D R J Chillingworth
email: drjc@maths.soton.ac.uk

Dynamics of an impact oscillator near a degenerate graze

We give a complete analysis of low-velocity dynamics close to grazing for a generic one degree of freedom impact oscillator. This includes nondegenerate (quadratic) grazing and minimally degenerate (cubic) grazing, corresponding respectively to nondegenerate and degenerate chatter. We also describe the dynamics associated with generic oneparameter bifurcation at a more degenerate (quartic) graze, showing in particular how this gives rise to the often-observed highly convoluted structure in the stable manifolds of chattering orbits. The approach adopted is geometric, using methods from singularity theory.

Introduction

The theory of forced nonlinear oscillations has well-known and widespread applications in science and engineering: see Hayashi [START_REF] Hayashi | Nonlinear Oscillations in Physical Systems[END_REF], Fidlin [START_REF] Fidlin | Nonlinear Oscillations in Mechanical Engineering[END_REF], for example, as well as landmark texts Andronov et al [START_REF] Andronov | Theory of Oscillators[END_REF], Schmidt and Tondl [START_REF] Schmidt | Non-Linear Vibrations[END_REF], Guckenheimer and Holmes [START_REF] Guckenheimer | Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields[END_REF]. However, there are many engineering contexts in which oscillations are a highly undesirable feature of performance, especially when these lead to impacts between moving parts or against rigid supports, generating noise and wear and ultimately leading to mechanical breakdown: for some examples see Alzate et al [START_REF] Alzate | Experimental and numerical verification of bifurcations and chaos in cam-follower impacting systems[END_REF], Mason et al [START_REF] Mason | Mathematical models of gear rattle in Roots blower vacuum pumps[END_REF], Stewart [START_REF] Stewart | Rigid-body dynamics with friction and impact[END_REF], Pavlovskaia and Wiercigroch [START_REF] Pavlovskaia | Modelling of vibro-impact system driven by beat frequency[END_REF] as well as those described in books such as Babitsky and Birkett [START_REF] Babitsky | Theory of Vibro-Impact Systems and Applications[END_REF], Brogliato [START_REF] Brogliato | [END_REF] and di Bernardo et al [START_REF] Di Bernardo | Piecewise-Smooth Dynamical Systems: Theory and Applications[END_REF].

A simple model for such dynamical impact phenomena (often called vibro-impact systems) is the one degree of freedom impact oscillator. This consists of a forced nonlinear oscillator ẍ + f (x, ẋ) = g(t) [START_REF] Alzate | Experimental and numerical verification of bifurcations and chaos in cam-follower impacting systems[END_REF] for x ∈ R, together with a constraint x c where c is a constant often called the clearance.

Here f and g are smooth 1 functions, and as usual the dot denotes d/dt. The model must include a prescription for how the dynamics are to proceed when x = c. This will typically be given by a restitution rule that whenever (x, ẋ) = (c, v) then the velocity v is instantaneously replaced by ṽ (a function of v and perhaps other variables) satisfying ṽv < 0. The simplest practical example is a linear rule with ṽ = -rv where 0 r 1, which includes the cases of perfect adherence (r = 0) and perfectly elastic restitution (r = 1). There are many variations on this model, involving more than one degree of freedom and/or more than one constraint or a noninstantaneous restitution rule. In this paper we shall require only that the linearization of ṽ at v = 0 takes the form -rv with 0 < r < 1; thus we exclude the extreme cases.

The fact that a simple impact system can give rise to extremely complicated dynamics has long been noted, and there are many analytical and numerical studies showing complicated periodicities or convoluted orbit structures in phase space: see for example Thompson and Ghaffari [44], Shaw and Holmes [START_REF] Shaw | A periodically forced piecewise linear oscillator[END_REF], Hindmarsh and Jeffries [START_REF] Hindmarsh | On the motion of the offset impact oscillator[END_REF], Peterka and Vacík [START_REF] Peterka | Transition to chaotic motion in mechanical systems with impacts[END_REF], Kleczka et al [START_REF] Kleczka | Local and global stability of a piecewise linear oscillator Phil[END_REF], Virgin and Begley [START_REF] Virgin | Grazing bifurcations and basins of attraction in an impact-friction oscillator[END_REF], Wagg and Bishop [START_REF] Wagg | Chatter sticking and chaotic impacting motion in a two-degree of freedom impact oscillator[END_REF][START_REF] Wagg | Dynamics of a two degree of freedom vibro-impact system with multiple motion limiting constraints[END_REF], Valente et al [START_REF] Valente | Hybrid dynamics of two coupled oscillators that can impact a fixed stop[END_REF]. Nevertheless, even a one degree of freedom linear oscillator with a linear restitution rule is still poorly understood from the overall dynamical point of view.

A more systematic attack on the global problem was undertaken by Whiston in a pioneering series of papers [START_REF] Whiston | The vibro-impact response of a harmonically excited and preloaded one-dimensional linear oscillator[END_REF][START_REF] Whiston | Global dynamics of a vibro-impacting linear oscillator[END_REF][START_REF] Whiston | Singularities in vibro-impact dynamics[END_REF]; this enterprise inspired much subsequent work including Budd and Dux [START_REF] Budd | The effect of frequency and clearance variations on single-degree-offreedom impact oscillators[END_REF], Lamba [START_REF] Lamba | Chaotic regular and unbounded behaviour in the elastic impact oscillator[END_REF], Nordmark [START_REF] Nordmark | Non-periodic motion caused by grazing incidence in an impact oscillator[END_REF] as well as this paper.

A key feature of impact oscillator dynamics that gives it a different flavour from (smooth) nonlinear dynamics is the phenomenon of grazing where a periodic orbit reaches the obstacle with zero velocity and nonzero acceleration: small perturbations of such orbits may or may not exhibit impacts. The resulting bifurcation phenomena have been studied by many authors including Budd and Dux [START_REF] Budd | Chattering and related behaviour in impact oscillators Phil[END_REF], Chin et al [START_REF] Chin | Grazing bifurcation in impact oscillators[END_REF], Dankowitz and Jerrelind [START_REF] Dankowicz | Control of near-grazing dynamics in impact oscillators[END_REF], Foale and Bishop [START_REF] Foale | Dynamical complexities of forced impacting systems Phil[END_REF], Ivanov [START_REF] Ivanov | Impact oscillations: linear theory of stability and bifurcations[END_REF], Nordmark [START_REF] Nordmark | Non-periodic motion caused by grazing incidence in an impact oscillator[END_REF][START_REF] Nordmark | Universal limit mapping in grazing bifurcations[END_REF], Szalai and Osinga [START_REF] Szalai | Unstable manifolds of a limit cycle near grazing[END_REF], Molenaar et al [START_REF] Molenaar | Mappings of grazing-impact oscillators[END_REF], Zhao and Dankowicz [START_REF] Zhao | Unfolding degenerate grazing dynamics in impact actuators[END_REF].

Another important aspect of impact oscillator dynamics that attracts particular interest is the low-velocity behaviour close to impact, characterized by the phenomenon of chatter. On leaving x = c with small positive velocity but negative acceleration the orbit (i.e. the solution to [START_REF] Alzate | Experimental and numerical verification of bifurcations and chaos in cam-follower impacting systems[END_REF]) may return to hit x = c again after a short time; after restitution at the impact the process will repeat, and under suitable conditions the orbit may come to rest at x = c after a finite time but an infinite number of such impacts. It will remain stuck at the obstacle x = c until such time as the acceleration (from the forcing) becomes positive and it once again lifts off into the region x > c. See for example Demeio and Lenci [START_REF] Demeio | Asymptotic analysis of chattering oscillations for an impacting inverted pendulum Q[END_REF] for a simple mechanical realization of this phenomenon or Stone and Askari [START_REF] Stone | Nonlinear models of chatter in drilling processes[END_REF], Valente et al [START_REF] Valente | Hybrid dynamics of two coupled oscillators that can impact a fixed stop[END_REF] for more complicated dynamical settings. The first systematic analytic and geometrical analysis of the dynamics of chatter was carried out by Budd et al [START_REF] Budd | Chattering and related behaviour in impact oscillators Phil[END_REF], using a combination of insightful numerics supported theoretically by first and second order approximations. In this paper we extend those results by providing a complete and rigorous analysis of chatter for generic impact oscillator systems [START_REF] Alzate | Experimental and numerical verification of bifurcations and chaos in cam-follower impacting systems[END_REF] and for generic 1-parameter families of such systems.

In many applications the function g in ( 1) is periodic with period T > 0. In this case a standard technique for studying a forced oscillator such as [START_REF] Alzate | Experimental and numerical verification of bifurcations and chaos in cam-follower impacting systems[END_REF] without the impacts is to focus on the the discrete dynamical system F that samples the solutions at times {nT : n ∈ Z}: then periodic solutions of (1) correspond to periodic orbits of F , and so on. When impacts are introduced, however, this is no longer appropriate since the influence of the obstacle on a given trajectory cannot be expected to be T -periodic. A common approach to dealing with this problem, used by Shaw and Holmes [START_REF] Shaw | A periodically forced piecewise linear oscillator[END_REF], Thompson and Ghaffari [44], Whiston [START_REF] Whiston | Global dynamics of a vibro-impacting linear oscillator[END_REF] and others, is first to regard (1) as an autonomous system in R 3 by including the equation ṫ = 1 and then to consider the plane x = c and the discrete dynamical system defined on it by the first return of trajectories after impacting the obstacle. This fits naturally with the geometry of the physical description of (1), but the resulting discrete system now has loci of discontinuity that arise from grazing trajectories. It is not easy in this setting to see how the discontinuities and the associated dynamical phenomena respond to change in initial conditions and/or governing parameters such as the clearance.

In this paper we use an alternative geometric approach, first set out in [START_REF] Chillingworth | Discontinuity geometry for an impact oscillator[END_REF], which allows greater insight into the geometry of discontinuities and their influence on dynamics by replacing the picture of curved orbits intersecting a flat obstacle with an equivalent picture of straightenedout orbits intersecting a curved obstacle: the impact surface. See section 2 for the precise description. From this point of view, the problem of analysing the geometry of orbits at and near grazing becomes more readily amenable to the tools of local differential topology, dynamical systems and singularity theory.

Taking this approach, we look again at the phenomena of grazing and chatter in the light of the geometry of the impact surface. We first use this to provide rigorous underpinning for the key results in [START_REF] Budd | Chattering and related behaviour in impact oscillators Phil[END_REF] that are based on approximations: by placing the results in a wider context we are able to give a full analysis of the local dynamical phenomena. We then extend this to cover the case of the discontinuous bifurcations that arise in the neighbourhood of a doubly degenerate graze as the clearance or some other parameter is varied and the degeneracy is unfolded. In particular, we exhibit the key mechanism for the creation of multiple-loop structures in stable manifolds of chatter points, which is a first step in understanding the multiple-loop patterns observed numerically for attractors and basins of attraction of periodic orbits in impact oscillators: see [50, figure 17], [14, figures 4 and 19], as well as [6, figure 3] and [START_REF] Di Bernardo | Piecewise-Smooth Dynamical Systems: Theory and Applications[END_REF] (in particular, the front cover). Taken together, the results of this paper provide a complete description of generic low-velocity impact dynamics and 1-parameter bifurcations.

Geometrical model for dynamics: the impact surface

We begin by recalling the construction described in Chillingworth [START_REF] Chillingworth | Discontinuity geometry for an impact oscillator[END_REF] for modelling the overall dynamics of a one degree of freedom impact oscillator. The model is based on a twodimensional surface in three dimensions that characterizes the impacts, together with certain natural maps associated with it that represent the dynamics. A similar geometrical construction was used by Sotomayor and Teixeira [START_REF] Sotomayor | Vector Fields Near the Boundary of a 3-Manifold[END_REF], albeit with a different dynamical interpretation, to classify local vector fields on a manifold with boundary.

The impact surface and associated maps

Let x(b, v, τ ; t) denote the unique solution to [START_REF] Alzate | Experimental and numerical verification of bifurcations and chaos in cam-follower impacting systems[END_REF]

with initial data (x, ẋ) = (b, v) when t = τ. For fixed choice of c ∈ R we then write x c (τ, v; t) := x(c, v, τ ; τ + t) so that x c (τ, v; 0) = c, ẋc (τ, v; 0) = v. ( 2
)
Definition 2.1. The impact surface V c is the locus of points among all the solutions where impact occurs, that is

V c := {(τ, v; t) ∈ R 3 : x c (τ, v; t) = c}.
Note that in [START_REF] Chillingworth | Discontinuity geometry for an impact oscillator[END_REF] the convention was to write (v, τ ; t) rather than (τ, v; t): we trust no confusion will arise from this notation reversal.

By definition V c contains the plane t = 0, which we shall denote by and which plays a key role in our formulation of the dynamics. It is proved in [16, lemma 1] that V c is indeed a surface (smooth 2-manifold), with the exception of the points on the τ -axis 0 and possibly certain other points where v = 0, characterized precisely in terms of the functions f and g in ( 1) and the clearance c. It is shown2 in [16, proposition 1] (see also [START_REF] Budd | Chattering and related behaviour in impact oscillators Phil[END_REF] where it is inherent in the analysis, and see the calculations below) that in a neighbourhood of 0 the impact surface V c consists of two sheets (smooth 2-manifolds) intersecting transversely along 0 . One of these is of course the plane while the other sheet V c is represented in a neighbourhood of 0 as the graph of a smooth function v = v c (τ, t) given implicitly by

y c (τ, v c ; t) = 0 (3)
when x c is written in the form

x c = c + ty c (τ, v; t) (4)
for a smooth function y c on a neighbourhood of 0 in R 3 . By implicit differentiation of (3) we find

∂ ∂t v c (τ, t)| t=0 = - 1 2 a c (τ ), (5)
where a c (τ ) denotes the initial acceleration ẍc (τ, 0; 0) (again see [START_REF] Budd | Chattering and related behaviour in impact oscillators Phil[END_REF] as well as [16, proposition 1]).

There are three maps which we use to build the dynamical model for the impact oscillator. We now describe each of these in turn. 3 to the plane of initial data, and let p c denote its restriction to V c :

The projection p c . Consider the projection

p : R 3 → : (τ, v; t) → (τ, v) from R
p c := p|V c : V c → .
Let H c be the set of singular points of p c : V c → (including those, if any, where V c itself may be singular). From the implicit function theorem it is immediate that

H c = {(τ, v; t) ∈ V c : ẋc (τ, v; t) = 0}
which in geometric terms is the horizon of V c as viewed along the t-direction. Dynamically, the points of H c represent graze points, that is points where the solution to (1) meets the obstacle x = c with zero velocity. See figure 1, where for diagrammatic simplicity the curve H c is not distinguished from the horizon of V c as seen from the viewpoint of the reader: in reality these are of course different. The subset P c := p(H c ) of is the apparent outline (classically the apparent contour) of the surface V c in the t-direction: see Bruce [START_REF] Bruce | Seeing-the mathematical viewpoint Math[END_REF], Bruce and Giblin [START_REF] Bruce | Curves and Singularities[END_REF].

The generic local geometry of smooth maps from a smooth surface M into R 2 is well understood since the work of Whitney [START_REF] Whitney | On singularities of mappings of Euclidean spaces: I. Mappings of the plane into the plane[END_REF]. The set of singular points is a smooth 1-manifold in M consisting of arcs of fold points (where the singular set maps locally to a smooth 1-manifold) and isolated cusp points (where the singular set maps locally to a 3 2 -power cusp). The precise characterizations are given in terms of partial derivatives, and local coordinates can be chosen so that at fold points or cusp points the map has the form (x, y) → (x, y 2 ) or (x, y) → (x, xy + y 3 ), respectively. These generic results apply also to the conceivably more restricted setting of projection of a surface in R 3 to a plane, and thus to the geometry of apparent outlines. In figure 1 the points w 1 , w 3 are images of fold points, while w 2 is the image of a cusp point.

As the surface or the view direction varies with one or more parameters the outline will typically undergo certain local transitions. The generic 1-parameter transitions in apparent outlines are also well understood: see [START_REF] Arnol'd V I | Catastrophe Theory[END_REF][START_REF] Bruce | Motion pictures: an application of singularity theory[END_REF][START_REF] Bruce | Outlines and their duals[END_REF] and figure 2.

In general, without any restriction on t, there may be regions of filled densely by arcs of P c . However, if V c is periodic in t (as may happen for undamped linear systems with periodic forcing) or we restrict to a compact subset of V c then this complication will not arise.

It is pertinent to ask whether for arbitrary choice of clearance c the apparent outline of the impact surface V c , which is determined by solutions of the differential equation ( 1), is indeed generic as just described, and whether the transitions in the apparent outline as c varies are also generic in this context. These questions may be posed both for a general system (1) and for a specific system such as the forced harmonic oscillator ẍ + x = cos ωt.

(

A detailed discussion of the generic geometry of V c and its outline for the general system (1) and for the linear system (6) can be found in [START_REF] Chillingworth | Discontinuity geometry for an impact oscillator[END_REF] where these questions are answered.

The re-set map φ c .

There is a natural smooth re-set map φ c : V c → obtained simply by re-setting the clock to t = 0 at the moment of impact: Let Z c denote the zero set of V c , that is the intersection of V c with the plane v = 0. The following facts about the singularity structure of φ c are proved in [16, propositions 6,7].

φ c : V c → : (τ, v; t) → (τ + t, ẋc (τ, v; t)).
Proposition 2.2. The singular set of φ c is Z c \ 0 . Points (τ, 0; t) ∈ Z c \ 0 where a c (τ ) = 0 are fold points of φ c , while points where a c (τ ) = 0 but a c (τ ) = 0 are cusp points.

2.1.3. The restitution map R c . The restitution map describes how the velocity immediately after impact depends on the velocity immediately before impact. Thus we write

+ = {(τ, v) ∈ : v > 0}, + 0 = + ∪ 0 , -= {(τ, v) ∈ : v < 0}, - 0 = -∪ 0 and we suppose that R c : - 0 → + 0 is the restriction to - 0 of a map R c : → : (τ, v) → (τ, ρ c (v)),
where ρ c : R → R is a differentiable function with ρ c (0) = 0 and ρ c (0) < 0. In many cases an appropriate choice is ρ c (v) = -rv with 0 < r < 1 independent of c, although all that we shall assume is that ρ c (0) = -r.

The first-hit map F c and the dynamical system G c

Since we are assuming the dynamics take place in the region x c it is only the upper halfplane + 0 that is of physical interest. Nevertheless, in much of what follows it will be useful first to consider as a whole and then restrict to + 0 later. The projection p c : V c → has a partial right inverse constructed as follows. Choose initial data (τ, v) ∈ + (so x = c and ẋ = v > 0 at t = 0), then proceed parallel to the t-axis in the direction of increasing t until the point F c (τ, v) of first intersection (first hit) with V c is reached: thus

F c (τ, v) = (τ, v; t 1 ) ∈ V c , where (τ, v; t) / ∈ V c for 0 < t < t 1 .
It is possible that no such t 1 exists (no future impacts occur), but for notational simplicity we disregard such points and write

F c : + → V c
in any case, with the understanding that F c may not be defined on the whole of + .

The construction of F c extends naturally to points (τ, 0) ∈ 0 for which x c (τ, 0; t) > 0 for all sufficiently small t > 0; if this does not hold (for example, if ẍc (τ, 0; 0) < 0) then the constrained orbit remains stationary at x = c and we define F c (τ, 0) = (τ, 0). The points of V c in the image of F c : + 0 → V c will be called visible points, as these are the points of V c that can be 'seen' from + 0 along the positive t direction. Finally, although it is not needed for dynamical purposes, for geometric convenience we define F c : -→ V c analogously, proceeding here in the direction of negative t.

The first-hit map F c : → V c defined in this way typically has discontinuities. Since p c is a local diffeomorphism at points z ∈ V c \ H c it follows that the discontinuities of F c must lie in the apparent outline P c = p(H c ) or on 0 . Not all points w ∈ P c are discontinuity points of F c , of course, but only those with F c (w) ∈ H c : thus in figure 1 the point w 1 is a point of discontinuity of F c while the point w 3 is not.

The correspondence principle

The task of describing the dynamics of the impact oscillator close to grazing is greatly helped by exploiting a correspondence, induced naturally by φ c , between geometric features of P c close to 0 and those away from 0 .

If z ∈ H c then p c (z) ∈ P c by definition, and since ẋc = 0 at z we have φ c (z) ∈ 0 . Likewise if z ∈ Z c then p c (z) ∈ 0 by definition, and it is also the case that φ c (z) ∈ P c . This is because z represents current data (on x = c) for a solution with initial velocity v = 0 so, by uniqueness of solutions of (1), after re-set the point φ c (z) represents initial data for a solution x(t) with x(t) = c and with ẋ(t) = 0 at some other time t. Thus

φ c (H c ) = 0 = p c (Z c ) and p c (H c ) = P c = φ c (Z c ).
In fact by extending this argument we obtain a more general result which we leave as an exercise:

Proposition 2.3.

p -1 c (P c ∪ 0 ) = φ -1 c (P c ∪ 0 ).
Furthermore, since re-setting the time clearly does not affect geometric features of the dynamics, any features of the outline P c in a neighbourhood of w = p c (z) ∈ P c where z ∈ H c must have counterparts in a neighbourhood of φ c (z) ∈ 0 . This correspondence principle between P c and 0 enables us to focus the analysis on points of 0 when studying local dynamics close to grazing, and (as we see below) provides a valuable framework for understanding the structure of the dynamics itself. A fuller description of the geometry associated with the correspondence principle can be found in [START_REF] Chillingworth | Discontinuity geometry for an impact oscillator[END_REF].

Construction of the dynamics

We are now ready to construct the (discontinuous) discrete dynamical system on the half-plane + 0 that we will use to model the dynamics of the impact oscillator.

Begin by applying the first-hit map F c : + 0 → V c . This corresponds to following a solution of (1) as it leaves the obstacle at x = c until it returns to x = c for the first time. Next, apply the re-set map φ c : V c → ; this is simply a re-calibration of the time variable. Since the initial data lie in + 0 the velocity at first hit must be negative or zero, so we have φ c • F c : + 0 → - 0 . Finally, apply the restitution map R c : - 0 → + 0 and consider the composition:

+ 0 F c → V c φ c → - 0 R c → + 0 .
The iteration of this process reconstructs the impact dynamics of the oscillator as a (discontinuous) discrete dynamical system on + 0 generated by the map

G c := R c • φ c • F c : + 0 → + 0 .
As indicated above, we may regard F c and R c and hence also G c as maps defined on the whole of whenever this is convenient.

The rest of this paper is devoted to describing fully the generic behaviour of iterations of G c close to the τ -axis 0 , both for fixed clearance c and as a 1-parameter family of dynamical systems when c or another parameter is varied. We take the partial results of [START_REF] Budd | Chattering and related behaviour in impact oscillators Phil[END_REF] further by giving a rigorous derivation of the dynamics for nondegenerate chatter at regular points of 0 (that is where a c (τ ) = 0) and degenerate chatter at tangency points of 0 (where a c (τ ) = 0 but the derivative a c (τ ) = 0). We also consider a yet more degenerate situation that generically arises only for certain values of the clearance c at so-called swan points of 0 (where a c (τ ) = a c (τ ) = 0 but a c (τ ) = 0), and we show how the dynamics unfold and undergo newly created discontinuities in a 1-parameter bifurcation as c passes through such values. The creation of new cusp points for p c gives rise to convoluted stable manifolds of nearby chatter points, with immediate implications for complicated global dynamical behaviour. Understanding the global structure of stable manifolds is one of the main challenges in formulating a dynamical theory for nonsmooth systems such as impact oscillators.

Nondegenerate chatter

Let the clearance c be fixed. To simplify notation in this section we temporarily write V , F, G, etc for V c , F c , G c , etc, although we retain v c for clarity and understand that p denotes p|V c . Using coordinates (τ, v) on and (τ, t) on V , we identify a point (τ * , 0; 0) of the τ -axis 0 = ∩ V with both w * = (τ * , 0) ∈ and z * = (τ * , 0) ∈ V .

Let w * = (τ * , 0) ∈ 0 be a regular point of 0 , so that a(τ * ) = 0. Since the 'free' dynamics take place in the region x > c, the behaviour of any solution that leaves x = c with zero velocity but positive acceleration will not be susceptible to our local (low-velocity) analysis close to 0 : therefore we assume a * := a(τ * ) < 0. On leaving x = c with small positive velocity the solution to (1) will slow down, then possibly reverse and hit x = c again after a short time. This is indeed the case as we see from (5): since z * / ∈ H the map p : V → is a local diffeomorphism at z * and therefore F is also a local diffeomorphism at w * . From proposition 2.2 we have that φ : V → is a local diffeomorphism at w * and so G = R •φ •F is (the restriction to + 0 of) a local diffeomorphism of at w * .

Local linearization along 0

As a first approximation to the dynamics we linearize at the equilibrium point w * ∈ 0 . Since by definition

G • p = R • φ τ τ w τ (a) (b) W s τ Π 0 Π 0 Π Π
(z * ) : (σ, t) → (σ, -1 2 a * t). Also, implicit differentiation of (3) shows that (∂v c /∂t)(z * ) = 1 2 a * so that (see [16, lemma 2]) Dφ(z * ) : (σ, t) → (σ + t, 1 2 a * t). Hence the derivative of the local diffeomorphism G = R • φ • F at w * is given by the 2 × 2 matrix DG(w * ) =   1 - 2 a * 0 r   , ( 7
)
where r = -ρ (0) is the restitution coefficient (compare [START_REF] Budd | Chattering and related behaviour in impact oscillators Phil[END_REF]). Eigenvectors of DG(w * ) are (1, 0) with eigenvalue 1 (since the τ -axis 0 is fixed) and (2, a * (1r)) with eigenvalue r. Therefore, to first order, we have the picture as illustrated in figure 3(a) (see also [START_REF] Budd | Chattering and related behaviour in impact oscillators Phil[END_REF]): orbits of the local linearization of the discrete dynamics G approach 0 along the directions with negative slope that tends to zero as a * → 0 or as r → 1.

This does not complete the story, however, as it remains to confirm that the linear description also accurately describes the nonlinear behaviour of G on close to 0 .

A fundamental component in the theory of smooth dynamical systems is the analysis of dynamics in the neighbourhood of a normally hyperbolic invariant manifold, that is an invariant manifold for which the rate of normal contraction or expansion overpowers any contraction or expansion along the manifold itself: main references are [START_REF] Hirsch | Invariant Manifolds[END_REF][START_REF] Takens | Partially hyperbolic fixed points[END_REF]. See also [START_REF] Hasselblatt | Partially hyperbolic dynamical systems Handbook of Dynamical Systems vol 1B[END_REF] for an encyclopaedic treatment.

In our case, let J be any open interval of the τ -axis 0 on which a(τ ) is negative and bounded away from zero. Then G is the identity on J and at each point w ∈ J the derivative DG(w) has a contracting eigenspace (since 0 < r < 1) transverse to 0 . Hence J is a normally hyperbolic invariant manifold for G, and from the standard theory [START_REF] Hirsch | Invariant Manifolds[END_REF] we immediately deduce the following description of the local dynamics of G on some neighbourhood U of J in . Recall, however, that for applications to the impact oscillator we are concerned only with forward iterations of G and with the region + 0 , that is where v 0.

Theorem 3.1. For each w τ = (τ, 0) ∈ J there is a smooth G-invariant curve W s τ in U ⊂ through w τ and tangent to the direction (2, (1r)a(τ )). The curve W s τ is a local stable manifold for w τ :

W s τ = {u ∈ U : G n (u) → w τ as n → +∞}. The curves {W s τ } τ ∈J provide a continuous foliation of U , in that there is a homeomorphism : J × R → U taking {τ } × R to W s
τ for each τ ∈ J . See figure 3(b). Thus the linear picture is indeed an accurate representation of the nonlinear dynamics close to 0 , although this result yields only that the stable manifold W s τ varies continuously with τ . Can we obtain more, and see that in fact W s τ varies smoothly with τ so that the union of the W s τ form a smooth foliation of a neighbourhood of 0 ? From a result of Takens [START_REF] Takens | Partially hyperbolic fixed points[END_REF] the answer turns out to be positive. Theorem 3.2. If the functions f and g in ( 1) are C ∞ then (for suitable U ) the homeomorphism in theorem 3.1 may be taken to be a C ∞ diffeomorphism.

Proof. This follows from the more general (although local) theorem in [START_REF] Takens | Partially hyperbolic fixed points[END_REF] concerning partially hyperbolic fixed points. The nonresonance conditions which are imposed in the more general setting of [START_REF] Takens | Partially hyperbolic fixed points[END_REF] are in our case vacuously satisfied since 0 < r < 1 and there is no expansion or contraction along the manifold J . Extension from the local result to the whole of J follows by usual compactness arguments applied to the closure of J .

Therefore, the dynamics of the local linearization as described in [START_REF] Budd | Chattering and related behaviour in impact oscillators Phil[END_REF] do indeed give an accurate picture of the true (nonlinear) dynamics up to smooth change of coordinates.

As a(τ ) → 0 the stable manifold W s τ approaches 0 with slope tending to zero. Our next step is to investigate the local dynamics in this limiting case.

Degenerate chatter

In this section we examine the nature of chatter close to a point w * = (τ * , 0) where now a c (τ * ) = 0. For generic choice of the functions f and g in [START_REF] Alzate | Experimental and numerical verification of bifurcations and chaos in cam-follower impacting systems[END_REF] and for an open dense set of choices of the clearance c the function a c (τ ) will have isolated zeros each with a c (τ ) = 0. In particular, this is the case for the linear system [START_REF] Di Bernardo | Bifurcations in nonsmooth dynamical systems[END_REF], since here a c (τ ) = -c + cos ωτ . However, for certain values of c the function a c (τ ) can typically be expected to undergo Morse transitions (fold bifurcations) where a c (τ ) = 0 and a c (τ ) = 0 while a c (τ ) = 0. We shall describe the local dynamics of the impact oscillator in both these situations.

We start by calculating the first few terms of the Taylor series of the maps

p c : (τ, t) → (τ, v c (τ, t)) (8) and φ c : (τ, t) → (τ + t, ẋc (τ, v c (τ, t)) (9) at points (τ, 0) ∈ 0 . Since from (4) we have v = ẋc (τ, v; 0) = y c (τ, v; 0) (10
) it follows that (∂y c /∂v) = 1 and all other derivatives of y c with respect to v and/or τ vanish on 0 . Using this, by implicit differentiation of (3) and writing τ = τ * + σ we find

v c (τ * + σ, t) = -1 2 at -1 2 a σ t + 1 2 (-1 3 a + 1 2 ab)t 2 -1 4 a σ 2 t + 1 2 (-1 3 a + 1 2 a b)σ t 2 -1 6 ( 1 4 a + 3 2 (-1 3 a + 1 2 ab)b)t 3 + O(4), ( 11
)
where a = a c (τ * ), b = (∂/∂v) ẍc (τ * , 0; 0) and the primes denote derivatives with respect to τ evaluated at τ * (which in general depend also on c). Hence, if a = 0 then

v c (τ * + σ, t) = -1 2 at + O(2) ( 12
)
while if a = 0 but a = 0 then

v c (τ * + σ, t) = -1 6 a t (3σ + t) + O(3). ( 13
)
Finally, if a = a = 0 but a = 0 then

v c (τ * + σ, t) = -1 24 a t (6σ 2 + 4σ t + t 2 ) + O(4). ( 14
)

Dynamics near a tangency point

A point (τ, 0) ∈ 0 where a c (τ ) = 0 and a c (τ ) = 0 is called a tangency point. In the context of section 3 this can be regarded as a point of 0 where the normal hyperbolicity fails in the least degenerate way. Suppose that w * = (τ * , 0) ∈ 0 is a tangency point. From [START_REF] Bruce | Outlines and their duals[END_REF] we see that v c has a saddle critical point at (σ, t) = (0, 0). The horizon H c is tangent to the line 3σ + 2t = 0 which projects from V c by p c to the graph in of a function v = 3 8 a c (τ * )σ 2 + O(3). We consider two cases separately, according to the sign of a c (τ * ). Again for convenience we drop the suffix c from the notation.

(a) The case a (τ * ) > 0. The points of H close to w * have v 0. The part of H relevant for us is the arc where t 0, that is σ 0, and so the relevant part of P = p(H ) here is a smooth arc P * quadratically tangent to 0 at w * from the direction τ < τ * and v > 0. This is the discontinuity set for the first-hit map F and hence for G locally: we call it the discontinuity arc. See figure 4.

Theorem 4.1. Let w * = (τ * , 0) be a tangency point with a (τ * ) > 0. There is a smooth arc γ ⊂ + 0 with an end-point at w * , quadratically tangent to 0 at w * from the side τ < τ * and otherwise lying between P * and 0 . The arc γ is forward invariant for G (that is G(γ ) ⊂ γ ) and is locally unique in the sense that if γ, γ are two such arcs then one is a subset of the other. The fixed point w * is exponentially attracting for G|γ , while γ is weakly normally hyperbolic (repelling) for G near w * . Orbits close to but not on γ either (if below γ ) are attracted exponentially to 0 in nondegenerate chatter or (if above γ ) cross the discontinuity arc P * after finitely many iterations of G, corresponding to loss of immediate contact with the obstacle. See figure 5.

By weakly normally hyperbolic near w * we mean that at points w ∈ γ the derivative DG(w) has an eigenvalue λ = 1 (in this case λ > 1) with eigendirection transverse to γ , although λ → 1 as w → w * along γ . The terms above and below correspond to values of v.

A proof of the existence of such an arc γ , approximately forward invariant up to second order, was given by Budd and Dux in [START_REF] Budd | Chattering and related behaviour in impact oscillators Phil[END_REF]. The analysis in [START_REF] Budd | Chattering and related behaviour in impact oscillators Phil[END_REF] is also for a particular linear system, although the authors observe that their results apply more generally. We provide here a complete proof that applies in full generality to nonlinear systems of type (1).

Proof of theorem 4.1.

From ( 8) and [START_REF] Bruce | Outlines and their duals[END_REF] we see that the projection p c : V → is given in local coordinates by where k = 1 6 a (τ * ) > 0, and from ( 9) and ( 13) the re-set map φ : V → is given by

p : (σ, t) → (σ, -k(3σ t + t 2 ) + O(3)) ( 15
)
+ t = 0 * H Z P G σ p R σ t v σ v σ v σ φ Π 0 + Π 0 _ Π 0 + Π 0 _ Π Π 0 0 + _
φ : (σ, t) → (σ + t, -k(3σ t + 2t 2 ) + O(3)). ( 16
)
Figure 4 indicates some local geometry of p and of φ. As already noted, to first order the horizon H is given by 3σ + 2t = 0, while to first order the zero set Z is given by 3σ + t = 0. Also, the line σ + t = 0 is by definition taken by φ to the v-axis σ = 0. Now take new coordinates (σ, η) in R 2 , where η = 3 2 σ + t. Then p : (σ, η) → (σ, -kη 2 + 9 4 kσ 2 + O( 3)). Composing with the diffeomorphism ψ : R 2 → R 2 : (u, w) → (u, w -9 4 ku 2 ) we obtain

ψ • p : (σ, η) → (σ, -kη 2 + O(3)),
which exhibits p explicitly as a fold map. By standard methods of singularity theory (see, e.g. Arnol'd et al [START_REF] Arnol'd V I, Gusein-Zade | Singularities of Differentiable Maps[END_REF], Golubitsky and Guillemin [START_REF] Golubitsky | Stable Mappings and their Singularities[END_REF]), after composing with local C ∞ diffeomorphisms of R 2 in range and domain that are the identity to second order, we may in fact assume that in a neighbourhood of (0, 0) we have

ψ • p : (σ, η) → (σ, -kη 2 ) ( 17
)
without any higher order terms. Let q : V → denote the composition R • φ. In the new (σ, η) coordinates we find

ψ • q : (σ, η) → (η -1 2 σ, -rk(2η 2 -3σ η) -9 4 k(η -1 2 σ ) 2 + O(3)) ( 18
)
where we recall that R(τ, v) = (τ, ρ(v)) and r = -ρ (0).

To say that γ is an invariant curve for G is equivalent to saying that the curve β = F (γ ) ⊂ V has the property that p(β) = q(β), which in turn is equivalent to saying that β is an invariant curve for the map h := p -1 • q whose domain and range are subsets of V . Explicitly we have

h = p -1 • q : (σ, η) → (η -1 2 σ, Q r (σ, η) + O(3)) (19)
where

Q r (σ, η) := r(2η 2 -3σ η) + 9 4 (η -1 2 σ ) 2 . ( 20
)
Our strategy for finding β will be to blow up the coordinates at (σ, η) = (0, 0) and then exhibit β in the new coordinates as the stable manifold of a hyperbolic fixed point of G. For the blow-up construction we write

(σ, η) = (s, sm) =: χ(s, m)
so that m is the slope of the ray from the origin on which (σ, η) lies, and s measures displacement along the ray. Then

h(s, sm) = ((m -1 2 )s, s 2 Q r (1, m) + O(s 3
)) and so using (s, m) as coordinates this becomes

h : (s, m) → (s , m ), where s = (m -1 2 )s, ( 21
) m = (m -1 2 ) -1 Q r (1, m) + O(s). ( 22
)
The domain D r of h is the subset of R 2 corresponding to the values of m for which m = 1 2 and the expression under the square root is nonnegative, for |s| sufficiently small. The following is straightforward to check.

0 < m -< 1 2 < m + < 3 2 and Q r (1, m) > 0 for m ∈ E r := (-∞, m -] ∪ [m + , ∞)
. Also m -, m + → 1 2 as r → 0. Since two distinct zeros of a quadratic function are necessarily simple zeros, the implicit function theorem yields the following result.

:= {(s, m) ∈ R 2 : |s| < , m < δ -(s)}, ( 23
)
D + r ( ) := {(s, m) ∈ R 2 : |s| < , m > δ + (s)} ( 24
)
and δ ± are smooth functions with δ ± (0) = m ± , respectively.

We now seek a fixed point for h on 0 × E r . This occurs where

m = (m -1 2 ) -1 Q r (1, m), that is where Q r (1, m) = m 2 (m -1 2 ) 2 . ( 25
)
Solutions of ( 25) are given by m = 3 2 and by the solutions of 2rm

= (m -1 2 ) 2 (m + 3 2 ). (26)
A simple sketch of the graph of the cubic function on the right-hand side of [START_REF] Hirsch | Invariant Manifolds[END_REF] shows that this equation has three solutions m 1 , m 2 , m 3 with m 1 < -3 2 and 0 < m 2 < 1 2 < m 3 . We now investigate these four solutions of [START_REF] Hindmarsh | On the motion of the offset impact oscillator[END_REF] and verify that only one of them is relevant here. Solution m = 3 2 : in the original (σ, t) coordinates this corresponds to t = 0, that is the τ -axis 0 . Clearly this is invariant under G (on one side of (τ * , 0)) but it represents equilibrium states and is not the arc γ representing chatter that we seek. In fact m = 3 2 implies r = 1 which is the ideal case of perfect restitution in the limit as v → 0. Solution m 1 < -3 2 : the image of this line is given opposite orientations by p and q, so that under the composition q • p -1 the orientation of the image is reversed. Such a curve cannot be a candidate for γ . Moreover, since the gradient of the line is less than -3 it maps under p (or q) to a curve tangent to the τ -axis but with v 0, not relevant here. Solution 0 < m 2 < 1 2 : the image of this line is also given opposite orientations by p and q (in particular, points with v > 0 are taken to points with v < 0) so this image cannot be a candidate for γ . Solution m 3 > 1 2 : here when t > 0 both p and q take this line into the region v > 0 with the same orientation, so if γ = p(β) exists as an invariant curve then the tangent direction to β at the origin must correspond to this ray. Note that m 3 increases from 1 2 to 3 2 as r increases from 0 to 1. We next study local linearization of h at (0, m 3 ). Using [START_REF] Demeio | Asymptotic analysis of chattering oscillations for an impacting inverted pendulum Q[END_REF] and [START_REF] Fidlin | Nonlinear Oscillations in Mechanical Engineering[END_REF] we find that the Jacobian matrix for h at a point (0, m) has the form

Dh (0, m) = m -1 2 0 * N r (m) with N r (m) = Q -1 2 ( 1 2 (m -1 2 ) -1 [r(4m -3) + 9 2 (m -1 2 )] -Q(m -1 2 ) -2 ) (27) = r(2m + 3)(2m -1) -2 Q -1 2 , ( 28
)
where Q stands for Q r (1, m) and where * denotes a term which depends on the O(3) term in φ.

In the case when m satisfies the fixed point equation [START_REF] Hindmarsh | On the motion of the offset impact oscillator[END_REF] this further simplifies (eliminating r) to give

N r (m) = (2m + 3) 2 8m 2 (2m -1)
.

As the eigenvalues of Dh (0, m) are λ = (m -1 2 ) and λ = N r (m) we easily conclude the following.

Proposition 4.4. The fixed point m 3 with 1 2 < m 3 < 3 2 has eigenvalues λ, λ with 0 < λ < 1 and λ > 1. Also

λ → 1, µ → 1 asr → 1 (m 3 → 3 2 ) while λ → 0, µ → ∞ as r → 0 (m 3 → 1 2
). The eigendirection for µ is the m-axis.

Note that the eigendirection for λ would be the s-axis if there were no O(3) terms in φ.

Since (0, m 3 ) is a hyperbolic fixed point for h it has a unique (local) smooth stable manifold W s transverse the m-axis. Then χ(W s ) is a smooth curve β through the origin in R 2 and tangent to the s-axis from above (η > 0) or below (η < 0) according as σ < 0 or σ > 0, and the same holds in the original (σ, t) coordinates. By construction, since h (W s ) ⊂ W s then h(β) ⊂ β which in turn implies forward invariance of γ := p(β) ∩ + 0 under G. The dynamical properties of h near the saddle point (0, m 3 ) carry over via χ to the stated properties of G on and close to γ . This completes the proof of theorem 4.1.

(b) The case a (τ * ) < 0. Here the horizon H for (τ, t) = (τ * , 0) = w * locally lies in the region v < 0 and so F : + 0 → V and hence also G : + 0 → + 0 are continuous maps in a neighbourhood of w * in + 0 . Moreover, in a neighbourhood of the origin G is a homeomorphism onto its image and is differentiable away from w * . In detail, with local coordinate τ = τ * + σ as before, we see that to first order F maps a neighbourhood T of the origin in + 0 onto a neighbourhood of the origin in the sector T : 3σ + t 0, t 0 in R 2 (coordinates on V ) which is then taken by φ to a neighbourhood of the origin in the region T bounded by the positive σ -axis and an arc tangent to it given by v = 3 8 a (τ * )σ 2 + O(3) in - 0 : see figure 6. Finally, R maps T to a similar if narrower region RT in + 0 . From this geometry it is reasonable to suppose that under forward iteration of G all points of T tend to the σ -axis 0 . We now look more closely to verify that this is the case.

First consider an arc α in V given in (σ, t) coordinates by t = σ with 0 and σ 0. From ( 15) and ( 16) we have

p(σ, σ ) = (σ, -k(3 + 2 )σ 2 + σ 3 µ( , σ )), ( 29
)
q(σ, σ ) = σ (1 + ), -rk(3 + 2 2 )σ 2 + σ 3 ν( , σ )), ( 30
)
where µ, ν are continuous (in fact smooth) functions for ( , σ ) sufficiently small. The images of α under p and q are locally the graphs of two smooth functions v = p (σ ) and v = q (σ ) for σ 0. Given constants K > 0, L > 0 we can choose δ > 0 such that

+ t = 0 σ H Z R p v v v σ σ σ σ φ G t R ′ Π 0 + Π 0 + Π 0 + Π Π 0 - Π Π 0 - Π Π 0 - T T T T ′ ′ ′′
p (σ ) > -k (3 + )(1 -K)σ 2 and q (σ ) < -rk (3 + 2 )(1 + ) -2 (1 + L)σ 2
for 0 δ and 0 σ δ, where as before k = 1 6 a (τ * ) and here k < 0. It then follows that

q (σ ) ph (σ ) < 3 + 2 (1 + ) 2 (3 + h ) 1 + L 1 -K r h
for any 0 < h < 1. Choose K, L sufficiently small (and δ accordingly) so that c := 1+K 1-L r < 1. Since (3 + 2 ) < (1 + 2 )(3 + h ) for all > 0 and h > 0, the above estimates show that when , σ < δ the arc F • R • φ(α ) lies below the arc α h in T when c < h < 1. In other words, when viewed in blown-up σ, coordinates in T the dynamics are dominated by exponential contraction in the -coordinate.

A similar argument applies to the derivatives p , q of p , q with respect to σ : for sufficiently small δ we can ensure that if 0 , σ δ then there exists 0 < h < 1 such that q (σ ) < p h (σ ) for 0 < σ < δ, and that therefore p (σ ) tends C 1 -uniformly to zero for 0 σ δ. Thus we obtain the anticipated result: Using this fact and the results from section 3 we can give a complete description of the local dynamics around the tangency point w * . Let D + denote the closed half-disc in R 2 given by D + = {(x, y) ∈ R 2 : x 2 + y 2 1, y 0}, and let E : D + → D + be the map described in polar coordinates (x, y) = (ρ cos θ, ρ sin θ) by θ → 1 2 θ. The dynamics of E are a topological model for the dynamics of G close to w * . More precisely: Theorem 4.6. There is a homeomorphism from D + onto a closed neighbourhood U of w * in + 0 with (0, 0) = w * and such that is a topological conjugacy between E and G, that is • E = G • . Moreover, can be chosen to be C ∞ off the x-axis. In particular, U is smoothly foliated (except perhaps on 0 ) by arcs of stable manifolds of points w = (τ, 0) ∈ 0 with τ > τ * , each arc having end-points on 0 on opposite sides of τ * . See figure 7.

Proof. Choose positive numbers 0 < δ 1 < δ ∈ R where δ is small enough to satisfy the conditions imposed in the proof of proposition 4.5. By theorem 3.2 there is a neighbourhood in + 0 of the interval J := [w * + δ 1 , w * + δ] ⊂ 0 that is smoothly foliated by stable manifolds of points of J . By proposition 4.5 there is some positive integer n and interval J := [w 1 , w * ] ⊂ 0 such that G n (J ) is sufficiently C 1 -close to J that it intersects the stable manifolds transversely. Let G n (w 1 ) ∈ W s (w 1 ), w 1 ∈ 0 . It follows by backward iteration of G that the stable manifold of each w ∈ J contains a closed arc with endpoints on opposite sides of w * in 0 , and in particular W s (w 1 ) contains an arc from w 1 to w 1 . Since this construction applies for arbitrary δ 1 with 0 < δ 1 δ we obtain a neighbourhood of w * in + 0 smoothly foliated (except perhaps on 0 ) by stable manifolds as claimed.

To construct a conjugacy we first construct a homeomorphism ψ from the sector D(n) of D + given by π 2 n+1 θ π 2 n to the region in + 0 bounded by an arc of G n+1 (J ), and an arc of G n (J ) together with an arc of W s (w 1 ). This can be done in such a way that arcs in D + of constant distance from the origin are taken by ψ to arcs of stable manifolds of points in [w * , w 1 ]. The homeomorphism ψ is then extended to on the whole of D + except for the positive x-axis by (u) = G -d •ψ •E d (u) where d is chosen to be the unique integer (positive or negative) such that E d (u) ∈ D(n). With care, ψ can be constructed at the boundary of D(n) in such a way that is a diffeomorphism away from the origin. Finally, extends to the positive x-axis (θ = 0) by continuity.

Further degeneracy: dynamics near a swan point

Finally, we consider the dynamics of G close to a swan point (τ * , 0); here a(τ * ) = a (τ * ) = 0 but a (τ * ) = 0.

As before, we work with coordinates τ = τ * +σ and we write (σ, η) = (s, sm) = χ(s, m). Using [START_REF] Budd | Chattering and related behaviour in impact oscillators Phil[END_REF] we see that the maps p and φ then become

p : (s, m) → (s, s 3 (6m + 4m 2 + m 3 + O(s))), (31)
φ : (s, m) → (s(1 + m), -s 3 (6m + 8m 2 + 3m 3 + O(s))), (32)
where = -1 2 4a (τ * ) and where the terms O(s) denote smooth functions of (s, m) that vanish when s = 0.

In (s, m) coordinates the map h becomes h : (s, m) → (s, m) where

s(1 + m) = s, (33)
rs 3 (6m + 8m 2 + 3m 3 + O(s)) = s3 (6 m + 4 m2 + m3 + O(s)) (34)
and substituting the first equation into the second for s = 0 we obtain

r(6m + 8m 2 + 3m 3 ) = (1 + m) 3 (6 m + 4 m2 + m3 ) + O(s).
The graphs of the two real-valued functions

f : m → r(6m + 8m 2 + 3m 3 )(1 + m) -3 , ( 35
)
g : m → 6m + 4m 2 + m 3 (36)
intersect at a unique point m * which lies in (-1, 0). This proves the following.

Proposition 5.1. The map h has a fixed point at (0, m * ) and no other fixed point on the m-axis.

It is easy to check from the graphs of f, g that f (m * ) > g (m * ), and also to verify that

D h(0, m * ) = 1 + m 0 * f (m * )/g (m * ) . ( 37
)
This implies the next result.

Proposition 5.2. The map h has a hyperbolic saddle fixed point at (0, m * ). The m-axis is the unstable manifold, while the stable manifold is a smooth curve W s through (0, m * ) transverse to the m-axis.

The interpretation of these results now depends on the sign of a (τ * ).

If a (τ * ) < 0 then a(τ ) has a local maximum at τ * with a(τ * ) = 0, and so from (5) we have ∂ ∂t v c (τ, t) > 0 for all (τ, t) in some neighbourhood of w * = (τ * , 0) in except at w * . Therefore, F is defined on a neighbourhood of w * in and is a smooth local homeomorphism onto a neighbourhood of w * in V . Hence if β = χ(W s ) then γ = p(β) ∩ + 0 is a smooth forward invariant curve γ for G terminating at the swan point. As is clear from [START_REF] Mason | Mathematical models of gear rattle in Roots blower vacuum pumps[END_REF] this curve γ has cubic tangency with 0 at w * . Theorem 5.3. Let w * = (τ * , 0) ∈ 0 be a swan point with a (τ * ) < 0. There is a smooth arc γ ⊂ + 0 with an end-point at w * , cubically tangent to 0 at w * from the side τ < τ * and otherwise lying in + . The arc γ is forward invariant for G and locally unique in the sense of theorem 4.1. The fixed point w * is exponentially attracting for G|γ , and is weakly normally hyperbolic (repelling) for G. All points in a neighbourhood of w * but not on γ lie on stable manifolds of regular points of 0 (nondegenerate chatter). See figure 8. Thus with a (τ * ) < 0 all initial data close to w * lead to nondegenerate chatter, with the exception of those on the curve γ which exhibit a form of highly degenerate chatter.

If, on the other hand, a (τ * ) > 0 then we have v c (τ, t) < 0 for sufficiently small t > 0 and τ close to τ * , and so the image under F of a neighbourhood of w * in + 0 will include no points close to w * . Therefore, the local analysis of low-velocity impacts which is the subject of this paper will not apply.

Unfolding the dynamics at the swan point

Typically (the precise conditions on the functions f and g in ( 1) are described in [START_REF] Chillingworth | Discontinuity geometry for an impact oscillator[END_REF]) swan points will occur for only certain isolated values of c or of some other parameter in the system: swan points are codimension-1 phenomena. Therefore, it is important to consider the bifurcations in the local dynamics as a generic transition through a swan point takes place.

We shall take the parameter to be c for simplicity, although this is not crucial for the discussion. For clarity we also now re-introduce the c suffix in notation for the maps p c , φ c , F c and G c and the surface V c . We suppose that a c ( Without loss of generality we suppose d 0 < 0. As c increases through c 0 an interval (τ 1 , τ 2 ) is created on which a c (τ ) reverses its sign, and a new component of P c (a swan configuration) is created out of the point w * as described in [START_REF] Chillingworth | Discontinuity geometry for an impact oscillator[END_REF]. See figure 9. The creation of the swan is in fact a version of the lips transition (figure 2(a)) with particular geometry dictated by the context, a characteristic geometry of singular sets noted earlier by Bruce [START_REF] Bruce | Geometry of singular sets Math[END_REF].

There are two cases to consider, according to whether a 0 (τ * ) := a c 0 (τ * ) > 0 (positive swan point) or a 0 (τ * ) < 0 (negative swan point). the low-velocity dynamical phenomena that are created when c > c 0 . Once again we now suppose such c fixed sufficiently close to c 0 and dispense with it as a suffix.

To help picture the dynamics of G (and in particular the geometry of F ) we show in figure 10 a sequence of sections through V = V c at constant τ for various choices of τ τ * . This information can be obtained from the Taylor expansion [START_REF] Bruce | Geometry of singular sets Math[END_REF], regarding v c (τ * + σ, t) as a function of t unfolded by the parameter σ . Analogous sketches for τ τ * are obtained by reversing the signs of both t and v.

In figure 11 we show the horizon H and the zero set Z in (τ, t)-coordinates on V , thus viewed from the 'vertical' v direction. Here A = (τ 1 , 0) and B = (τ 2 , 0) where τ 1 < τ 2 are the two zeros of a(τ ) near τ * for c > c 0 and close to c 0 . The shaded area S indicates the connected component of the visible part of V that contains the interval AB (see section 2.2).

As already seen in section 4, the dynamics of G close to A and B are as in figures 5 and 7, respectively. These combine to give the swan dynamics in the way that we now describe. maximal extension (as far as the apparent outline LB) of the arc ε in figure 7. The G-invariant curve γ at the degenerate chatter point B is tangent to 0 at B and transverse to 0 at its other point of intersection. It separates S into two regions: one foliated by stable manifolds of points of AB (nondegenerate chatter), the other consisting of points that eventually leave S . See figure 13.

Negative swan point: a

0 (τ * ) < 0
For c < c 0 we have a(τ ) < 0 for all τ in a neighbourhood of τ * and so theorem 3.1 applies: every point lies in the stable manifold of some point of 0 and nondegenerate chatter takes place locally as in figure 3(b). For c = c 0 theorem 5.3 applies: there is a smooth stable manifold for (τ * , 0) that is cubically tangent to 0 and corresponds to highly degenerate chatter, while all other orbits experience nondegenerate chatter becoming increasingly degenerate as τ → τ * from either side as in figure 8. We therefore focus on the case c > c 0 .

The analogue of figure 10 for this case is obtained by inverting the v-axis. In figure 14 we again show the horizon H and the zero set Z as viewed from the v-direction, with A, B as before corresponding to the values τ 1 , τ 2 of τ where a(τ ) = 0. Now, however, it is a different region T = T 1 ∪ T 2 ∪ T 3 that is visible, bounded by the arc BC of Z, the arc DA of H , and the (dotted) arc DC which is the shadow of DA, that is the set of 'next-hit' points starting from DA: more precisely it is the set of points (τ, v; t 1 ) ∈ V for which there exists t 0 > 0 with (τ, v; t 0 ) ∈ DA but (τ, v; t) / ∈ V for all t 0 < t < t 1 . The visible part of V is the region lying outside the loop ADCBA (including AD but not including the open arc DC).

′ (a) ″ A=C ′ D K A=A K=E C E ′ (b) Π 0 B=D L ″ Π 0 ″ τ τ B=B ′ ″ T 2 T 3 ′ ′ T 2 ′′ T 1 ′′ T 3 ′′ T′ 1 Π 0 _ Π + 0 Π Π 0 0 + _ Figure
Under the correspondence principle in figure 14 the points A, B each correspond to cusp points elsewhere on H : in fact B corresponds to D while A corresponds to its counterpart (not shown) with t < 0. Figure 15 These pictures repay contemplation in order to form a clear mental picture of the dynamics of G. Figure 16 shows schematically how φ • F takes each segment of the swan region into the next segment; remember, however, that the map G also incorporates the restitution R at each step and takes each of + 0 , - 0 into itself. We now describe the dynamics. In section 4.1 we have already obtained local models for the dynamics of G close to the two tangency points A and B (figures 5 and 7, respectively). As there are no discontinuities of G close to B, the neighbourhood of B that is foliated by stable manifolds W s τ of points (τ, 0) with τ > τ 2 (by theorem 4.6) must extend at least as far as the stable manifold W s τ 3 = W s (X) that contains the point A, where X = (τ 3 , 0) with τ 3 > τ 2 . See figure 17. Proposition 6.1. The stable manifold W s (X) is tangent to 0 at A.

Proof. We argue that F W s (X) ⊂ V is tangent to Z at C, from which the result follows by applying p.

Since the map φ has a cusp singularity at C by proposition 2.2, if F W s (X) were not tangent to Z at C then φ • F W s (X) would be aligned with the direction of the cusp at C in figure 15. In that case R • φ • F (W s (X)) = G(W s (X)) = W s (X) would be tangent to the arc ε = R(D C ) at its end-point R(C ), and this is inconsistent with the geometry of chatter close to the point B where a(τ ) = 0 and a (τ ) < 0 as in figure 7: stable manifolds of points of 0 to the right of B are transverse to ε .

Let λ denote the arc of W s (X) from A to Q = W s (X) ∩ ε . The inverse image G -1 (λ) is a closed loop around the discontinuity arc δ = AD and with both ends at A. The inverse images of δ under iterates of G form a sequence of arcs tangent to 0 at the degenerate chatter point A and accumulating on its invariant arc γ , and therefore the inverse images of λ form a sequence of loops also accumulating on γ . Stable manifolds W s (Y ) of points Y to the right of X on 0 (i.e. with τ > τ 3 ) and close to X oscillate more and more wildly under backwards iteration of G as they attempt to adhere to the loops of G -n (λ). Again see figure 17.

Global dynamics

The dynamical phenomena that become apparent in the unfolding of a negative swan point give important clues about the global geometry for dynamics of an impact oscillator. In particular, we see that the particular geometry of the smooth map p • φ -1 close to a tangency point w * = (τ * , 0) ∈ 0 (so that z = φ -1 (w * ) is a cusp point of H ) plays a key role. At a tangency point w * where a (τ * ) > 0 the nearby stable manifolds W s τ with τ > τ * have infinitely many loops accumulating on the forward invariant curve γ that is the stable manifold of w * Therefore, we see that the global behaviour of the stable manifolds of degenerate chatter points and, in particular, their intersections with unstable manifolds of fixed or periodic points elsewhere in the phase space + 0 are crucial in organizing the global effects of the complicated dynamics unleashed at a (negative) swan bifurcation. The swan configuration created at a negative swan point will persist until variation in the parameter c causes it to impinge on some other relevant branch of the apparent outline P . Careful study of the generic ways in which this can happen will provide important tools for understanding the complicated global structure of discontinuous planar discrete dynamical systems modelling one-degree of freedom impact oscillators.

Figure 1 .

 1 Figure 1. The impact surface V c = ∪ V c with horizon H c that is the set of singular points of the projection p|V c → . Points w 1 , w 2 , w 3 belong to the apparent outline P c = p(H c ) (not shown).

Figure 2 .

 2 Figure 2. The three generic transitions in apparent outlines: (a) lips, (b) beaks, (c) swallowtail.

Figure 3 .

 3 Figure 3. Local dynamics of G in the (τ, v)-plane close to 0 : nondegenerate chatter for a(τ ) < 0. (a) Local linearization, (b) local foliation by stable manifolds.

Figure 4 .

 4 Figure 4. The actions of p, φ and R close to (τ * , 0) with a (τ * ) > 0: here τ = τ * + σ . The shaded area in the top left diagram represents the visible part of V , and the other shaded areas are its images under the relevant maps as indicated. The arc P * ⊂ p(H ) is the discontinuity arc.

Figure 5 .

 5 Figure 5. Degenerate chatter at w * = (τ * , 0) with a (τ * ) > 0. Invariant curves below the invariant curve γ represent nondegenerate chatter; invariant curves above γ carry orbits across the discontinuity arc P * .

Proposition 4 . 2 .

 42 The quadratic function m → Q r(1, m) has zeros at m -, m + where

Corollary 4 . 3 .

 43 For sufficiently small > 0 the domain D r of h contains the region D r ( ) = D - r ( ) ∪ D + r ( ) where D - r ( )

Figure 6 .

 6 Figure 6. The actions of p,φ and R close to (τ * , 0) with a (τ * ) < 0: here τ = τ * + σ . The shaded area in the top left diagram represents the visible part of V , and the other shaded areas are its images under the relevant maps as indicated.

Figure 7 .Proposition 4 . 5 .

 745 Figure 7. Chatter close to w * = (τ * , 0) with a (τ * ) < 0. Points of 0 with τ > τ * are limit points of nondegenerate chatter that becomes increasingly degenerate as τ → τ * from above. The arc [w 1 , w * ] on 0 is taken by G to an arc ε tangent to 0 at w * .

Figure 8 .

 8 Figure 8. Stable manifolds of chatter points close to a swan point (τ * , 0) where a (τ * ) = 0 and a (τ * ) < 0.

  τ * ) = a c (τ * ) = 0 when c = c 0 and let d(c, τ ) := a c (τ ) ∂a ∂c (c, τ ) where a(c, τ ) = a c (τ ). As c increases through c 0 the function a c (τ ) undergoes a Morse transition at which a pair of zeros τ 1 , τ 2 of a c (τ ) is created or annihilated according to the sign of d 0 := d(c 0 , τ * ). Note that under the correspondence principle the creation of two tangency points at a swan point corresponds to the creation of two cusps at a swallowtail (figure 2(c)).

6. 1 .Figure 9 .

 19 Figure 9. Graphs of a(τ ) and associated swan configurations in the plane in two cases: (a) positive, (b) negative.

Figure 10 .

 10 Figure 10. Sections through V c in the (t, v)-plane for increasing values of τ with τ τ * .

Figure 12 (Figure 11 .

 1211 Figure 11. Positive swan configuration: the visible part S of V is bounded by an arc of the horizon H and an arc of the zero locus Z as well as the interval AB of 0 . See text for details.

Figure 12 .Figure 13 .Figure 14 .

 121314 Figure 12. The images under (a) the projection p and (b) the re-set map φ of various regions of V in figure 11 for an unfolded positive swan point: see text for details.

15 .

 15 Figure 15. The images under (a) the projection p and (b) the re-set map φ of various regions of V for an unfolded negative swan point: see text for details.

  (a) shows the images A , . . . , E of the points A, . . . , E of V under projection p, while figure15(b)shows their images A , . . . , E under the re-set map φ. The only visible part of H is the arc from A to D: under p this projects to the arc AD in + 0 while under φ it is taken to the interval A D on the τ -axis 0 .At points of the shadow arc CD apart from D the velocity ẋc is nonzero (in fact it has to be negative) and so φ takes CD to an arc C D with v < 0 except at D ; moreover, C is a cusp point of P by proposition 2.2. From this we see the nature of the discontinuity of the map φ • F (and hence of G): cutting + along the arc AD then mapping one edge (heavy) to the interval A D of 0 and the other edge (dotted) to the arc D C . Meanwhile the interval AB of figure15(a) is mapped by F to the arc CB of Z which is then taken by φ to the arc C B of figure 15(b).

Figure 16 .Figure 17 .

 1617 Figure 16. Schematic representation of the action of φ • F within the swan region. Matching arrows indicate arcs and their images. The third step is discontinuous along AB.

For simplicity we take smooth to mean C ∞ , although many of the results will clearly hold with less differentiability.

In the statement r 2 there is an error: it originally referred to degree of differentiability, and has no connection with the restitution coefficient here.

Acknowledgments

The author is most grateful to the Centre de Recerca Matemàtica, Barcelona, for hospitality during the preparation of the main part of this paper as part of the Research Programme on Complex Non-Smooth Dynamical Systems organized there by the Bristol Centre for Applied Nonlinear Mathematics in Spring 2007. The author thanks colleagues at BCANM for hospitality and continuing interest in and support for this research.