1 Introduction

Because of the essential characteristics of non-smooth dynamics in various impacting oscillators, discontinuity induced bifurcations, especially grazing bifurcations^[1-34], have been widely studied. Many published papers focused on low dimensional dynamic models with theoretical, numerical, and experimental methods. Brzeski et al.^[2] presented a bifurcation analysis in the dynamic model of a church bell, and revealed the phenomenon of bifurcation through impact. Jiang et al.^[3] presented a numerical study comparing the grazing-induced bifurcations in impact oscillators with one-sided elastic and rigid constraints. Yi and Du^[4] investigated the degenerate grazing bifurcation in a smooth and discontinuous oscillator. Li et al.^[5] investigated the chaotic behavior of a universal Nordmark map from the probabilistic point of view. Kryzhevich^[6] and Kryzhevich and Wiercigroch^[7] studied the topological features and occurrence of a chaotic invariant set in impact oscillators near the grazing point. Mason and Piiroinen^[8] investigated the novel interaction between global boundary crises and grazing bifurcation in the impact model of spur gears. Meanwhile, the useful theory of discontinuity geometry^[11-12] is extended to analyze the existence of elementary near-grazing impact periodic motion^[13] and the relationship between the saddle-node and grazing bifurcations^[14], and many new insights into the rich dynamic behavior in a periodically forced impact oscillator with two discontinuity surfaces are presented^[15]. From the perspective of experimental studies corresponding to grazing bifurcation, Piiroinen et al.^[16] performed numerical and experimental studies on the grazing bifurcation of an excited pendulum with one degree of freedom. Ing et al.^[17] experimentally and semi-analytically studied the near-grazing dynamics of a linear impact oscillator with secondary elastic support. Chakraborty and Balachandran^[18] investigated the non-smooth dynamics of impacting cantilevers at different scales, and observed period-doubling events close to the grazing point.

In the context of grazing bifurcations, the discontinuity mapping technique^[19-22] is an effective tool for analytically deriving the normal form of the Poincaré map near grazing. Using the derived Poincaré map, the corresponding near-grazing bifurcations and chaos can be further analyzed and simulated. Nordmark^[19] originally introduced the concept of local discontinuous mapping, and revealed that a square root term could exist in the Poincaré map in hard impacting oscillators near grazing. His pioneering work laid the foundation for many subsequent studies. Near the grazing point, the square root term can induce an infinite eigenvalue from the Jacobian matrix and lead to instability of the elementary near-grazing impact periodic motions. Kundu et al.^[23-24] investigated the character of the normal form of the Poincaré map in the neighborhood of a grazing periodic orbit, and demonstrated how the excitation frequency could be chosen such that the infinite eigenvalue of the Jacobian matrix vanished.

The bifurcations of elementary near-grazing impact periodic motion and the stabilization of near-grazing dynamics have also been widely studied. In Refs. [25]-[30], it has been shown that the bifurcation curves of grazing period-doubling saddle nodes could meet at certain degenerate grazing points in the dynamic model of impact oscillators with one degree of freedom. Regarding the control in near-grazing dynamics, Dankowicz and Jerrelind^[31] initially proposed an interesting idea of using temporally discrete linear feedback control to change the conditions at the grazing point according to the grazing stability criterion proposed by Fredriksson and Nordmark^[32]. Misra et al.^[33] used a temporally discrete feedback controller to suppress the discontinuity-induced fold bifurcations of limit cycles in a soft impact oscillator. Xu et al.^[34] suppressed the discontinuous grazing bifurcations in an impact system with rigid clearance and two degrees of freedom.

Since grazing bifurcation can induce instability in elementary near-grazing impact periodic motions and lead to some unwanted dynamic phenomena, e.g., jump phenomena, subsequent discontinuous transitions, and dynamic uncertainty of chaotic motion, in this paper, we aim to investigate a control strategy for the grazing-induced instability in an impact oscillator with one degree of freedom. This paper extends the results from Refs. [10] and [30]. The primary contributions of this paper are as follows:

(ⅰ) What results could be obtained from an extended analysis of discontinuity mapping is an open question since the higher order terms in discontinuity mapping are often omitted in the bifurcation analysis and control of grazing in the aforementioned publications. In this paper, the commonly-used lower order zero time discontinuity mapping is extended to higher order terms to perform the necessary analysis, and the necessity of using higher order discontinuity mapping is also illustrated.

(ⅱ) The elimination of the infinite eigenvalue from the Jacobian matrix for the control goal in this paper requires high precision estimation of all eigenvalues in the neighborhood of certain grazing bifurcation points. Compared with previous papers^{[10, 30]}, we perform a perturbation analysis for the eigenvalues with higher order discontinuity mapping. As will be shown, the eigenvalues predicted from the perturbation analysis can agree very well with the results obtained from the shooting method^[35]. Degenerate grazing bifurcation occurs when the infinite eigenvalue is eliminated.

The remainder of this paper is organized as follows. In Section 2, the model describing an impacting oscillator with one degree of freedom is presented. In Section 3, local leading zero time discontinuity mapping is briefly extended to higher order. Based on the higher order zero time discontinuity mapping, the perturbation analysis of eigenvalues corresponding to near-grazing period-one impact motion is performed, and the control precondition for grazing-induced instability is shown in Section 4. In Section 5, the analytical conditions corresponding to the control goal are fully established, and the numerical results are presented for better understanding. Finally, the conclusions are summarized in Section 6.

2 Model for an impacting oscillator with one degree of freedom

The dynamic model of an impacting oscillator with one degree of freedom and harmonic excitation can be found in many published papers^{[3-4, 6, 8-17, 22-30, 36-39]}. Though this model is quite simple, many dynamical behaviors can be exhibited due to the non-smooth nature of impacts. The aforementioned papers have explored many different aspects, e.g., local grazing bifurcation analyses^[22-30], global dynamics analyses^{[8-9, 36-37]}, and experimental investigations^[16-18]. In addition to harmonic excitation, few papers are devoted to non-harmonically forced impacting oscillators^[38-39]. Guzek et al.^[38] analyzed the behavior of the bilinear oscillator under constant preload with the assumption that the spring had lower stiffness in tension than in compression. They pointed out that the use of impact oscillator as an asymptotic system at low stiffness ratios should be only possible for sufficiently small preload values.

In this paper, the model from Ref. [30] is adopted to present some new results. Assume that the equations governing the impact and non-impact motions are

(1)

where x=d is the position of a rigid wall, the overdot symbol denotes differentiation with respect to the time t, 0 < ζ < 1 is the damping coefficient, 0 < r < 1 is the coefficient of restitution, ω is the harmonic excitation frequency, and τ₀ is the phase angle. In terms of the regular notation in this paper, different periodic impact motions are characterized by the symbol p/n, where p is the number of impacts, and n is the number of the excitation periods over the entire period. For example, the symbol 0/1 denotes the non-impact period-one motion, and the symbol 1/1 denotes the period-one motion with one impact.

When the system parameters keep changing, e.g., the rigid wall position d or the excitation frequency ω, the grazing contact between the mass and rigid wall might arise as special events. Under such circumstances, the critical non-dimensional clearance value can be written as follows:

(2)

where

Furthermore, the regular non-sticking condition at grazing requires that the acceleration of the oscillator should satisfy

(3)

where τ^* is the critical value when grazing events arise during the 0/1 motion. In what follows, the rigid wall position d will be taken as a bifurcation parameter.

3 Higher order zero time discontinuity mapping

The discontinuity mapping technique initially derived by Nordmark^[19] has been widely used in dealing with the stability and bifurcation analyses of near-grazing 1/1 impact periodic motions. To apply the corresponding technique in the dynamic model of an impact oscillator with one degree of freedom, the Poincaré sections should be defined to construct the corresponding Poincaré maps. The Poincaré sections with constant phase are required for the analysis of zero time discontinuity mapping. In this paper, the first Poincaré section is defined as follows:

(4)

where τ =mod(ωt+τ₀, 2π) is the phase angle of the external harmonic excitation. Under the theory of discontinuity mapping, the smooth return map on the Poincaré section Π₁ can be specified as follows:

(5)

This smooth map can be analytically expressed as follows:

(6)

where N_g is the linearization matrix, and is defined as is the grazing point on the Poincaré section Π₁.

As for the local map defined on the Poincaré section Π₁, the generalized local zero time discontinuity mapping can be expressed as follows^[21]:

(7)

where X is the state vector, the function J(X) is the discrete impact event determined by the distance function H_max (X, d), and d is the bifurcation parameter. The time t denotes a small quantity here, and ϕ(X, t) is the smooth flow. For the non-impact case H_max (X, d) < 0, the local zero time discontinuity mapping is an identity mapping, while the local zero time discontinuity mapping is often approximated by an expansion of the smooth flow in the vicinity of the grazing point for the impact case H_max (X, d)≥ 0. In most cases, the approximate local discontinuity mapping is truncated with only the leading order term in common use. However, higher order truncation is strongly required for special cases, e.g., degenerate or other co-dimension two grazing bifurcations. In this paper, the commonly-used zero time discontinuity mapping is extended to higher order to perform the necessary analysis. To achieve the goal of expanding local discontinuity mapping, the second-order terms in the series expansion of the smooth flow are kept during the derivation. It should be noted that the variable t in the series expansion of the smooth flow ϕ(X, t) is considered as an independent variable for error estimation. Therefore, the derivatives with respect to X and t are well-defined. Although the newly added second-order terms can increase the calculation complexity and other analyses. The requirement for the higher order discontinuity mapping close to the degenerate grazing bifurcation points will be demonstrated in the next section.

For the impact case, i.e., H_max (X, d)≥ 0, the local second-order zero time discontinuity mapping is given as follows without providing many details for brevity (see more details about the theory of local zero time discontinuity mapping in Ref. [21]):

(8)

where , and h.o.t. denotes the omitted higher order terms with respect to the variables (X-X^*), δ₁, and δ₂, in which , and . Note that δ₂ is a small positive quantity close to the grazing point. The involved scalar function H_max (X, d) can be defined as follows:

where H_X =[1,0], v_X =[0, 1], and H_d =-1. Based on Eq. (8), the global normal form of the Poincaré map near the grazing orbit can be written as follows:

(9)

Further analyses can be performed accordingly.

4 Perturbation of the eigenvalues of the near-grazing 1/1 motion 4.1 Fixed point of the 1/1 motion on the Poincaré section

For suppressing grazing-induced instability, the stability analysis of the near-grazing 1/1 motion is required to establish a proper control strategy. The perturbation analysis of the characteristic equation can be performed with the aid of the higher order discontinuity mapping presented in Eq. (8). Before the perturbation procedure can be finished, the fixed point of the near-grazing 1/1 motion on the Poincaré section Π₁ should be determined to derive the corresponding Jacobian matrix. Based on the newly derived higher order zero time discontinuity mapping, the global Poincaré map for the 1/1 impact periodic motion can be written as follows:

(10)

The fixed point of the 1/1 motion on the Poincaré section Π₁ can be derived as follows:

(11)

Suppose |I-N_g-N_gβ₃(δ₁-δ₂)|≠ 0. Then, we have

(12)

where D₀ =(I-N_g)^-1, |I-N_g|≠ 0, and D₁ =(I-N_g)^-1Nβ₃(I-N_g)^-1. Thus, the fixed point of the 1/1 motion on the Poincaré section Π₁ can be written as follows:

(13)

To establish the ordering relationship between δ₁ and δ₂, we define

(14)

where H_X =[1,0], and v_X =[0, 1]. According to the definitions of δ₁ and δ₂ presented below Eq. (8), it is clear that

(15)

Thus, the perturbation solution of δ₁ in terms of δ₂ can be written as follows:

(16)

Substituting Eq. (16) into Eq. (15) yields the following scalar equation:

(17)

where

Let the sufficiently small quantity (d-d^*) be a perturbation parameter. Then, the unperturbed case corresponding to Eq. (17) can be written as 0=2ξ₁δ₂ +a^*γ₁δ₂² +O(δ₂³). It is clear that δ₂ =0 would be a double root of the above unperturbed equation, which could provide that ξ₁ =0 and γ₁ ≠ 0. Under such a condition, the grazing bifurcation is called degenerate grazing^[30] and the corresponding degenerate grazing points of the elementary near-grazing 1/1 motion can be analytically expressed as ω^*=2ω_d/m, where ω_d is the natural frequency of the oscillator, and m is an integer. It should be noted that there exists more than one way to analytically determine the degenerate grazing points in the model of an impacting oscillator with one degree of freedom. More details can be found in Refs. [23]-[27] and [30]. In a sufficiently small neighborhood of degenerate grazing points, Eq. (17) will have at most two real perturbed solutions according to the perturbation theory. Therefore, the O(δ₂³) terms in Eq. (17) can be omitted and the scalar equation (17) can be further simplified in a sufficiently small neighborhood of degenerate grazing points as follows:

(18)

4.2 Perturbation analysis for characteristic equation

When applying perturbation analyses to the characteristic equation, the Jacobian matrix corresponding to the near-grazing 1/1 motion becomes required. According to Eq. (10), the required Jacobian matrix can be expressed as follows:

(19)

where

After substituting Eq. (13) and Eq. (16) into Eq. (19), an expansion of the Jacobian matrix D_P in terms of δ₂ can be obtained as follows:

(20)

where

(21)

and S₂ is too tedious to be analytically expressed. Based on Eqs. (20) and (21), we can express the characteristic equation of the near-grazing 1/1 motion as follows:

(22)

where λ corresponds to the eigenvalue of the 1/1 motion, N_{g, 12} indicates an element of the matrix N_g, and χ₁, χ₂, χ₃, and χ₄ are bounded constants related to the matrices S₁ and S₂.

From the perspective of perturbation, Eq. (22) is singularly perturbed in terms of δ₂ when N_{g, 12} ≠ 0. In such a situation, a singular eigenvalue will arise and lead to instability of the 1/1 motion branching from the grazing bifurcation point. This eigenvalue can be infinite at the grazing bifurcation point. Obviously, the singular term in Eq. (22) can be eliminated only when N_{g, 12} =0. Therefore, Eq. (22) turns out to be regularly perturbed, and bounded eigenvalues can be obtained. N_{g, 12} =0 is the precondition of the control goal, i.e., the suppression of the grazing-induced instability. What should be emphasized is that N_{g, 12} =0 can lead toξ₁ =0 in this model for an impact oscillator with one degree of freedom, i.e., N_{g, 12} =0 means ω^*=2ω_d/m. This feature can be easily obtained from the definition of ξ₁ in Eq. (14).

Considering the fact that Eq. (22) is regularly perturbed under the precondition of degenerate grazing bifurcation, it shall be quite clear that the eigenvalue λ is restricted in a sufficiently small neighborhood of λ₀, i.e., λ =λ₀ +O(δ₂), where δ₂ is sufficiently small. As a result, the stability corresponding to the near-grazing 1/1 motion can be determined from the modulus of λ₀, i.e., if |λ₀| < 1, |λ| < 1 can be guaranteed for sufficiently small δ₂ and the near-grazing 1/1 motion is stable. Otherwise, the 1/1 motion is unstable. Here, λ₀ is the solution of the following unperturbed equation:

(23)

where the analytical expression of χ_i (ω^*) in this model for an impact oscillator with one degree of freedom can obtained as follows:

(24)

It should be noted that the higher order truncation of local zero time discontinuity mapping is required in order to guarantee the correctness of these important coefficients χ_i (ω^*). A misunderstanding of the stability analysis may be resulted in if the higher order terms in the local mapping are omitted. Practical computation of the eigenvalue λ at the fixed bifurcation parameter (d-d^*) can be conducted by the following truncation of Eq. (22):

(25)

For a given parameter value (d-d^*), Eq. (18) can help determine the value of δ₂. Then, the required eigenvalue λ can be obtained by substituting δ₂ into Eq. (25). The validity of Eqs. (23) and (25) will be verified numerically in the following section.

Regarding non-degenerate grazing bifurcation, the eigenvalue in Eq. (22) can also be approximated by truncating the nonlinear equation as follows:

(26)

The reason why the above truncation can be performed is that λ in the O(δ₂) and O(1/δ₂) terms in Eq. (22) are both of order one. Thus, the O(δ₂) term can be regarded as smaller than the O(1/δ₂) term and can be further neglected. Though the omitted O(δ₂) term in Eq. (22) might still be large with respect to zero, it should be noted that the key information regarding the eigenvalues has been captured in Eq. (26). The validity of this truncation will be also verified in the following section.

5 Suppression of grazing-induced instability 5.1 Control strategy

As introduced in the previous section, the grazing-induced instability of the near-grazing 1/1 motion can lead to some unwanted dynamic phenomena, such as the jump phenomena close to grazing. Though the existing grazing control strategies^[31-34] can effectively suppress the jump phenomena at the grazing point, they cannot avoid the unwanted dynamic phenomena of subsequent discontinuous transitions and the dynamic uncertainty of chaotic motion. Therefore, a new control strategy is required to suppress grazing-induced instability. It has been revealed in Subsection 4.2 that degenerate grazing bifurcation is the precondition for suppressing the grazing-induced instability. This subsection aims to provide the details on the control strategy by using a temporally discrete feedback controller^[31]. To make use of the previous result, i.e., the truncated eigenvalue approximation in Eq. (23), the temporally discrete feedback controller applied in this paper must not increase the dimension of the control system. Under such conditions, the adopted temporally discrete feedback controller is designed to change the velocity of the impact oscillator on a selected Poincaré section. In this paper, the feedback controller is applied on a new Poincaré section Π₂. The new section Π₂ for numerical simulation is

(27)

The temporally discrete controller is

(28)

where X^**=(x^**, )^T is the fixed point of the grazing periodic orbit on the Poincaré section Π₂, and c₁ and c₂ stand for the control parameters.

In order to apply the discontinuity mapping technique in this control system, the following two new maps are required:

(29)

Then, the new return map on the Poincaré section Π₁ that involves the discrete controller G(X, c_i) can be written as follows:

(30)

where the above linearization matrices are

(31)

Therefore, the matrix N_g can be defined as follows:

(32)

In such a situation, one can see that the control parameters c₁ and c₂ will enter the linearization matrix N_g and change the dynamics of the control system.

According to the perturbation analysis presented in Subsection 4.2, we know that degenerate grazing bifurcation is the precondition for suppressing any grazing-induced instability, i.e., the control parameters c₁ and c₂ must be set to force the grazing bifurcation to become degenerate so as to eliminate the infinite eigenvalue. The first requirement on the control parameters c₁ and c₂ is

(33)

As mentioned before, N_{g, 12} indicates the element inside the matrix N_g. Meanwhile, the stability corresponding to the near-grazing 1/1 motion can be determined from the modulus λ₀ in degenerate grazing. Therefore, the extra required stability condition for the controlled 1/1 motion sufficiently close to grazing is

(34)

According to Ref. [40], the above stability conditions are equivalent to

(35)

Meanwhile, to guarantee the appearance of a continuous transition between non-impact and controlled stable impact motions, the sign of γ₁ (c_i) for the control system shall remain positive. Thus, γ₁ (c_i) should be positive to ensure that the controlled impact 1/1 motion exists in the post-grazing region, i.e., d < d^*. Thus, the last condition is

(36)

At this stage, all required conditions for the suppression of grazing-induced instability corresponding to the 1/1 motion have been established. Once the control parameters c₁ and c₂ are chosen to fulfill the established conditions, i.e., Eqs. (33), (35), and (36), grazing-induced instability will be suppressed and continuous transition between non-impact and controlled stable impact motions will be observed at the grazing point.

5.2 Numerical illustration

This subsection aims to provide the details on the control procedure of grazing-induced instability corresponding to the 1/1 motion. In this section, the following fixed system parameters are chosen as an example for further illustration: ζ =0.01, r=0.68, and ω =2ω_d/3+5× 10^-4, where . The grazing-induced instability under this set of parameters is presented in Fig. 1 firstly. The corresponding bifurcation diagram on the Poincaré section Π₂ is shown in Fig. 1(a), where all black dots are the results obtained by the direct numerical simulation. From the figure, one can see that chaotic motion arises and acts as a stable attractor when grazing bifurcation occurs. The unstable 1/1 motion branching from the grazing point is obtained from the shooting method^[40] and marked by blue dots. Figure 1(b) shows a partial enlargement of Fig. 1(a). From the figure, one can see that the unstable 1/1 motion eventually becomes stable through period-doubling bifurcation at d-d^* ≈ -1.005 27× 10^-5, where the corresponding eigenvalues can be obtained as λ₁ =-1 and λ₂ ≈ -0.383 01 from the shooting method. The continuation corresponding to the branch of the 1/1 motion is presented separately in Fig. 1(c). In this paper, we denote the horizontal axis label Δd as (d-d^*). The modulus of the eigenvalues corresponding to the 1/1 motion is presented in Fig. 1(d). It is clear that the largest modulus |λ₁| tends to infinity while the bifurcation parameter Δd approaches zero.

We use the MAPLE software for perturbation computation. In Fig. 1, the perturbation results of eigenvalues are presented by discrete blue circles for comparison. From the figure, one can see that the perturbation results agree well with the shooting results. The corresponding period-doubling bifurcation point, where the grazing-induced 1/1 motion becomes stable, is d-d^* ≈ -1.005 28× 10^-5, and the eigenvalues obtained at this bifurcation point are almost the same as the above shooting results. Obviously, this example indicates that the truncated eigenvalue approximation expressed in Eq. (26) is valid close to grazing.

To suppress the above grazing-induced instability phenomenon, the temporally discrete feedback controller under the already established control strategies (see Eqs. (33), (35), and (36)) will be used here. Under the effect of the chosen system parameters, the first condition (degenerate grazing) can be numerically expressed as follows:

(37)

Note that finite decimal places are given here for convenience. Let c₂ be a variable and c₁ be determined as follows:

(38)

After substituting Eq. (38) into Eqs. (35) and (36), we can obtain the results shown in Fig. 2. Figure 2(a) shows the graphs for the control constraints. Figure 2(b) shows a partial enlargement of Fig. 2(a). As can be seen, the selection of the control parameter c₂ (∈ [-2, 0]) is admitted to suppress the grazing-induced instability. c₂ =-1 is chosen for further illustration, c₁ ≈ -1.020 60 can be calculated from Eq. (38), and the continuation of the controlled 1/1 motion can be obtained correspondingly. As shown in Fig. 2(c), the controlled stable 1/1 motion is marked by black dots. For comparison, the continuation of the controlled 1/1 motion overlaps with the continuation of the original system. From this figure, one can see that the period-doubling bifurcation of the original system now disappears under the effect of the temporally discrete control input. The corresponding eigenvalues obtained from the shooting method are presented in Figs. 2(d) and 2(e). As can be seen, the largest eigenvalue in the modulus λ₁ is bounded near the grazing point. The unbounded eigenvalues in the uncontrolled case in Fig. 1(d) are eliminated, and the grazing induced instability is suppressed successfully. Meanwhile, to verify the validity of Eqs. (23) and (25), which lay the foundation of the established control strategies, the perturbation results are also presented in Fig. 2(e) with blue circles. Obviously, the truncated eigenvalue approximation expressed in Eqs. (23) and (25) are valid. Figures 2(f) and 2(g) are presented here to show the control effects corresponding to independent values of the bifurcation parameter d-d^*, where black dots indicate the uncontrolled system responses on the Poincaré section Π₂, while red dots indicate the controlled responses. In Fig. 2(f), the bifurcation parameter d-d^* is chosen as d-d^* =-5× 10^-6, and the given initial value converges to the chaotic attractor, which can be found in Fig. 1(b) during the first 1 000 iterations on the Poincaré section Π₂. Subsequently, the temporally discrete controller begins functioning and the chaotic motion is suppressed to the controlled 1/1 motion. In Fig. 2(g), the bifurcation parameter is chosen as d-d^* =-9× 10^-6. The uncontrolled attractor (2/2 motion) originates from the period-doubling bifurcation discussed in the previous section. Similarly, when the controller begins functioning, the control responses will converge to a stable 1/1 motion.

6 Conclusions

In this paper, the grazing-induced instability is suppressed by using temporally discrete control in the model of an impacting oscillator with one degree of freedom. The analytical conditions corresponding to suppressing bifurcation are fully established by applying a perturbation analysis to the characteristic equation of the near-grazing 1/1 motion with the aid of higher order zero time discontinuity mapping. As shown in the perturbation analysis results, the elimination of the singular term in the characteristic equation requires degenerate grazing bifurcation. In such a situation, higher order truncation of the local discontinuity mapping is required to guarantee the correctness of some important coefficients in the characteristic equation. Finally, the numerical results demonstrate the detailed control procedure and verify the above analysis.

Although the proposed control strategy is successful, one should note that the perturbation analysis in this paper is limited to the impacting oscillators with one degree of freedom. In such a simple model, the elimination of a singular eigenvalue from the characteristic equation is related to degenerate grazing bifurcation. However, when the system dimension increases, the expression for the singular term in the characteristic equation will become more complicated, and the degeneracy point of grazing bifurcation will no longer be associated with the elimination of the singular eigenvalue. In such a situation, more complicated near-grazing bifurcations may arise in the vicinity of a degenerate grazing point in higher dimensional impacting oscillators. In addition, the conjugate eigenvalues in Fig. 1(d) are induced by the interaction between the singular and regular eigenvalues. This indicates that the transition of eigenvalues in higher dimensional impacting oscillators will be more complicated and the interactions among eigenvalues may lead to other novel dynamic phenomena, e.g., Neimark-Sacker bifurcation. Therefore, future work should focus on applying perturbation analyses to the characteristic equation in higher dimensions.