Co-dimension-two Grazing Bifurcations in Single-degree-of-freedom Impact Oscillators

Transcription

1 Co-dimension-two razing Bifurcations in Single-degree-of-freedom Impact Oscillators Phanikrishna Thota a Xiaopeng Zhao b and Harry Dankowicz c a Department of Engineering Science and Mechanics MC 219 Virginia Polytechnic Institute and State University Blacksburg Virginia 261 USA thota@vt.edu b Department of Biomedical Engineering Duke University Durham NC 2778 USA xzhao@duke.edu c Department of Mechanical and Industrial Engineering University of Illinois at Urbana-Champaign Urbana IL 6181 USA danko@uiuc.edu Abstract razing bifurcations in impact oscillators characterize the transition in asymptotic dynamics between impacting and non-impacting motions. Several different grazing bifurcation scenarios under variations of a single system parameter have been previously documented in the literature. In the present paper the transition between two characteristically different co-dimension-one grazing bifurcation scenarios is found to be associated with the presence of certain co-dimension-two grazing bifurcation points and their unfolding in parameter space. The analysis investigates the distribution of such degenerate bifurcation points along the grazing bifurcation manifold in examples of single-degree-of-freedom oscillators. Unfoldings obtained with the discontinuitymapping technique are used to explore the possible influence on the global dynamics of the smooth codimension-one bifurcations of the impacting dynamics that emanate from such co-dimension-two points. It is shown that attracting impacting motion may result from parameter variations through a co-dimension-two grazing bifurcation of an initially unstable limit cycle in a nonlinear MEMS oscillator. 1 Introduction Structurally stable features of dynamical systems are those that are expected to persist on open sets in suitable spaces of vector fields. For example as is wellknown for smooth systems conditions of hyperbolicity guarantee the persistence and retained stability characteristics of equilibria or periodic orbits under small variations in system parameters [1]. The loss of structural stability has implications both in terms of model uncertainty as well as for example in detecting and designing against catastrophic system failure [2]. When structural stability is lost on hypersurfaces of one dimension smaller than the dimension of parameter space one speaks of co-dimension-one bifurcations namely changes in system features that can be observed under variations in a single system parameter. Similarly co-dimension-two bifurcations are those that would generically only occur under suitable simultaneous variations of two system parameters. As has been extensively explored over the past decades [1 3] the associated co-dimension-two surfaces in parameter space serve as organizing centers for co-dimension-one bifurcations. Indeed co-dimension-one bifurcation surfaces are often found emanating from the co-dimension-two surfaces in ways that are characteristic of some associated degeneracy at the corresponding point of origin in parameter space. Furthermore the dynamical characteristics of the corresponding dynamical system for parameter choices in the vicinity of such co-dimension-two points take a number of forms that can be characterized through particular normal forms [1]. In impact oscillators the presence of a discontinuity surface in the associated state space corresponding to the occurrence of impact naturally induces codimension-one bifurcations. As an example a nonimpacting periodic oscillation of the dynamical system may under variations in a single system parameter increase in amplitude until beyond some critical value the amplitude would normally exceed that available to the oscillator. The ensuing change in system behavior for example to an impacting system response is certainly a change in system feature. As the point of loss of structural stability corresponds to the existence of a periodic trajectory in state space that grazes the discontinuity surface this type of discontinuity-induced bifurcation is commonly referred to as a grazing bifurcation [-6]. 1

2 In addition to changes in the presence or absence of contact as a result of grazing bifurcations the latter are typically associated with dramatic changes in stability properties of the dynamical system for example the sudden appearance of robust chaos or a so-called period-adding sequence. Indeed a number of different grazing bifurcation scenarios have been documented in the literature [5-1]. As was the case for smooth dynamical systems a study of co-dimension-two grazing bifurcations would thus be expected to provide an understanding of the relationship between these different bifurcation scenarios. It would also lead to insight into the dynamical characteristics of impact oscillators away from the co-dimension-one grazing bifurcation surfaces. The purpose of this paper is to investigate the distribution of such co-dimension-two grazing bifurcations in single-degree-of-freedom impact oscillators and to inquire into the possible dynamical characteristics of the system response on neighborhoods of such bifurcation points. Here our emphasis is on features that appear generic to this class of oscillators. As an example through the use of the so-called discontinuitymapping approach initially conceived of by Nordmark [9] it is demonstrated how stable impacting oscillations may originate in co-dimension-two grazing bifurcations of initially unstable nonimpacting periodic oscillations. 2 Mathematical preliminaries 2.1 Definitions Consider a periodically forced single-degree-of-freedom oscillator with angular excitation frequency. Let x v and θ = t mod 2π denote the displacement velocity of the oscillator and the phase of the excitation respectively. In terms of the state vector x = x v θ T it follows that dx dt = v a x v θ T def = f x 1 where a equals the acceleration of the oscillator as a function of x v and θ. Denote by Φ smooth x t the corresponding flow function. Now suppose that the movement of the oscillator is limited by a unilateral rigid constraint placed at x = δ corresponding to a state-space discontinuity surface D described by the zero-level surface of the event function h D x def = δ x such that h D x during the motion of the oscillator. Let h P x def = h D x x f x = v. A transversal intersection of a state-space trajectory with D then corresponds to a point x such that h D x = and h P x since given a trajectory x t with x t = x for some time t it follows that d dt hd x t t=t = h D x x f x = h P x. Indeed if h P x < a collision occurs between the oscillator and the constraint. We model the collision as an instantaneous impact with a characteristic coefficient of restitution r i.e. the state immediately after impact relates to that immediately before impact according to the jump map g D x = x r v θ T. 2 In contrast a point of simple grazing contact of a statespace trajectory with D corresponds to a point x such that h D x = h P x = and h P x x f x >. It follows that x corresponds to a transversal intersection of a state-space trajectory with the surface P given by the zero-level surface of h P. Indeed since d 2 dt h D x t 2 t=t = h P x x f x > it follows that x is a local minimum in the value of h D i.e. the distance to D along a state-space trajectory of 1. By transversality it follows that nearby trajectories achieve locally unique points of intersection with P corresponding to local minima in the value of h D. Now suppose that x is a point of simple grazing contact of a periodic trajectory of 1 with D such that Φ smooth x t + T = Φ smooth x t. Let a superscript denote quantities evaluated at x for example a = a x. Then ignoring the effects of the jump map and using the transversality of the intersection with P it is possible to uniquely define a local Poincaré map P smooth : P P such that P smooth = x. Moreover it is straightforward to show that P smoothx = Id f h P x f h P x Φ smoothx x T. 3 From the semi-group property of the smooth flow Φ smooth Φ smooth x t s = Φ smooth x t + s it follows by differentiation with respect to t and evaluation at t = and s = T that and consequently Φ smoothx x T f = f 5 P smoothx f = 6 It follows by induction on the integer n that P n smoothx = Id f h P x f h P x Φ n smoothx x T 7 where integer superscripts denote matrix powers. When reintroducing the effects of the jump map the dynamics of the vibro-impact oscillator under perturbations in initial conditions away from the grazing periodic orbit and in parameter values away from those corresponding to the existence of the grazing periodic orbit may be analyzed using the discontinuity-mapping approach originally introduced by Nordmark [9] see also [ ]. Here a discontinuity mapping D may be introduced on a neighborhood of the point x such that the Poincaré surface P is invariant under D and 2

3 such that the Poincaré mapping P associated with the Poincaré surface P for the flow near the grazing trajectory including the effects of the jump map can be written as P = P smooth D. 8 Here D provides a correction to the smooth flow that accounts for differences in the times-of-flight from nearby initial conditions to the discontinuity surface D and the effects of the jump map. Since points on P near x correspond to local minima in the distance to D along state-space trajectories of 1 it follows that for points x P such that h D x the discontinuity mapping equals the identity mapping. In contrast a nontrivial expression for D applies for x P such that h D x <. Here x and D x are points of intersection with P of trajectory segments leading up to and emanating from D when ignoring the jump map such that the corresponding intersections with D are related through the jump map. 2.2 Stability at grazing Stability of the grazing periodic trajectory when ignoring the effects of the unilateral constraint can to lowestorder be determined by the eigenvalues of the matrix P smoothx. In contrast as shown by Fredriksson and Nordmark [11] assuming that the grazing periodic trajectory is asymptotically stable in the absence of the constraint the stability properties of the grazing periodic trajectory in the presence of the unilateral constraint are determined by the geometry of the sequence of vectors where β = h P x v n = P n smoothx β 9 gx D f h P x f Id gx D f. 1 Indeed a sufficient condition for the stability of the grazing periodic trajectory is that the sequence ξ n = h D x v n n = 1... be positive and that this remains true under small perturbations in system parameters whereas the trajectory is unstable if ξ j < for some j 1. Here for small deviations from x in the direction of negative values of h D the discontinuity mapping results in a large stretching in a direction given by the image of the vector β under the jacobian P smoothx. The positiveness of ξ thus implies that every trajectory that remains in the vicinity of the grazing periodic trajectory achieves at most one intersection with D in the vicinity of x. Since P n smoothx f = it follows that P n smoothx = P n11 P n12 a P n12 11 P n31 P n32 a P n32 and thus ξ n = 1 + r a P n In the special case when P i12 = for some integer i P mi smoothx = m = where denotes a nonzero entry. By the above formula for ξ n it follows that ξ i = for some i ξ mi = for m = From 7 it follows that h D x since h D x where P n smoothx = h D x Φ n smoothx x T 15 f =. Thus ξ n = h D x Φ n smoothx x T β = 1 + r a [ Φ n smoothx x T ] [ ] Φ n smoothx x T of Φ n smoothx x T. 2.3 Post-grazing dynamics 12 denotes the 1 2 element Suppose that x is a transversal point of intersection with D of a curve of hyperbolic asymptotically stable fixed points of P smooth under variations in some system parameter µ such that P smooth = x for µ = µ. It follows that there exists a unique local attractor in the vicinity of the grazing periodic trajectory for µ µ when the effects of the jump map are ignored. By transversality and without loss of generality consider the case where for µ < µ the corresponding periodic trajectories remain in the h D > region. When reintroducing the effects of the jump map as shown in [11] these assumptions together with the conditions guaranteeing the stability of the grazing periodic trajectory discussed in the previous section actually imply the persistence of a local attractor near the grazing trajectory for µ µ. The bifurcation scenario is continuous in parameter space in the sense that there exists a one-parameter family of impacting attractors for µ > µ that emanates continuously from the grazing periodic trajectory. Here the repulsion away from the near-grazing region due to occasional impacts is balanced by the attraction to the virtual i.e. for which h D < stable fixed point of P smooth. From an applications-point-of-view the transition across the critical parameter value µ is safe as the system response continues to closely resemble the pregrazing dynamics. In contrast if ξ i is negative for some i a discontinuous bifurcation scenario results corresponding to a jump to a different system attractor as the critical parameter value is crossed. From the analogous 3

4 behavior in smooth bifurcations one might thus refer to the continuous bifurcation scenario as a supercritical grazing bifurcation while the discontinuous scenario has features in common with subcritical smooth bifurcations. In the special case when ξ i = for some i and ξ j > for j i the sustained stability of the grazing periodic trajectory in the presence of the unilateral constraint cannot be conclusively determined from the lowest-order analysis employed in [11]. Indeed a variety of co-dimension-one bifurcation curves can be shown to emanate from such co-dimension-two bifurcation points [7 8]. Moreover persistent local attractors may be traced in selected one-parameter variations through such co-dimension-two grazing bifurcation points. 3 The Distribution of Co-dim-2 Bifurcations 3.1 A linear impact oscillator Consider the linear impact oscillator given by the acceleration function a x v θ = α cos θ 2ζ v x where α is the excitation amplitude and ζ is the damping coefficient. Then a unique limit cycle of period T = 2π exists for the smooth flow given by x t = α γ cos t φ where γ = α ζ 2 2 and tan φ = 2ζ It follows that a grazing periodic trajectory is achieved for an excitation amplitude α = α = δγ and θ = φ such that a = δ 2. Solving the corresponding variational equation it is straightforward to show that Φ n smoothx x T = φ n11 φ n12 δφ n12 φ n21 φ n22 δ φ n where in the underdamped case when < ζ < 1 φ n11 = e 2nπζ φ n12 = 2 2nπζ e sin nπω Ω φ n21 = 2 2nπζ e Ω φ n22 = e 2nπζ cos nπω + 2ζ nπω sin Ω sin nπω cos nπω 2ζ nπω sin Ω and Ω = 2 1 ζ 2 2 but see also the appendix for a general approach. It follows that ξ n = 2 Ω 1 + r δ2 e 2 n πζ sin nπω. 21 This expression implies that except for points where ξ 1 = and consequently ξ n = for all n all grazing bifurcations that occur in the linear impact oscillator with < ζ < 1 are discontinuous in the sense described above. Indeed if ξ n > for some n then π 2 k nπω π mod 2π < 2 k 1 22 for some k 1 and thus ξ 2 k n where equality only holds when π = n πω 2 k mod 2π. In the latter case unless n = 1 there exists a relative prime n such that n πω mod 2π π 2 k 23 for all integers k i.e. at least one of the ξ n s must be negative. For fixed ζ co-dimension-two grazing bifurcation points corresponding to ξ n = are thus given by nm = n Ω m for arbitrary integers n and m. In particular for each n ξ n > for > n1 = nω. Also for each n the sequence nm has a unique limit point γ sin t φ t mod 2π T at =. As an example Figure 1 shows the curve of 17 points in the α parameters space corresponding to the existence of a grazing periodic trajectory the grazing curve. Here points for which ξ 1 = are indicated by circles. These are the same points as those found by Foale [13] and Foale and Bishop [1]. For fixed codimension-two grazing bifurcation points corresponding to ξ n = are given by ζnm = 1 m2 2 n 2 for arbitrary integers n and m with m < 2n. In particular for each n ξ n > for ζ > ζn1 = 1 2 n. 2 Also for each m the sequence ζnm has a unique limit point at ζ = 1 corresponding to the point where the eigenvalues of Φ smoothx x T change from complex conjugate to real. Indeed for any three integers n m and ñ there are ñ n m co-dimension-two points ξñ = for which ζñ m ζ nm where denotes the integer part of its argument. In the critically damped case when ζ = 1 the n-th power of the monodromy matrix takes the same form as that given in Equation 19 with φ n11 = 2nπ + 1 e 2nπ φ n12 = 2nπ 2nπ e φ n22 = and thus φ n21 = 2nπ 1 2nπ e 2nπ e 2nπ ξ n = 2n 1 + r δπe 2nπ > n. 2

5 α 2 o oo ooo o o 1 2 variations in ξ n. To this end let ε = 1 and consider the dynamical system dx dτ = f x = εv 2ζεv εx + εv 2 1 cos2θ 21 x 2 1 where τ = t and make the following ansatz T 27 xτ = x + εx 1 τ + ε 2 x 2 τ + 28 v τ = v 1 τ + εv 2 τ + ε 2 v 3 τ + 29 θ τ = τ mod 2π 3 Figure 1: razing curve for the linear impact oscillator in α-space. Here ζ =.5 and δ = 1. Similarly in the overdamped case when ζ > 1 where and thus φ n11 = λ 1e 2nπλ2 λ 2 e 2nπλ 1 λ 1 λ 2 φ n12 = e 2nπλ1 e 2nπλ 2 λ 1 λ 2 φ n21 = e 2nπλ 2 e 2nπλ 1 λ 1 λ 2 λ 1 λ 2 φ n22 = λ 1e 2nπλ1 λ 2 e 2nπλ 2 λ 1 λ 2 λ 1 = ζ ζ 2 1 λ 2 = ζ + ζ ξ n = 1 + r δ 2 e 2nπλ1 e 2nπλ 2 > n. 26 λ 1 λ 2 In contrast to the case when < ζ < 1 all grazing bifurcations of the linear impact oscillator with ζ 1 are continuous in the sense described above. 3.2 A nonlinear MEMS oscillator Now consider a nonlinear impact oscillator given by the acceleration function a x v θ = V 2 sin 2 θ 2ζv x corresponding to an applied excitation voltage with ampli- 1 x 2 tude V across a parallel-plate capacitor with a stationary electrode at x = 1. Such a parallel-plate capacitor can act as the actuation element in an impact microactuator see e.g. experimental results in [15] and theoretical analysis in [ ]. In contrast to the linear case no closed form expression is available for ξ n as a function of the forcing frequency. For excitation frequencies 1 as might occur when operating the microelectromechanical oscillator by radio-frequency transmission regular perturbation analysis may be employed to investigate the for large. Here the constant x accounts for the static offset due to the nonzero time-average of the excitation. Equating terms of the same order in ε and eliminating secular terms then yields and x 1 x 2 V 2 = 2 31 x 1 τ = 32 x 2 τ = 1 x cos2τ 33 x 3 τ = 1 ζx sin 2τ. 3 The above equations describe a periodic motion with period π whose maximum displacement during each period is x 1 + ε2 + O ε 35 and occurs at τ = εζ + O ε Since the maximum displacement of a grazing periodic motion is δ it follows that x = δ 1 ε2 + O ε 37 for a grazing periodic trajectory. Similarly the acceleration of the oscillator at the point of grazing contact is found to equal a = εδ + O ε Finally substituting Equation 37 into Equation 31 yields the critical excitation voltage V corresponding to a grazing periodic trajectory for a given frequency V = 2δ 1 δ ε 2 1 3δ δ 2 + O ε 39 5

6 Thus when V 2δ 1 δ. By making use of a similar perturbation ansatz see the appendix to O ε 2 the n-th power of the associated monodromy matrix is found to equal Φ n smoothx x π 1 3δ δ 1 1 εnπ εnπ 1 2εnπζ 1 explored in Figure 3 the number of co-dimension-two points ξñ = for n again appear to go as ñ n. The figure also indicates the existence of an additional limit point for the set of co-dimension-two points at Again this turns out to be a point where the eigenvalues of the corresponding monodromy matrix change from complex conjugate to real..97 and thus ξ n = ε 2 δ 1 + r nπ + O ε 3. 1 For each n and sufficiently large ξ n >. It follows that no co-dimension-two bifurcations associated with ξ n = for fixed n will occur for large independently of δ and ζ in agreement with the observation made in the case of the linear oscillator. Although closed-form expressions are unavailable both for the grazing curve in V -space as well as for the ξ n s along this curve grazing periodic trajectories and the corresponding loci for which ξ i = for some i can be obtained numerically using a Newton- Raphson-based continuation scheme see Zhao et al [16] and Dankowicz and Zhao [1]. A segment of the corresponding grazing curve is shown in Figure 2. Again points for which ξ 1 = have been indicated by circles..6 V.2.5 T 1 Figure 2: razing curve for the nonlinear MEMS oscillator in V -space. Here ζ =.2 and δ =.5. Here and in later figures solid lines refer to stable orbits and dashed lines refer to unstable orbits. As in the case of the linear oscillator numerical computation indicates the existence of infinitely many ξ n = co-dimension-two points and that for each n they have a limit point at =. From the perturbation analysis it follows that for each n there exists an n such that ξ n > for > n. Numerical computations indicate that lim n n.9796 corresponding to the point where the eigenvalues of the corresponding monodromy matrix change from complex conjugate to real again in qualitative agreement with the situation observed in the linear oscillator. Indeed as further T n 1 18 Figure 3: Co-dimension-two points ξ n = for 2. The Influence of Co-dim-2 Bifurcations That all grazing bifurcations are continuous in the case of the linear oscillator with ζ 1 and discontinuous except possibly at points where ξ 1 = in the case when < ζ < 1 is in agreement with the predictions of an exact analysis of the near-grazing dynamics carried out by Chin et al. [] using a first-order truncation of the discontinuity mapping D. These authors also establish the presence of distinct boundaries between different discontinuous grazing bifurcation scenarios corresponding to the loci where ξ n = sin nπω =. However as the chosen form for the truncated discontinuity mapping used in their paper assumes that ξ 1 > their results provide no obvious insight into the bifurcation scenario that results from a parameter sweep through a ξ 1 = co-dimenstion-two point nor the possibility of additional bifurcation curves emanating from such a point. Using the full discontinuity-mapping approach alluded to in previous sections and developed further in the references higher-order truncations of the discontinuity mapping may be obtained that apply on neighborhoods of any point on the grazing curve including ξ n = co-dimension-two points cf. Dankowicz and Zhao [1] and Zhao and Dankowicz [17]. For small deviations from the grazing curve in parameter space the predictions of such higher-order truncations may be employed to investigate the near-grazing dynamics as well as to analytically approximate bifurcation curves emanating from such co-dimension-two points. As an example Figure shows a continuous grazing bifurcation obtained through application of the higher-order 6

7 approach when increasing the excitation amplitude α through the ξ 1 = co-dimension-two grazing bifurcation point shown in Figure 1 with * * * 1-6 Figure : A continuous grazing bifurcation is achieved in the linear oscillator with = 2 1 ζ 2 ζ =.5 and δ = 1 for which ξ 1 =. Here and in later figures the of a point on P is defined as h D and denotes the grazing bifurcation. Initial studies of co-dimension-two bifurcation scenarios have previously appeared in the literature see [1 17] in the case of impact oscillators and [7] for a related analysis of systems with sliding solutions. In all these scenarios and including all possible grazing scenarios for the linear oscillator the grazing periodic trajectory is locally asymptotically stable when the effects of the jump map are ignored. In contrast the results presented below focus on bifurcation analyses in the vicinity of grazing periodic trajectories that are unstable in the absence of the unilateral constraints. By an analysis analogous to that presented in Fredriksson and Nordmark [11] there can be no local attractor that emanates continuously from the grazing periodic trajectory except possibly for the case when ξ n = for all n. As shown previously such points are common and coincide with ξ 1 = co-dimension-two points in the single-degree-of-freedom oscillators discussed here. In the discussion below three distinct unfoldings are presented of the bifurcation behavior on a neighborhood of a ξ 1 = co-dimension-two grazing bifurcation point for the MEMS oscillator at which the grazing periodic trajectory is unstable in the absence of the unilateral constraint cf. Figure 5. Here compositions of higher-order truncated series expansions of the smooth Poincaré mapping P smooth and the discontinuity mapping D have been numerically analyzed to locate and continue branches of stable or unstable fixed points of the composite Poincaré mapping P. Similarly repeated iteration of P has been used to capture periodic of at least twice the period of the excitation or chaotic system attractors. Qualitatively and for small deviations away from the grazing curve quantitatively similar bifurcation diagrams showing system attractors could be obtained using direct simulation and continuation using the original differential equations albeit at tremendous computation expense. As an example bifurcation diagrams that can be obtained in a matter of minutes on a desktop computer using iterations of the discrete maps require hours and days using direct numerical simulation. Indeed the approximate maps provide a useful means for locating appropriate initial conditions to converge onto such attractors in direct simulation..1 Scenarios 1 and 2 Scenario 1 is exemplified by the bifurcation behavior in the vicinity of the co-dimension-two grazing bifurcation point corresponding to.95 and V.3986 and agrees qualitatively with that found for at least three other ξ 1 = co-dimension-two bifurcation points in the MEMS oscillator. Figure 5 top panel shows a sketch of the co-dimension-one bifurcation curves associated with period-one impacting trajectories that emanate from this point under variations in and V. Sample grazing bifurcation diagrams obtained through variations in the excitation amplitude across points on either side of the co-dimension-two point are shown in Figure 6. These establish the absence of a local system attractor emanating continuously from the grazing periodic trajectory. Nevertheless as shown in Figure 6 bottom panel a branch of stable impacting periodic trajectories is born in a regular saddle-node bifurcation and persists over a small interval of excitation amplitudes for excitation frequencies >. These stable periodic trajectories undergo a period-doubling sequence to a chaotic attractor that disappears in an exterior crisis as V decreases. We note that in the limit as the saddle-node period-doubling and crisis bifurcation curves appear to have a quadratic tangency with the grazing curve. Thus although the impacting system attractors found here are nonlocal in the codimension-one sense it is reasonable to refer to them as local in the co-dimension-two sense even though the range of values over which they exist shrinks to zero as. Scenario 2 here exemplified by the bifurcation behavior in the vicinity of the co-dimension-two grazing bifurcation point corresponding to.19 V.361 similarly exhibits a small interval of excitation amplitudes bounded by a saddle-node and a crisis bifurcation for > in which an impacting attractor consisting of period-doubling sequence to chaos is found see a sketch of bifurcation curves in Figure 5 middle panel and sample grazing bifurcation scenarios in Figure 7..2 Scenario 3 As with the scenarios 1 and 2 above no local impacting attractor exists under variations in V on either side of the co-dimenstion-two grazing bifurcation point corresponding to.1392 V.39 see a sketch of 7

8 bifurcation curves in Figure 5 bottom panel and sample grazing bifurcation scenarios in Figure 8. In contrast to the previous scenarios however for on either side of there exists an interval of excitation amplitudes in which an impacting system attractor may be found. Here the repulsion away from the near-grazing region due to occasional impacts is balanced by the repulsion from the unstable fixed point of P smooth. Indeed the left boundary of this interval limits on V as. In contrast to the previous scenarios the size of the interval does not shrink to zero as. In fact the nonlocal chaotic attractor observed under variations in V for is local in the original codimension-one sense under variations in V for =. 5 Discussion Two fundamental contributions have been made in this paper to the study of discontinuity-induced bifurcations in impact oscillators. Firstly the analysis has identified and explored the distribution of co-dimension-two grazing bifurcation points along the grazing curve in two example single-degree-of-freedom oscillators. Although closed-form expressions for the location of such co-dimension-two bifurcation points were only available for the linear oscillator numerical results were used to illustrate some striking similarities between the linear and nonlinear oscillators in as far as the clustering characteristics of these bifurcation points at points where the eigenvalues of the monodromy matrix associated with the grazing periodic trajectory change from complex conjugate to real. Indeed near any such point the distribution of ξ n = points is determined by the collection of values of s for which the argument of the complex eigenvalues of a matrix of the form a + sa sc 1 sb 1 a + sd 1 2 is a rational multiple of π note that for sufficiently small s the real and imaginary parts of the eigenvectors limit on constant directions. The type of clustering documented above then follows from the observation that the argument of the eigenvalues is proportional to sȧs shown in a previous section degeneracies present themselves in the single-degree-of-freedom case that would not be expected to persist in higher dimension. In particular it is straightforward to construct higherdegree-of-freedom oscillators for which co-dimensiontwo points may be found with ξ k = and ξ n for n k. Initial attempts however have failed to produce such an example in which ξ n > for n k although this should be relatively straightforward. As for the clustering of the ξ n = points in such higher-degree-offreedom systems this is naturally an interesting avenue for further study. Secondly the bifurcation scenarios shown above are remarkable in as much as they involve the occurrence of impacting attractors in the near-grazing dynamics of an originally unstable nonimpacting periodic orbit. It appears quite reasonable to argue that these system attractors find their origin at the isolated co-dimensiontwo bifurcation points. As they subsequently persist over some region of parameter space which may be quite large as obvious already in the near-grazing region in the third scenario the co-dimension-two bifurcation points thus serve as organizing centers for some essential components of the impacting system response. That this should be the situation even when the grazing periodic trajectory is originally unstable is certainly not immediate from any previous analysis of the neargrazing region. Finally we emphasize the need for careful analysis of the convergence of the predictions of the discontinuitymapping approach as the truncation order is increased. For example while lowest-order approximations to D are able to capture the qualitative asymptotic behavior for parameter values and trajectories in the vicinity of accurately found co-dimension-one grazing bifurcation points they predict incorrect results when formulated at co-dimension-two grazing bifurcation points. We are at this time not aware of a systematic approach to determining the appropriate truncation order although a heuristic approach can be deduced from the analysis in the co-dimension-one case. 6 Acknowledgements The authors gratefully acknowledge support for this work from the National Science Foundation Division of Civil and Mechanical Systems grant number Appendix A: Exact analysis of the monodromy matrix for a linear oscillator Consider the linear nonhomogeneous differential equation ẋ = f t x = Ax + b t 3 where b t + 2π = b t for all t and A is a constant matrix. In terms of the augmented state vector x = x θ T where θ = t mod 2π the associated autonomous dynamical system is given by x = f x = à x + b θ A where à = and b θ = b θ T. As the general solution to the θ-equation is θ t = θ + t mod2π it follows that a solution to the above 8

9 autonomous dynamical system is a solution to the nonhomogeneous dynamical system x = Ã x + b t + θ 5 when Φ is evaluated at x τ with x = δ 1 ε2 + O ε. 53 i.e. x t = and thus Φ t x = e ta x + e ta t e sa b s + θ ds θ + t mod 2π e ta x + e ta t e sa b s + θ ds θ + t mod 2π 6. 7 Suppose in particular that x is a point on a periodic trajectory of period 2π of the autonomous dynamical system i.e. such that x = e 2π A x + e 2π A 2π It follows that Φ x t x = e sa b s + θ ds. 8 e ta e ta t e sa b θ s + θ ds 1. 9 From this expression the eigenvalues of the monodromy matrix Φ x 2π x are found to equal 1 and the eigenvalues of e 2π A i.e. e 2π λi where the λ i are the eigenvalues of A. We recover the expression in the main text by considering the case where 1 A = 1 2ζ b t = α cos t. 5 Appendix B: Perturbation analysis for the monodromy matrix of the MEMS oscillator Consider the vector field f x = εv 2εζv εx + εv 2 1 cos2θ 1 21 x 2 T 51 in the state vector x = x v θ mod 2π T with ε = 1. Then as shown in the main text there exists a periodic trajectory of period π through the point x = δ εζ + O ε 3 T of the form Φ = x + ε 2 x ε x 2 ζx cos 2τ ε3 sin 2τ + O ε ζx sin 2τ ε2 2 cos 2τ + O ε 3 τ mod 2π 52 The corresponding monodromy matrix Φ x x π satisfies the initial-value problem d dτ Φ x x τ = f x Φ x τ Φ x x τ 5 Φ x x = Id. 55 Using the ansatz Φ x x τ = Φ x x τ + εφ 1 x x τ + O ε 2 56 where Φ x x = Id Φ 1 x x = 57 and collecting terms of equal order in ε then yields Φ x x π = Id 58 π Φ 1 x x π = 1 3δ δ 1 π 2πζ. 59 For example for large the nontrivial eigenvalues of the monodromy matrix Φ x x π are given by 1 3 δ 1 + επ ζ ± δ 1 + ζ2 + O ε 2. 6 Since δ > 1 3 ζ+ 3 δ 1 1 δ + ζ2 > it follows that for δ > 1 3 and sufficiently large the grazing periodic trajectory is unstable when disregarding the effects of the jump map. In contrast when δ < 1 3 both eigenvalues lie inside the unit circle for sufficiently large i.e. the grazing periodic trajectory is stable when disregarding the effects of the jump map. References [1] uckenheimer J. and Holmes P Nonlinear Oscillations Dynamical Systems and Bifurcations of Vectorfields Springer-Verlag New York. [2] Zeeman E. C Catastrophe Theory: Selected Papers Addison-Wesley: Reading MA. [3] olubitsky M. and Schaeffer D Singularities and roups in Bifurcation Theory Springer-Verlag New York. [] Chin W. Ott E. Nusse H. E. and rebogi. C. 199 razing bifurcations in impact oscillators Physical Review E 5 pp

10 [5] Dankowicz H. and Nordmark A. B. 2 On the origin and bifurcations of stick-slip oscillations Physica D 136 pp [6] Di Bernardo M. Budd C. J. and Champneys A. R. 2 Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems Physica D 16 pp [7] Nordmark A. B. and Kowalczyk. P. 26 A codimension-two scenario of sliding solutions Nonlinearity 19 pp [8] Zhao X. and Dankowicz H. 26 Unfolding degenerate grazing dynamics in impact actuators Nonlinearity 19 pp [9] Nordmark A. B Non-periodic motion caused by grazing incidence in an impact oscillator Journal of Sound and Vibration 15 pp [1] Dankowicz H. and Zhao X. 25 Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators Physica D 22 pp [11] Fredriksson M. H. and Nordmark A. B Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators Proceedings of the Royal Society of London series A 53 pp [12] Molenaar J. de Weger J.. and Van de Water W. 21 Mappings of grazing impact oscillators Nonlinearity 12 pp [13] Foale S. 199 Analytical Determination of Bifurcations in an Impact Oscillator Proceedings of the Royal Society of London series A 37 pp [1] Foale S. and Bishop R. 199 Bifurcations in impacting systems Nonlinear Dynamics 6 pp [15] Mita M. Arai M. Tensaka S. Kobayashi D. and Fujita H. 23 A micromachined impact microactuator driven by electrostatic force IEEE Journal of Microelectromechanical Systems 121 pp [16] Zhao X. Dankowicz H. Reddy C. K. and Nayfeh A. H. 2 Modelling and simulation methodology for impact microactuators Journal of Micromechanics and Microengineering 1 pp [17] Zhao X. Reddy C. K. and Nayfeh A. H. 25 Nonlinear dynamics of an electrically driven impact microactuator Nonlinear Dynamics pp Scenario 1 a Scenario 2 a Scenario 3 B a A COD-2 B B COD-2 A D COD-2 A b b b E D C PD C PD D E PD E Figure 5: Sketch of the bifurcation curves associated with impacting trajectories emanating from the codimension-two bifurcation point at.95 Scenario 1.19 Scenario 2 and.1392 Scenario 3 denoted by COD-2. Here and in the following figures PD denotes period-doubling bifurcations denotes saddle-node bifurcations denotes crisis bifurcations e.g. homoclinic bifurcations involving an impacting chaotic attractor. The V -parameter space is divided into five regions according to the type of solutions: A- no stable or unstable period-1 orbits B- coexistence of an unstable period-1 impacting orbit and an unstable nonimpacting period-1 orbit C- coexistence of two unstable impacting period-1 orbits D- coexistence of an impacting chaotic attractor or periodic attractor of period > 1 and unstable impacting period-1 orbits E- coexistence of a stable impacting period-1 orbit and an unstable impacting periodic orbit and F- coexistence of a stable impacting period-1 orbit and an unstable nonimpacting periodic-1 orbit. F 1

11 -5*1-6 -1*1-6 V-V * -.6*1-5 V-V* 2.965* * *1-6 PD V-V* 9* *1-6 Figure 6: Bifurcation diagrams showing impacting and nonimpacting solutions corresponding to sweeps a top panel = and b middle and bottom panels = for Scenario 1. Bottom panel is the blow-up of the middle panel near the point. -*1 - V-V*.5*1-5 1*1 - V-V* 5* * *1-5 V-V* 1*1-6 V-V* 2.96*1-5 PD Figure 8: Bifurcation diagrams showing impacting and nonimpacting solutions corresponding to sweeps a top panel = and b middle and bottom panels = 1 5 for Scenario 3. Bottom panel is the blow-up of the middle panel near the grazing point. -1*1-2.3*1 - V-V* Figure 7: Bifurcation diagrams showing impacting and nonimpacting solutions corresponding to sweeps a top panel = and b bottom panel = for Scenario 2. 11