitions in this class of systems. We show that such a singularity cannot occur in classical singleinput singleoutput systems in the Lur’e form. It is, however, a potentially dangerous phenomenon in multipleinput multipleoutput switched control systems. The theoretical derivation is illustrated by means of a representative example.
Twofolds in nonsmooth dynamical systems
"... Abstract: In a three dimensional dynamical system with a discontinuity along a codimension one switching manifold, orbits of the system may be tangent to both sides of the switching manifold generically at isolated points. It is perhaps surprising, then, that examples of such ‘twofold ’ singulariti ..."
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Cited by 3 (1 self)
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Abstract: In a three dimensional dynamical system with a discontinuity along a codimension one switching manifold, orbits of the system may be tangent to both sides of the switching manifold generically at isolated points. It is perhaps surprising, then, that examples of such ‘twofold ’ singularities are difficult to find amongst physical models. They occur where the relative curvature between the flow field and the switching manifold is nonsymmetric about the discontinuity. Here we motivate their study with a local form model of nonlinear control that exhibits the twofold singularity, where the flow is constant either side of a curved switching manifold. We describe the local dynamics around general twofold singularities, then consider their effect on global dynamics via one parameter bifurcations of limit cycles.
The twofold singularity of nonsmooth flows: leading order dynamics in ndimensions
, 2013
"... A discontinuity in a system of ordinary differential equations can create a flow that slides along the discontinuity locus. Prior to sliding, the flow may have collapsed onto the discontinuity, making the reverse flow nonunique, as happens when dryfriction causes objects to stick. Alternatively, a ..."
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A discontinuity in a system of ordinary differential equations can create a flow that slides along the discontinuity locus. Prior to sliding, the flow may have collapsed onto the discontinuity, making the reverse flow nonunique, as happens when dryfriction causes objects to stick. Alternatively, a flow may slide along the discontinuity before escaping it at some indeterminable time, implying nonuniqueness in forward time. At a twofold singularity these two behaviours are brought together, so that a single point may have multiple possible futures as well as histories. Twofolds are a generic consequence of discontinuities in three or more dimensions, and play an important role in both local and global dynamics. Despite this, until now nothing was known about twofold singularities in systems of more than 3 dimensions. Here, the normal form of the twofold is extended to higher dimensions, where we show that much of its lower dimensional dynamics survives. 1
Hidden dynamics in models of discontinuity and switching
, 2014
"... Sharp switches in behaviour, like impacts, stickslip motion, or electrical relays, can be modelled by differential equations with discontinuities. A discontinuity approximates fine details of a switching process that lie beyond a bulk empirical model. The theory of piecewisesmooth dynamics descri ..."
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Sharp switches in behaviour, like impacts, stickslip motion, or electrical relays, can be modelled by differential equations with discontinuities. A discontinuity approximates fine details of a switching process that lie beyond a bulk empirical model. The theory of piecewisesmooth dynamics describes what happens assuming we can solve the system of equations across its discontinuity. What this typically neglects is that effects which are vanishingly small outside the discontinuity can have an arbitrarily large effect at the discontinuity itself. Here we show that such behaviour can be incorporated within the standard theory through nonlinear terms, and these introduce multiple sliding modes. We show that the nonlinear terms persist in more precise models, for example when the discontinuity is smoothed out. The nonlinear sliding can be eliminated, however, if the model contains an irremovable level of unknown error, which provides a criterion for systems to obey the standard Filippov laws for sliding dynamics at a discontinuity. 1 Dynamics at a jump It is common to assume that underlying any physical system are a set of welldetermined, and moreorless smoothly varying, physical laws. Nevertheless, smooth variations can give rise to discontinuities by means of, for example, bifurcations, shocks, or singular perturbations. Discontinuities are a common feature of empirical models in engineering and biology particularly, for example in rigid body impact, stickslip due to friction, and switches in electrical, biochemical, or social dynamics. The question arises: if an observer is able to reconstruct a set of physical laws only at the piecewisesmooth level, i.e. to the extent that they involve a discontinuity, to what extent can the system dynamics be uniquely determined? The key to handling switches in dynamical systems lies in recognising that a discontinuous vector field places certain restrictions on the flow it generates.
Link to publication record in Explore Bristol Research
"... Early version, also known as preprint Link to published version (if available): ..."
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Early version, also known as preprint Link to published version (if available):