Grazing bifurcations in impact oscillators

Impact oscillators demonstrate interesting dynamical features. In particular, new types of bifurca tions take place as such systems evolve from a nonimpacting to an impacting state (or vice versa), as a system parameter varies smoothly. These bifurcations are called grazing bifurcations. In this paper we analyze the different types of grazing bifurcations that can occur in a simple sinusoidally forced oscilla tor system in the presence of friction and a hard wall with which the impacts take place. The general picture we obtain exemplifies universal features that are predicted to occur in a wide variety of impact oscillator systems.

Impact oscillators demonstrate interesting dynamical features. In particular, new types of bifurcations take place as such systems evolve from a nonimpacting to an impacting state (or vice versa), as a system parameter varies smoothly. These bifurcations are called grazing bifurcations. In this paper we analyze the different types of grazing bifurcations that can occur in a simple sinusoidally forced oscillator system in the presence of friction and a hard wall with which the impacts take place. The general picture we obtain exemplifies universal features that are predicted to occur in a wide variety of impact oscillator systems.

I. INTRODUCfiON
We say that a system is an impact oscillator if it has an oscillating object that impacts frequently with some other object [1,2). Impact oscillators occur in many technological situations. For example, mechanical devices are often engineered with loose fitting joints to accommodate thermal expansion, and the dynamical behavior of such systems often leads to impacts in the joint. In addition, many machines inevitably suffer from effects of vibroimpacts. A common feature shared by models of these systems is the smoothness of the systems between the impacts. Shaw and Holmes studied a piecewise linear, sinusoidally forced impact oscillator for various values of the forcing frequency [3][4][5][6]. Whiston originally showed the importance of grazing impacts (i.e., zero velocity impacts) of the global dynamics [7,8). Recently, Nordmark expanded (to first order) solutions in the neighborhood of a grazing orbit for a simple physical system (described below) and obtained a two dimensional map representing the dynamics of lm orbit in the neighborhood of the grazing state [9]. Nordmark also studied the dynamics of this map, obtaining several important results [9,10]; we will give details later. Nusse, Ott, and Yorke [11) obtained results for the dynamics of the one dimensional limit of a two dimensional map equivalent to the map derived by N ordmark. Budd, Dux, and Lamba considered sinusoidally forced impact oscillators, studying such features as chattering, intermittency, the effect of frequency and clearance variations, and the scaling of Lyapunov exponents at nonsmooth bifurations [12].
In this paper, we use the simple physical system shown in Fig. 1 as a prototype impact oscillator. This is the system considered in [9,10). A mass m is attached to a linear spring with spring constant k that is fixed to the wall on the right hand side. There is a sinusoidal external force F 0 sinwt acting on the mass. The friction force is proportional to the velocity of the mass with coefficient Jl· Here s represents the position of the mass m, and t is the time derivative of 5, which is the velocity of the mass m. A hard wall stands at the position Sc· When the amplitude of oscillation is sufficiently small, there are no impacts between the mass m and the wall at Sc• and the dynamics of the system is the same as that of a forced damped harmonic oscillator without the wall at Sc· As the amplitude of oscillation is increased, the mass m begins to have impacts with the wall, first with very low velocity. The bordering state between the impacting and nonimpacting is called a grazing impact, i.e., when the mass contacts with the wall at Sc with zero velocity. Interesting new bifurcations are observed at grazing, and they are called grazing bifurcations [9,10) considered in [11,13] and discussed in Sec. VI). The purpose of this paper is to present an analysis of grazing bifurcations for the system in Fig. 1. It is anticipated that these results are universal in that they apply to many systems in which impacts occur.
The two dimensional map derived for the system in Fig. 1 by Nordmark in [9] is equivalent to the following map, which will henceforth be referred to as the Nordmark map: { Xn +I =axn +yn +p Y = _ yx for xn ::S 0 , n +I n (1) { xn+I_= -vxn +yn +p _2 for Xn >0. Yn +I--yrxn (2) Here xn and Yn are transformed coordinates in the position-velocity space (s,t> evaluat.ed at times tn, where wt n = 2n 1T, and w is the frequency of the external forcing (see Fig. 1). The quantity 71 is the restitution coefficient of the impacts. The relation of r and a to the intrinsic properties of the oscillator such as the quantities k,m,w,ll in Fig. 1 is given in Sec. II. The parameter pis related to F 0 • Equations (1) govern the system if there is no impact between time tn and tn +I· Otherwise, if an impact takes place between tn and tn + I• then Eqs. (2) govern the system. Note that the Nordmark map is continuous at xn =0, but that its Jacobian matrix of partial derivatives is singular _at xn =0 [in particular, axn + ,Jaxn = -1 /(2y Xn) for Xn > 0]. This singularity at xn =0 is responsible for the new bifurcations studied in this paper. The map is normalized so that for fixed r and a, the long-time behavior is such that the orbit does not impact with the wall at Sc for p < 0, is in the grazing state for p=O, and may impact with the wall at Sc for p>O.
Thus if we vary p through zero with fixed r and a, the Nordmark map describes the dynamics of an orbit in the neighborhood of the grazing state if lpl << 1. Since the map is obtained by expansion of solutions in the neighborhood of the grazing state, its dynamics is related to the physical system only for lp I << 1. However, since we are interested in the bifurcations at p=O (i.e., the grazing bifurcations), the map is expected to capture the universal properties of impact oscillators near grazing. That is, other, physically different systems, when suitably normalized and expanded about the grazing state, should also yield Eqs. (1) and (2).
In what follows we shall be concerned with the bifurcation phenomena for the Nordmark map that occur as the bifurcation parameter p is increased through p=O (grazing incidence) with r and a held fixed. Depending on the values of r and a ( 0 < r < l' a < l + r for physically admissible systems), we observe three basic bifurcation scenarios listed as cases l-3 below. One of our goals will be to give an analysis to delineate the ( r, a) parameter space into regions in which the bifurcations in each case take place.
Case 1: Bifurcation from a stable period-] orbit in p < 0 to a reversed infinite period adding cascade as p increases through zero. Depending on r and a, there are two possible forms such a cascade can take: (a) a cascade where chaos appears in bands between successive windows of periodic behavior, and (b) a cascade with hysteresis. Subcase (a) is illustrated by the example shown in Fig. 2(a), while subcase (b) is illustrated by the example shown in Fig. 2(b). (See the figure caption for Figs. 2 for a description of how the bifurcation diagrams are made.) The line in the diagrams occurring for p < 0 represents the x location of an attracting period-1 orbit for the map. Since this period-I orbit is located in x < 0, it is determined  (1). In terms of the system in Fig. 1, this period-1 attractor of the map corresponds to a forced periodic orbit where the mass never impacts the wall. Referring to Fig. 2(a) we see that for subcase (a), as p is decreased from positive values, we encounter windows of stable periodic behavior, and each such window is followed by a band of chaos and then a window of stable periodic behavior whose period is one higher than the period in the previous window. Asp decreases, there is an infinite cascade of such windows of ever decreasing width in p and ever increasing period, accumulating on p=O+. To make this phenomenology clearer we plot again in Fig. 3(a) the bifurcation diagram for the same values of ( r, a) as in Fig. 2(a), but now using the variables x /p vs lnp. We clearly see in this figure that there are six successive period addings with period 3 occurring on the right of the figure and period 9 occurring on the left. Numerically, we find no evidence of any stable periodic orbits other than those in the reversed period adding cascade. Currently we believe that the p intervals between a period m window and a period m + 1 window are occupied entirely by a chaotic attractor. Now refer to Fig.  2(b), which illustrates subcase (b). We see that the p intervals of stable period m and period m + 1 orbits overlap, and chaotic attractors are not present in the cascade. Again, this occurs as a reversed infinite period adding cascade. Figure 3(b) is a bifurcation diagram using the variables x I p vs lnp for the same ( r, a) as for Fig. 2 We see three successive period addings in this figure, with period 2 occurring at the right of the figure and period 5 occurring at the left. We have derived a scaling rule for the widths of the periodic windows in terms of r and a, applicable to both subcases (a) and (b). The stable periodic orbits in our period adding cascades are numerically observed to be of a very special type. In particular, if the period of the orbit is m, then the orbit spends one iterate in x > 0 and the other m -1 iterates in x < 0. We call such a periodic orbit maximal. In terms of the system in Fig. 1, a maximal periodic orbit of the map corresponds to a forced periodic orbit where the mass impacts with the wall exactly once per period.
Case 2: Bifurcation from a stable period-I orbit in p < 0 to a chaotic attractor as p increases through zero. An example of a bifurcation diagram for this case is shown in Fig. 4(a). We see that as soon asp is increased through zero (corresponding to the occurrence of impacts in Fig.  1), chaos appears. Numerically, we find for Fig. 4(a) that there is no evidence of any window of stable periodic behavior throughout the entire range between p=O and the value of p at which the stable period-2 orbit first ap- pears. In general, case 2 is defined as follows: as p increases from zero there is an interval of p values occupied entirely by a chaotic attractor, and this interval terminates at the appearance of a periodic orbit of some period M 0 • In Fig. 4(a), M 0 =2, but other values of the period M 0 occur depending on the values of r and a. Figure 4(b) shows a case where M 0 = 3. Indeed, we observed numerically that M 0 ~ oo as the boundary in ( r, a) space between case 1 and case 2 is approached from the case 2 side. Case 3: Collision of an unstable period-M maximal orbit (which is a regular saddle, and is created, together with a stable period-M maximal orbit, in a saddle-node bifurcation in p < 0) and the period-] orbit at p = 0. When plotting a bifurcation diagram, the regular saddle, of course, does not show up. One observes that the attractor is a stable period-1 orbit for p < 0 and it is a stable period-M maximal orbit (which is created in the saddle-node bifurcation) for p > 0. Loosely speaking, we will say that there is a (discontinuous) "bifurcation" from a stable period-1 orbit to a stable period-M maximal orbit as p increases through zero. To explain the basic phenomenology of this case, imagine that the orbit is initialized on the period-1 attractor for some negative value of p, and p is then increased very slowly with time. While p remains negative, the orbit tracks the location of the period-1 orbit since the period-1 orbit is attracting for p < 0. However, when p increases through zero, the period-1 orbit becomes unstable and the orbit goes to some other attractor away from the period-1 orbit. We find that this other attractor is always a stable period-M maximal orbit, which is created in a saddle-node bifurcation in p < 0. The unstable period~M maximal orbit created in the same saddle-node bifurcation collides with the period-1 orbit at p=O. Furthermore, we :find that at p=O, depending on the parameters ( r, a), there exists either only one stable maximal periodic orbit or two stable maximal periodic orbits. When two stable maximal periodic orbits coexist, their periods differ by l. In the cases where two stable maximal orbits coexist, it is always the maximal orbit of lower period that the orbit goes to from the period-1 orbit as p increases slowly from negative to positive values. We call this the "observed" maximal orbit and we say that the period-1 orbit "bifurcates" to this observed maximal periodic orbit as p increases through zero. Figure   5(a) shows a bifurcation diagram for ( r, a) in the region where only a single period-3 stable maximal orbit exists at p=O (this is typical of what happens for other periods). We see that the period-3 stable maximal orbit is born in a saddle-node bifurcation at some negative p value, p = p 3 < 0. (The location of the period-3 saddle is indicated by the dashed lines in the figure.) For p 3 < p < 0, the stable period-1 orbit coexists with the pair of stable and unstable maximal period-3 orbits created at p=p 3 • As p~o-, the unstable period-3 maximal orbit collapses onto the period-1 orbit. The stable period-3 maximal orbit continues to exist in p > 0 and the period-1 orbit becomes a flip saddle in p > 0. In addition, we want to point out that the period-3 maximal saddle and the period-1 orbit are involved in the local bifurcation that occurs at p=O, while the stable period-3 maximal orbit is not (since it is bounded away from the origin). On the other hand, for p > 0, the solutions will converge to the stable period-3 maximal orbit that is created at p 3 • Therefore, we call the bifurcation a "bifurcation" from a period-1 attractor to a period-3 attractor. Figure 5(b) shows a bifurcation diagram for ( r, a) in the region where period-3 and period-4 stable maximal orbits coexist at p=O. Now two stable maximal orbits are created in p < 0, the period 3 in a saddle-node bifurcation at p = p 3 < 0, and the period 4 in a saddle-node bifurcation at p=p 4 <0, where p 4 <p 3 • Both stable maximal orbits continue to exist asp becomes positive, but, as already discussed, only the period 3 will be observed to bifurcate from the period-one orbit as p increases through zero. Later on in Sec. VA, it will be explained why this bifurcation to the lower period orbit is observed.
It should be noted that in all three cases above, the stable period-1 orbit that exists in p < 0 becomes a flip saddle in p > 0. That is, suppose 11 and K are eigenvalues -? = 1. A pair of stable and unstable period-3 maximal orbits are created in a saddle-node bifurcation at p = p3 < 0, and a pair of stable and unstable period-4 maximal orbits are created in a saddle-node bifurcation at p=p4 <0. The unstable periodic orbits are not shown. of the Jacobian matrix at the period-one orbit; then both 1771 < 1 and I Kl < 1 for p < 0 and 71 < -1 < K < 1 for p > 0.
The region of ( r, a) space corresponding to systems with non-negative friction [JL ?:: 0 in Fig. 1 and v?:: 0 in Eq. (3)] is shown in Fig. 6, where the parameter values corresponding to the various cases in Figs. 2-5 are labeled as points. (The region shown shaded is unphysical and corresponds to negative friction.) As shown subsequently, the requirement of positive friction leads to the restrictions 0 < r < 1 and a < I + r. This region is divided into two parts by the parabolic curve K given by r =a 2 /4. The part above curve K (i.e., regions I and II) corresponds to overdamping (i.e., the linear harmonic oscillator that results from Fig. 1 with the wall removed is overdamped). This leads to real eigenvalues for the Jacobian matrix of the linear map in Eqs. (1). The part below curve K (i.e., region Ill) corresponds to underdamped systems [or systems that have complex conjugate eigenvalues for the Jacobian matrix of the linear map in Eqs.  stable period-M maximal orbits exist at p=O for M = 3, 4, 5, 6 with -? = 1. As already noted, when two such orbits coexist, only the one of lower period will be observed to bifurcate from the period-1 orbit with slowly increasing p. Figure 7(b) [obtained by assigning the overlap regions of the (y,a) space in Fig. 7(a) to the lower period] shows regions for which the observed bifurcating orbit has period M. Regions corresponding to higher M appear in a similar way and accumulate on the curve K as Moo. It will become clear in Sees. IV and V that the delineation of the regions in Fig. 6 is valid for all 0 ~ ?-~ 1, while the results presented in Figs. 7 is obtained with?-= 1.
Nordmark [9,10] has previously discussed scaling for case 1 and obtained case 2. The existence of the two subcases within case 1, like our treatment of the existence  and stability of maximal orbits for cases 1 and 2 (see Sec. IV), is new. All the results reported for case 3, and the delineation of the ( r, a ) parameter space corresponding to each case, are also new.
In Sec. II, explicit relations between y, a, and the physical parameters of the model (Fig. 1) are obtained. Section III derives expressions for maximal periodic orbits. Sections IV and V contain an analysis of the Nordmark map for lp I << 1. This includes the existence and stability conditions of maximal periodic orbits for all ( r, a) located in the physically allowed regions in Fig. 6. The analysis results in the division of ( r, a) space into regions corresponding to the different types of grazing bifurcations in the system as explained above, as well as a scaling law for widths of the windows with high periods. Special attention is devoted to the limiting behavior as one approaches the boundaries in Fig. 6. This will, for example, show how an infinite period adding cascade results as one approaches the boundary to region I from one of the other regions. In Sec. VI, we discuss the results of this paper on grazing bifurcations in view of some general results on border-collision bifurcations obtained by Nusse and Yorke in [13]. Conclusions are presented in Sec. VII.

II. RELATION BETWEEN PHYSICAL QUANTITIES AND PARAMETERS OF THE NORD MARK MAP
In this section we study the relation between the parameters r and a and the physical parameters of Fig. 1, namely the mass m, the spring constant k, the frequency of the external forcing {t), and the friction coefficient J.L· With these expressions we will be able to understand the physical meaning of results obtained from our analysis of the Nordmark map, which is in terms of y and a. For the physical system in Fig. 1, the equation of motion without impacts with the wall at 5, is d 2~ +v!!l_ +0 2 5=F 0 sin21Tf, (3) dT dT where we have introduced the quantities v=21TJ.L/m{t), 0 2 =41T 2 k/m{t) 2 , and F 0 =41T 2 F 0 /m{t) 2 , and normalized time t so that the external forcing has frequency 21T and 21Tt=(t)t. The mapping from T=n to T=n + 1 for integer n is a Poincare return map on the plane (5,~) with constant phase, and thus has the same set of eigenvalues as the Jacobian matrix of the linear map in Eqs. (1). Let P be a particular solution of the differential equation (3).
Then the general solution of (3) is given by and where C 1 ,C 2 are real numbers and From now on, we assume that ~-40 2 *0. Then we also have the time derivative of 5 (i.e., velocity of the mass) from Eq. (4): 5=P+C1s1e +C 2 s 2 e .
Hence for T=n, the state vector in the (5,~) space is and for T=n + 1, we have The matrix B has the same set of eigenvalues as the Jacobian matrix (6) of the linear map in Eqs. (1). We denote the eigenvalues of matrices A and B by A. 1 From (6) we have Combining these relations, we obtain explicit expressions of the parameters r and a in terms of the physical parameters (10) a=A.t+A.z=e'l+/2=2e-vl2cosh [ v'v2~402]. (11) For positive friction v > 0, we have from (10) and (11) that O<r<1, O<a<1+r. (12) This also yields I A. 1 1 < 1 by (8) and (9) and corresponds to the unshaded region of the ( y, a) space in Fig. 6. Points on the curve K in Fig. 6 satisfy the relation a 2 -4y =0 [or, equivalently, v-40 2 =0 by Eqs. (7)- (11)] and correspond to systems with critical damping. Points above the curve K correspond to overdamped systems (i.e;, systems with real eigenvalues A 1 and A 2 ) and points below the curve K correspond to underdamped systems (i.e., systems with complex conjugate eigenvalues A 1 and A 2 ).
Also notice that y is related to the friction coefficient by Eq. (10). In the limit of large friction coefficient, v~ 00' we have r ~o. and the two dimensional map in Eqs. (1) and (2) reduces to the one dimensional map studied in [11], { axn +p for xn ::SO, Case 1 [along with subcases (a) and (b)] and case 2 were found [ 11] to occur for this one dimensional map for the a value ranges evident by examining the a axis (i.e., r =0) in Fig. 6 [i.e., case l(a) occurs for t <a< f; case l(b) occurs for 0 <a< f; and case 2 occurs for t <a< 1 ].
In the opposite limit of zero dissipation (i.e., v=O and -?= 1 ), the map given by Eqs. (1) and (2) becomes area preserving. This case has been studied in [12].

III. MAXIMAL PERIODIC ORBITS
We study the grazing bifurcations at p=O for Eqs. (1) and (2) in the physically admissible region of the ( r, a) parameter space as characterized in (12). For all these values of ( r, a), the system has a stable period-1 orbit for small negative p values, which becomes a flip saddle for small positive p values.
Our numerical experiments indicate that only one type of stable periodic orbit is involved in the bifurcations at p=O. We call such orbits the maximal periodic orbits. Here a maximal periodic orbit is a periodic orbit for which exactly one point per period is in the region x > 0. Our strategy is to find the range of p values in which a period-m maximal orbit exists and the range of p values in which the same orbit is stable.
By dividing both sides of (20) by Ar(k-u and using the notation &'kl=a/Ar(k-ll, with a standing for any variable, Eq. (20) takes the form From Eqs. (8) and (9), we have 0 < A 2 < A 1 < 1 for points that are not on the curve K. Thus (A 2 /A 1 )k and (A 1 )k both approach zero ask goes to infinity. Also in the expression of\P'k -I l [cf. Eq. (17)],

1-Ak
The expressions (28) and (29) are key results for our subsequent discussions. We see that a period-m maximal orbit exists for p < p";, and is stable for p > p";,. Let  There are two distinct situations: case 1 in which Im exists for all large m, and case 2 in which for every integer m > M 0 (for some threshold M 0 ) the interval I m does not exist. The first case implies bifurcations from the period-1 attractor in p < 0 to a reversed infinite period adding cascade in p > 0, and the second case implies bifurcations from the period-1 attractor in p < 0 to a chaotic attractor in p > 0. We discuss these two cases separately as follows.
A. Case 1: bifurcation from a period-1 attractor to a reversed infinite period adding cascade The interval I m exists for all large m if p"; > p";, for all large m. Then for any period m, there is an interval Im for which the period-m window appears if p E Im. Hence there is a reversed infinite cascade of period adding windows as p---+0+. Using the expressions of p";, and p"; in Furthermore, from (28) and (29), we can deduce a scaling law for the window widths as p---+0+. In particular, where I I m I = p";-p";, (assuming p"; > p";, for all large m) is the width of the period-m window. This scaling agrees accurately with our numerical results, and it applies (for large m) to all systems with ( y, a) in region I in Fig. 6.
This scaling law was also obtained in [9].
As indicated in Figs. 2 and 3, there are two different types of reversed cascades of period adding windows. For the first type, the system is chaotic between successive periodic windows in the bifurcation diagram, as in Figs. 2(a) and 3(a). Numerical experiments show no evidence of stable periodic orbits for the p values between the successive maximal periodic windows. For the second type, successive periodic windows overlap, and the system presumably does not have chaotic attractors, as in Figs. 2(b) and 3(b). The first type corresponds to the case in which the neighboring intervals I m and I m + 1 have no intersection, as schematically shown in Fig. 8(a). The system is presumably chaotic for p"; + 1 < p < p~. The second type corresponds to the case in which the neighboring intervals I m and I m + 1 overlap for large m, as schematically illustrated in Fig. 8(b). The period m and m + 1 orbits coexist for p~ <p <p"; +I• and we call this hysteresis. The border between these two types of cascades is p";, =p"; +I• which by (28) and (29) reduces to A. 1 = t· By Eq. (8) we find that this border is given by the segment of the line a=4r+t, II as shown in Fig. 6, i.e., the border between regions I and II.

V. ANALYSIS FOR CASE 3
The matrix A has complex conjugate eigenvalues if ( y, a) falls below the curve K in Fig. 6. Let Note that 8=0 on the curve K (given by y =a 2 /4).
In the region below the curve K in Fig. 6, we observe grazing bifurcations from a stable period-1 orbit to a stable period-M maximal orbit as p increases through zero (as described in case 3 in Sec. 1). Recall that the actual bifurcation is a collision of an unstable period-M maximal orbit and the period-1 orbit. For ( r, a) values below but very close to the curve K, the grazing bifurcations involve orbits with high periods. In particular, M--+ oo as 8--+0 (i.e., as the curve K is approached from below). When e is not small (i.e., when M is not large), we concentrate only on the local bifurcation that occurs at x =y =0 as p--+0-; while for small e (i.e., near the curve K) we are able to do more. Asp increases from zero, the grazing bifurcation to the period-M maximal orbit is either followed by chaos or by a reversed period adding cascade starting with a period M -I window.
For large M, we thus also investigate the occurrence and scaling properties of stable maximal periodic orbits in p > 0. This allows us to obtain an understanding of how phenomena below the curve K match on to those above the curve K (in particular, how the bifurcation from a stable period-1 orbit to a stable period-M maximal orbit of case 3 goes over to the bifurcation to the reversed infinite period adding cascade of case 1 and the interval of chaos extending from p=O of case 2 as the curve K is crossed from below.) A. When 6 is not necessarily small Our goal in this section is to find the regions under the curve Kin Fig. 6 corresponding to different values of the integer M. In this case of complex conjugate eigenvalues, we divide both sides of Eq. (20) by r 2 <k -I l instead of by A.i<k-u. This time we use the notation a<kl=a/r 2 <k-n, where a stands for any variable. Thus Eq. Since the integer M need not be large, we cannot make the approximation rM-1 -o. Thus the terms of order rk-l may not be dropped, and the quantity f/!~k-ll=1+A.;+A.r+ · · · +A.~-1 (where i=1,2) in the expression w<k-1) [cf. Eq. (17)] may not be approximated by 1 / ( 1 -A;). Then, substituting Eq. (34) into Eq. (19), we find that for a period-m maximal orbit with orbit points (x 1 ,y 1 ),(x 2 ,y 2 Notice that the right hand sides of the solutions [Eqs. (40) and (41)] are required to be real and positive.
For the part of' ( r ,a) parameter space in Fig. 6, numerical computations show that the product c:c; is nonpositive. With this in mind, we discuss the two kinds of period-m maximal orbits depending on the sign of the quantity C;' as follows: In this case, both solutions x\';'l and x~2l can exist.
The expressions for ~ and ~ in (40) and (41) indicate that a pair of period-m orbits, corresponding to x~';'l and x\2l, respectively, are created in a saddle-node bifurcation at some negative p value satisfying l-4C;'C;'tfm>=o.
The orbit corresponding to x\';'l only exists for p <0 [since the right hand side of Eq. (40) is negative for p > 0] and is numerically observed to always be unstable. In particular, it collapses onto the origin as p-0-. On the other hand, the orbit corresponding to x\2l continues to exist up to some positive p value and is observed to remain stable. Figure 5 in p > 0 that correspond to V x\';'l in the limit that the curve K is approached from below. From cases (i) and (ii) we see that a pair of stable and unstable period-M maximal orbits are created in p < 0 in systems with ( r, a) satisfying C~ < 0 and xl!' 1 < 0 for 1 ~ k ~M -1 in Eq. (35) for p=O. To delineate the regions of ( r, a) space satisfying these conditions for fixed M with -r2 = 1, we take a grid in the region below the curve K in Fig. 6 and numerically determine from Eqs. Claim 2. When the parameters (y,a) are in the region where only one maximal stable periodic orbit is created in p < 0, that orbit is the one that will be observed to bifurcate from the stable period-1 orbit asp increases slowly through zero.
To explain what we mean by "observed" in Claim 2, assume that p is initially negative and that the orbit is initially on the period-1 orbit. Now imagine that p is allowed to drift slowly upward with time. For p < 0 the orbit will track the location of the period-1 orbit because the period-1 orbit is stable. However, when p becomes positive, the period-1 orbit becomes unstable, and the orbit will go to some other attractor. Claim 2 is that the other attractor to which the orbit goes is always the stable maximal periodic orbit. Claim 3. When the parameters (y,a) are in an overlap region such that maximal stable orbits of period M and M + 1 are both created in p < 0, the lower period stable maximal orbit (i.e., period M) is the one that will be observed to "bifurcate" from the stable period-1 orbit asp increases slowly through zero.  (3 2 for the angle between the half line through 0 and 4>M and the negative y axis (see Fig. 9 for M=5). Now, let 0<(3 1 <()and 0<(3 2 <fJ be given. This implies the following. If at most two pairs of maximal orbits can be created at negative p values. Support for Claims 2 and 3. From now on, U 0 denotes a suitable region that includes the origin in its interior. We observed numerically that after its birth at PM <0, the stable period-M maximal orbit and its basin of attraction (.13M) are embedded in a region U 0 • This region U 0 was originally occupied by the basin 13 1 of the period-1 orbit. In particular, for PM <p <0, 13M and 131 share the region U 0 that was occupied entirely by 13 1 before the birth of 13M. (Here, if two maximal orbits coexist, M denotes the lower of the two periods.) As p increases toward zero, the area occupied by 13M increases and the area occupied by 13 1 diminishes. Roughly speaking, as p--.0-the region that was originally occupied by 13 1 is gradually taken over by 13M. In particular, 13 1 shrinks to a finite number of curves emanating from the origin as p--.0-. Meanwhile, the region U 0 occupied by 13M and 13 1 combined, as well as the basins of attraction of other stable periodic orbits, are not significantly altered. A region U that includes U 0 may also have points that belong to basins of stable nonmaximal periodic orbits. As an example illustrating Claim 3, the point (y,a)=(0.9,0.5) with ,-2= 1 falls in the overlap of regions M =3 and M =4 in Fig. 7(a). A pair of period-3 maximal orbits are created in a saddle-node bifurcation at p = p 3 and a pair of period-4 maximal orbits are created in a saddle-node bifurcation at p=p 4 , where p 4 <p 3 <0. A stable period-7 orbit (which is not a maximal orbit) also exists.    Fig. 11(a), (y,a)=(0.9,0.7) in Fig. 1l(b), and (y,a)=(0.9,0.8) in Fig. ll(c). The points (y,a) in Figs. 1l(a)-1l(c) all fall in the overlap of regions M = 3 and M =4 in Fig. 7(a), so both stable period-3 and period-4 maximal orbits exist at p=O. Comparing Fig. lO(b), Fig.  ll(a), Fig. 11(b), and Fig. ll(c), we see that at p=O the territories occupied by .23 3 and .23 7 combined are reduced and the area of .23 3 decreases to zero as ( r, a) moves upwards approaching the upper boundary of the overlap of the regions M = 3 and M =4 in Fig. 7(a). Thus we see how M = 4 supplants M = 3 as the stable maximal orbit observed to bifurcate from the period-1 orbit as p increases through zero. Also notice that .23 7 no longer exists in Figs  p (1-a+y) r2<m-ll [Notice that Qm and p always have the same signs; see also (12).] Using the approximated expressions above, we reexamine the two kinds of period-m maximal orbits depending on the sign of C;' (as in Sec. VA).
where If:. is the existence threshold in the complex case.  Fig. 6 with y < f, which is the right hand side border of region I. Moreover, the neighboring intervals J m and J m + 1 do not overlap if {J',: + 1 < p~, which is similar to the schematic illustration in Fig. 8(a), while the intervals J m and J m + 1 overlap if {J'; + 1 > p~, which is similar to the schematic illustration in Fig. 8(b). By the expressions of p~ and if:. in (55) and (56) Similarly, in the limit of 8-o, (57) and the opposite of (58) combined correspond to the right hand side border of region l(a), where the system is chaotic between successive windows with high periods; while the opposite of (57) corresponds to the right hand side border of region II, where the system is chaotic in an interval in p > 0 extending from zero (recall that the period-M maximal orbits collapse to the origin as e-o).
The expressions (55) and (56)  Thus we see how the phenomena below the curve K approach those above K as 8-0.

VI. DISCUSSION OF THE GRAZING BIFURCATION IN VIEW OF ORBIT INDEX
In the bifurcation theory for maps, attention has been focused on differentiable maps when one or more eigenvalues of a fixed point (or periodic point) cross the unit circle. When this occurs, the nature of the fixed point changes. For example, a fixed point attractor becomes a saddle (possibly a flip saddle) or a repeller. The Nordmark map, however, is piecewise smooth and is not differentiable at x =0. In particular, the Jacobian matrix of the Nordmark map changes discontinuously at x =0 and becomes singular as . This singularity is responsible for the new bifurcation phenomena studied in this paper. According to [13], the fixed point of the Nordmark map, which is an attractor located in x < 0 for p < 0 and a flip saddle located in x > 0 for p > 0, is a border crossing fixed point; and the grazing bifurcations that take place at p = 0 are examples of border-collision bifurcations. In the rest of this section, we present the precise definitions of the terms used above, introduce the border-collision bifurcation theorem obtained by Nusse and Yorke in [13], and discuss the grazing bifurcations analyzed in this paper in the context of the more general results on border-collision bifurcations.
A map is smooth if it has a continuous derivative. Here we examine maps that are piecewise smooth, and restrict our attention to those that are smooth in two regions of the plane with the border between these regions being a smooth curve. Let r be a smooth curve that divides the plane into two regions denoted by R 1 and R 2 • We say that a map F from the phase space R 2 to itself is piecewise smooth if (i) F is continuous and (ii) F is smooth in both the regions R 1 and R 2 • Let F(·,ll)=FI-' be a one-parameter family of piecewise smooth maps from the phase space R 2 to itself, which depends smoothly on the parameter ll• where 1l varies in a certain interval on the real line. Let El-' denote a fixed point of Fl-' defined on -E < 1l < E for some E > 0. The position of E 1-' depends continuously on ll· We say Ell is a border crossing fixed point if it crosses the border r between the two regions R 1 and R 2 as 1l is varied. Assume that the crossing occurs at ll=O. A periodic orbit P is a border crossing orbit if it includes a point that is a border crossing fixed point under some iterate of the map. If, furthermore, there exists a neighborhood U of the orbit P such that P is the only periodic orbit in U at 1l = 0, then P is an isolated border crossing orbit.
For a general approach we need the concept of the "orbit index" of a periodic orbit [15]. The orbit index is a number associated with a periodic orbit, and this number is useful in understanding the allowable patterns of bifurcations the orbit undergoes. We say an orbit of period p is typical if its Jacobian matrix (i.e., the Jacobian matrix of the pth iterate of the map at a point on the orbit) exists and neither + 1 or -1 is an eigenvalue (of the Jacobian matrix). For typical orbits, the orbit index is -1, 0, or + 1. The orbit index is a bifurcation invariant with respect to, as in our cases, the periodic orbits that collapse onto the fixed point E ~-' as ll-o. That is, the sum of the orbit indices of the periodic orbits that collapse onto the fixed point Ell as ll-ois equal to the corresponding sum as ll-o+. Suppose a typical periodic orbit P of a map F has (minimum) period p. The orbit index of P depends on the eigenvalues of the Jacobian matrix AP of the map FP at one ofthe points on P. Let m be the number of real eigenvalues of AP smaller than -1, and let n be the number of real eigenvalues of AP greater than + 1. The orbit index I p of P is defined by lp=O if m is odd, I P = -l if m is even and n is odd , I P = + 1 if both m and n are even .
If the orbit itidex = -1, then the orbit is a regular saddle.
If the orbit index =0, then the orbit is a flip saddle. The typical orbits with orbit index + 1 are repellers and attractors and orbits with nonreal eigenvalues. The definition of the drbit index is technical when a point of the orbit lies on the boundary r since the Jacobian matrix of the map does not exist for points on the boundary.
It is unnecessary to define the orbit index on r since we consider orbits for J.L=foO.
For a moment, assume that E J. L is in the interior of the region R 1 (or the region R 2 ), and denote 71 and K for the eigenvalues of the Jacobian matrix DFJ.L(EJ.L). If neither of the two eigenvalues 71 and K is on the unit circle, then the fixed point E J. L is a flip saddle (and has index 0) if 71 < -1 <K < 1; E~' is a regular saddle (and has index -1) if -1 < 71 < 1 < K; E is a repeller (and has index + 1 ) if both l11l > 1 and IKI> 1; and El' is an attractor (and has index + 1) if both l11l < 1 and IKI < 1. (Note that E~' has orbit index + 1 if the eigenvalues are not real.) Hence, a typical fixed point is a flip saddle, a regular saddle, a repeller, or an attractor. Now we introduce the border-collision bifurcation theorem. Let the regions R 1 and R 2 , the map F J.L, and the fixed point (or periodic point) E I' be as above. Suppose there exists a number E > 0 such that (i) E 0 is on the border of the two regions R 1 and R 2 , (ii) for -E < J.L < 0 the fixed point E J. L is in the region R 1 , and its index is I 1 , and (iii) for 0 < J.L < E the fixed point E J. L is in the region R 2 , and its index is I 2 • If I 1 and I 2 are different, then (as to be stated next) some bifurcation must occur at E 0 so that the sum of the orbit indices is invariant as J.L crosses zero. The following "border-collision bifurcation theorem" is obtained in [13]: Border-collision bifurcation theorem. For each two dimensional piecewise smootp map depending smoothly on a parameter J.L, if the index of an isolated border crossing orbit changes as J.L crosses zero, then at J.L =0 a bifurcation occurs at this point, a bifurcation involving at least one additional periodic orbit.
This result says that additional fixed points or periodic points must bifurcate from E 0 at J.L =0 if the index of E J. L changes as J.L crosses zero. Since this bifurcation occurs while the fixed point (or periodic point) collides with the border of the regions R 1 and R 2 , we call it a bordercollision bifurcation. In other words, a border-collision bifurcation is a bifurcation at a fixed point (or periodic point) on the border of the two regions, when the orbit index of the fixed point (or periodic point) before the collision with the border is different from the orbit index of the fixed point (or periodic point) after the collision.
Therefore, if the orbit index of a fixed point (or periodic orbit) is different before and after it crosses the border r, the following two things can possibly occur.
(i) There are additional periodic orbits which collapse onto E 0 as J.L~O+ and/or J.L~O-, whose individual indices are such as to make the total orbit index conserved.
(ii) One or more chaotic sets collapse onto E 0 as J.L~O+ and/or J.L~O-. (Since the orbit index for a chaotic set is not defined, conservation of the index is no longer an issue.) The border separating the two regions in which the Nordmark map is smooth is the line x =0. The fixed point in the Nordmark map crosses the border x =0 at p=O; thus it is a border crossing fixed point. For p<O, the fixed point is in the region x < 0 and is stable (thus it has orbit index I 1 = + 1 ); for p > 0, the fixed point is in the region x > 0 and is a flip saddle (thus it has orbit index I 2 =0). We find numerically that it is an isolated saddle at p=O. Since I 1 =foi 2 , by the border-collision bifurcation theorem, there must be other orbits (periodic or chaotic) that collapse onto the fixed point at p=O and are involved in the bifurcation there.
For systems with ( y, a) in the region I (Case 1) in Fig.  6, a reversed infinite period adding cascade collapses onto the fixed point [located at (0,0)] as p~o+. Conservation of the orbit index before and after the bifurcation at p = 0 is not violated in this case, because the orbit index of the stable periodic orbits in the infinite cascade is + 1 and the orbit index of the chaotic sets in the infinite cascade is not defined. For systems with ( y, a) in the region II (Case 2) in Fig. 6, a chaotic set collapses onto the fixed point as p~o +, whose orbit index is not defined. As for systems with (y,a) in the region III (Case 3), which is below the curve K in Fig. 6, an unstable period-M maximal orbit (which is a regular saddle and thus has orbit index -1) collapses onto the fixed point as p~o-. The value of M is determined by Eq. (45). Therefore the sum of the orbit indices is invariant (and equals zero) as the fixed point crosses the border x =0. Hence the grazing bifurcations studied in this paper are border-collision bifurcations involving different types of orbits that collapse onto the fixed point as p~O+ or p~o-.

VIII. CONCLUSION
We have observed three major types of grazing bifurcations: (i) bifurcations from a stable period-1 orbit to a reversed infinite period adding cascade; (ii) bifurcation from a stable period-! orbit to attracting chaos occupying a full interval of the bifurcation parameter; and (iii) collision of an unstable maximal periodic orbit and a period-1 orbit, which is observed to be a local bifurcation from a stable period-1 orbit to a stable maximal periodic orbit. These bifurcations are "unconventional" in that they do not occur in smooth systems. Since the Nordmark map represents the dynamics of typical systems that have low-velocity impacts and that are smooth between the impacts, the bifurcations studied in this paper are expected to be universal for such systems.