# Codimension-Two Grazing Bifurcations in Three-Degree-of-Freedom Impact Oscillator with Symmetrical Constraints

Authors: Qunhong Li; Pu Chen; Jieqiong Xu
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

This paper investigates the codimension-two grazing bifurcations of a three-degree-of-freedom vibroimpact system with symmetrical rigid stops since little research can be found on this important issue. The criterion for existence of double grazing periodic motion is presented. Using the classical discontinuity mapping method, the Poincaré mapping of double grazing periodic motion is obtained. Based on it, the sufficient condition of codimension-two bifurcation of double grazing periodic motion is formulated, which is simplified further using the Jacobian matrix of smooth Poincaré mapping. At the end, the existence regions of different types of periodic-impact motions in the vicinity of the codimension-two grazing bifurcation point are displayed numerically by unfolding diagram and phase diagrams.


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## Body

## 1. Introduction

Impacting phenomena exist in a large number of mechanical systems. Because the collision introduces essential nonlinearity and discontinuity, the vibroimpact systems can exhibit rich and complicated dynamical behavior. There is rich literature on the analysis of the dynamics for impact oscillator systems. The early work mainly focuses on the single-degree-of-freedom impact oscillators, for example, [1–3]. For multidegree-of-freedom vibroimpact systems, detailed studies of dynamics (including stability and bifurcations) using numerical simulations and qualitative analyses were carried out in decades, for example, [4–8]. Aidanpaa and Gupta [4] analyzed a two-degree-of-freedom vibroimpact system and obtained the expression of periodic motion, which is too complex to analyze the dynamical behavior. Leine [5] presented an asymptotic approximation method for the critical restitution coefficient of a parametrically excited impact oscillator and described its dynamics by a unilaterally constrained Hill's equation. Yue and Xie [6] researched the symmetric period n-2 motion and bifurcations of a two-degree-of-freedom vibroimpact system. Luo [7] developed a method to investigate the symmetry of solutions in nonsmooth dynamical systems and obtained all possible stable and unstable motions. Luo et al. [8] considered multiperformance, multiprocess coupling, and multiparameter simulation analysis for dynamics of a two-degree-of-freedom periodically forced system with a clearance represented by two symmetric rigid stops.A special situation arises when an impact with zero velocity occurs, namely, grazing impact. Grazing impact gives a nondifferentiable Poincaré mapping, which is important for the bifurcation when stable nonimpact motion changes to impact motion. The pioneer work in this field was done by Nordmark [9], who developed systematic method that is so-called discontinuity-mapping approach to investigate grazing dynamics and its attendant bifurcations, providing the results which laid the foundation for many subsequent studies, for example, [10–13]. Li et al. [14, 15] investigated the existence and stability of the grazing periodic trajectory in a two-degree-of-freedom vibroimpact system with unilateral constraint and symmetric constraints, respectively. Dankowicz and Zhao [16] analyzed the codimension-one and codimension-two grazing bifurcations in impact microactuators. Thota et al. [17] investigated the distribution of such codimension-two grazing bifurcations in single-degree-of-freedom impact oscillators and inquired into the possible dynamical characteristics of the system response on neighborhoods of such bifurcation points. Csaba and Champneys [18] analyzed nonsmooth bifurcations in both one and two parameters for a simple mechanical model of a pressure relief valve which is an autonomous impact oscillator. Dankowicz and Katzenbach [19] collected four distinct instances of grazing contact of a periodic trajectory in a hybrid dynamical system under a common abstract framework and established selected general properties of the associated near-grazing dynamics. Mason et al. [20] analyzed a model of a periodically forced impact oscillator with two discontinuity surfaces and provided new insights into the extremely rich dynamical behavior including codimension-one, codimension-two, and codimension-three bifurcations by discontinuity-geometry methodology.Despite that much work has been carried out to analyze nonsmooth codimension-two bifurcation of impact system, little work has been reported on the analysis of such bifurcation in multidegree-freedom with two discontinuity surfaces. In this paper, we investigated codimension-two grazing bifurcations in three-degree-of-freedom impact oscillator with symmetrical constraints. This paper is organized as follows. A three-degree-of-freedom vibroimpact system with proportional damping property is considered and an existing criterion of double grazing period-n motion is proposed in Section 2. The Poincaré mapping is obtained by combination of discontinuity map and smooth Poincaré mapping of double grazing periodic motion in Section 3. The Poincaré mapping will be used to analyze the sufficient conditions of stability of double grazing periodic trajectories and codimension-two grazing bifurcations in Section 4. Using the above result, the dynamical features near critical points of grazing codimension-two bifurcation are displayed by numerical simulation in Section 5. Finally, some conclusions are drawn in Section 6.


## 2. Double Grazing Periodic Motion in Three-Degree-of-Freedom Impact Oscillator

### 2.1. Mechanical Model

The mechanical model for a three-degree-of-freedom vibrator with massesM1, M2, and M3 is shown in Figure 1. Displacements of masses M1, M2, and M3 are represented by X1, X2, and X3, respectively. The masses are connected to linear springs with stiffness K1, stiffness K2, and stiffness K3. The excitations on the three masses are harmonic with amplitudes P1, P2, and P3, respectively. The excitation frequency Ω and the phase τ are the same for three masses. Mass M1 moves between rigid stops A and C. When the displacement X1 of mass M1 is B or  -B, mass M1 will hit rigid stop A or C. The impact is described by a coefficient of restitution R, and it is assumed that the duration of impact is negligible compared to the period of the force. Damping in the mechanical model is assumed as proportional damping of the Rayleigh type, which in this case implies C1/K1=C2/K2=C3/K3.Figure 1
Schematic of the there-degree-of-freedom impact oscillator with symmetrical constraints.Between consecutive impacts, for|X1|<B, the differential equations of motion are(1)M1000M2000M3X¨1X¨2X¨3+C1-C10-C1C1+C2-C20-C2C2+C3X˙1X˙2X˙3+K1-K10-K1K1+K2-K20-K2K2+K3X1X2X3=P1P2P3sinΩT+τ.When the impact occurs, for|X1|=B, the velocity of the impacting mass is changed according to the impact law, and the impact equations of mass M1 are given by(2)X˙2A+=-RX˙2A-,X1=B,X˙2C+=-RX˙2C-,X1=-B,where − and + denote the values just before and after impact, respectively.Equations (1) and (2) are rewritten in nondimensional form for |x1|<b:(3)m1000m2000m3x¨1x¨2x¨3+2ζ1-2ζ10-2ζ12ζ1+2ζ2-2ζ20-2ζ22ζ2+2ζ3x˙1x˙2x˙3+k1-k10-k1k1+k2-k20-k2k2+k3x1x2x3=f10f20f30sinωt+τ,(4)x˙2A+=-Rx˙2A-,x1=b,x˙2C+=-Rx˙2C-,x1=-b,where a dot (·) denotes differentiation with the nondimensional time t. Let M1≠0, K1≠0, and F0=P12+P22+P32. The nondimensional quantities mi=Mi/M1, ki=Ki/K1, fi0=Pi/F0, t=TK1/M1, ω=ΩM1/K1, b=BK1/F0, xi=XiK1/F0, ζi=Ci/2K1M1, and γ=ζi/ki have been introduced, where i=1,2,3.Equation (3) is amenable to analytical treatment due to the special relation between stiffness and damping. Let Ψ represent the canonical model matrix of (3). ω1, ω2, and ω3 denote the eigenfrequencies of the system as impacts do not occur. Taking Ψ as a transition matrix, the motion equation (3) under the change of variables x=Ψξ is(5)Iξ¨+Cξ˙+Λξ=F¯sinωt+τ,where X=(x1,x2,x3)T, ξ=(ξ1,ξ2,ξ3)T, I is an unit matrix of degree 3×3, C and Λ are diagonal matrixes, and C=diag2ζω12,2ζω22,2ζω32, Λ=diagω12,ω22,ω32, F¯=f¯1,f¯2,f¯3T=ΨTPk, and Pk=f10,f20,f30T. The general solutions of (3) are given by(6)xit=∑j=13‍φije-ηjt-t0ajcos⁡ωdjt+bjsinωdjt+Ajsin⁡(ωt+τ)+Bjcos⁡⁡ωt+τ,(i=1,2,3),where  t0 denotes the time when mass M1 collides with constraint A or C, ψij are the elements of the canonical modal matrix Ψ, ηj=ζjωj2, ωdj=ωj2-ηj2, aj and bj are the constants of integration which are determined by the initial conditions and modal parameters of the system, Aj and Bj are the amplitude parameter, j=1,2,3, and(7)Aj=ωj2-ω2ωj2-ω22+2ζjωj2ω2fj-,Bj=2ζjωj2ωωj2-ω22+2ζjωj2ω2fj-.


### 2.2. The Condition for Existence of Grazing Periodic Motion

If oscillatorM1 impacts each rigid constraint with zero velocity and the direction of the acceleration is opposite to the motion, then we say that the system is undergoing grazing motion. A grazing period motion may be denoted by n-p which means that oscillator M1 grazes with each constraint for P times in n periodic external excitation force. In the following, we will derive an existence condition of grazing motion with period nT, where T is the period of external excitation. Assume that the grazing periodic motion begins from the grazing point on constraint A.The initial conditions of grazing period-n motion are(8)x10=x10=b,x20=x20,x30=x30,v10=v10=0,v20=v20,v30=v30.The periodic conditions of grazing period-n motion are(9)x12nπω=x10=b,x22nπω=x20,x32nπω=x30,v12nπω=v10=0,v22nπω=v20,v32nπω=v30.If the grazing periodic motion begins from the grazing point on constraintC, similar to the case above, the initial conditions and the periodic conditions are(10)x10=x10=-b,x20=x20,x30=x30,v10=v10=0,v20=v20,v30=v30,x12nπω=x10=-b,x22nπω=x20,x32nπω=x30,v12nπω=v10=0,v22nπω=v20,v32nπω=v30.Substituting the above condition into the general solutions of (6), we can obtain the expression of aj and bj as(11)aj=ejbjsj1-ejcj,j=1,2,3,where(12)ej=e-ηj(2nπ/ω),sj=sin2nπωdjω,cj=cos⁡2nπωdjω,j=1,2,3.Substituting (11) into the initial conditions and the periodic conditions yields(13)φ11l¯1b1+φ12l¯2b2+φ13l¯3b3=0,φ21l¯1b1+φ22l¯2b2+φ23l¯3b3=0,φ31l¯1b1+φ32l¯2b2+φ33l¯3b3=0,where(14)l¯j=ωdj1-ej(cj-ej)1-ejcj,j=1,2,3.Thus, if(15)φ11l¯1φ12l¯2φ13l¯3φ21l¯1φ22l¯2φ23l¯3φ31l¯φ32l¯2φ33l¯3≠0,we have b1=0, b2=0, and b3=0. Hence a1=0, a2=0, and a3=0. For simplicity, assume that the parameters are chosen such that the integral constants b1, b2, b3, a1, a2, and a3 are vanishing. Inserting integral constants into (6) gives(16)d1sinτ+d2cos⁡τ=b,-d2sinτ+d1cos⁡τ=0,as the grazing periodic motion sets off from the grazing point on constraint A or(17)d1sinτ+d2cos⁡τ=-b,-d2sinτ+d1cos⁡τ=0as the grazing periodic motion sets off from the grazing point on constraint C. Then it follows that(18)b2=d12+d22,τ=arctand1d2,where  d1=A1φ11+A2φ12+A3φ13 and d2=B1φ11+B2φ12+B3φ13. Denote the acceleration of oscillator M1 as Ag1(orAg2) for the case in which the periodic grazing motion begins from the grazing point on constraint A(orC) with Ag1=x¨(t)t=0=-ω2b and Ag2=x¨(t)t=0=ω2b. If b>0, we have Ag1<0 and Ag2>0.Based on the analysis above, if there exists a double grazing periodic trajectory in the system with initial condition and periodic conditions, then system parameters must satisfy the following condition:(19)φ11l¯1φ12l¯2φ13l¯3φ21l¯1φ22l¯2φ23l¯3φ31l¯φ32l¯2φ33l¯3≠0,b=d12+d22,d12+d22≠0,τ=arctand1d2.Using the condition for existence of double grazing periodic motion for the three-degree-of-freedom impact system, the curve of points in the(ω,b) parameters space corresponding to the existence of a double grazing periodic trajectory with k2=k3=5, m2=m3=10, ζi=0.05, and R=0.8 is shown in Figure 2, where i=1,2,3.Figure 2
Grazing curve.In order to verify the existence condition obtained, numerical simulation of the original system equations (3)-(4) will be given. For fixed k2=k3=5, m2=m3=10, ζi=0.05, R=0.8, f10=1, f20=0, and f30=0, a double grazing period-1 motion is obtained with ω=0.1, b=1.44159 as shown in Figure 3. Figures 3(a) and 3(b) are the phase portrait and time history of oscillator M1. Figure 3(a) shows that oscillator M1 collides with constraints A and C with zero velocity. It illustrates the validity of the condition for existence of grazing periodic motion.Figure 3
Double grazing period-1 motions of the system: (a) the phase portraits of oscillatorM1; (b) the time response of oscillator M1.
(a)
(b)


## 2.1. Mechanical Model

The mechanical model for a three-degree-of-freedom vibrator with massesM1, M2, and M3 is shown in Figure 1. Displacements of masses M1, M2, and M3 are represented by X1, X2, and X3, respectively. The masses are connected to linear springs with stiffness K1, stiffness K2, and stiffness K3. The excitations on the three masses are harmonic with amplitudes P1, P2, and P3, respectively. The excitation frequency Ω and the phase τ are the same for three masses. Mass M1 moves between rigid stops A and C. When the displacement X1 of mass M1 is B or  -B, mass M1 will hit rigid stop A or C. The impact is described by a coefficient of restitution R, and it is assumed that the duration of impact is negligible compared to the period of the force. Damping in the mechanical model is assumed as proportional damping of the Rayleigh type, which in this case implies C1/K1=C2/K2=C3/K3.Figure 1
Schematic of the there-degree-of-freedom impact oscillator with symmetrical constraints.Between consecutive impacts, for|X1|<B, the differential equations of motion are(1)M1000M2000M3X¨1X¨2X¨3+C1-C10-C1C1+C2-C20-C2C2+C3X˙1X˙2X˙3+K1-K10-K1K1+K2-K20-K2K2+K3X1X2X3=P1P2P3sinΩT+τ.When the impact occurs, for|X1|=B, the velocity of the impacting mass is changed according to the impact law, and the impact equations of mass M1 are given by(2)X˙2A+=-RX˙2A-,X1=B,X˙2C+=-RX˙2C-,X1=-B,where − and + denote the values just before and after impact, respectively.Equations (1) and (2) are rewritten in nondimensional form for |x1|<b:(3)m1000m2000m3x¨1x¨2x¨3+2ζ1-2ζ10-2ζ12ζ1+2ζ2-2ζ20-2ζ22ζ2+2ζ3x˙1x˙2x˙3+k1-k10-k1k1+k2-k20-k2k2+k3x1x2x3=f10f20f30sinωt+τ,(4)x˙2A+=-Rx˙2A-,x1=b,x˙2C+=-Rx˙2C-,x1=-b,where a dot (·) denotes differentiation with the nondimensional time t. Let M1≠0, K1≠0, and F0=P12+P22+P32. The nondimensional quantities mi=Mi/M1, ki=Ki/K1, fi0=Pi/F0, t=TK1/M1, ω=ΩM1/K1, b=BK1/F0, xi=XiK1/F0, ζi=Ci/2K1M1, and γ=ζi/ki have been introduced, where i=1,2,3.Equation (3) is amenable to analytical treatment due to the special relation between stiffness and damping. Let Ψ represent the canonical model matrix of (3). ω1, ω2, and ω3 denote the eigenfrequencies of the system as impacts do not occur. Taking Ψ as a transition matrix, the motion equation (3) under the change of variables x=Ψξ is(5)Iξ¨+Cξ˙+Λξ=F¯sinωt+τ,where X=(x1,x2,x3)T, ξ=(ξ1,ξ2,ξ3)T, I is an unit matrix of degree 3×3, C and Λ are diagonal matrixes, and C=diag2ζω12,2ζω22,2ζω32, Λ=diagω12,ω22,ω32, F¯=f¯1,f¯2,f¯3T=ΨTPk, and Pk=f10,f20,f30T. The general solutions of (3) are given by(6)xit=∑j=13‍φije-ηjt-t0ajcos⁡ωdjt+bjsinωdjt+Ajsin⁡(ωt+τ)+Bjcos⁡⁡ωt+τ,(i=1,2,3),where  t0 denotes the time when mass M1 collides with constraint A or C, ψij are the elements of the canonical modal matrix Ψ, ηj=ζjωj2, ωdj=ωj2-ηj2, aj and bj are the constants of integration which are determined by the initial conditions and modal parameters of the system, Aj and Bj are the amplitude parameter, j=1,2,3, and(7)Aj=ωj2-ω2ωj2-ω22+2ζjωj2ω2fj-,Bj=2ζjωj2ωωj2-ω22+2ζjωj2ω2fj-.


## 2.2. The Condition for Existence of Grazing Periodic Motion

If oscillatorM1 impacts each rigid constraint with zero velocity and the direction of the acceleration is opposite to the motion, then we say that the system is undergoing grazing motion. A grazing period motion may be denoted by n-p which means that oscillator M1 grazes with each constraint for P times in n periodic external excitation force. In the following, we will derive an existence condition of grazing motion with period nT, where T is the period of external excitation. Assume that the grazing periodic motion begins from the grazing point on constraint A.The initial conditions of grazing period-n motion are(8)x10=x10=b,x20=x20,x30=x30,v10=v10=0,v20=v20,v30=v30.The periodic conditions of grazing period-n motion are(9)x12nπω=x10=b,x22nπω=x20,x32nπω=x30,v12nπω=v10=0,v22nπω=v20,v32nπω=v30.If the grazing periodic motion begins from the grazing point on constraintC, similar to the case above, the initial conditions and the periodic conditions are(10)x10=x10=-b,x20=x20,x30=x30,v10=v10=0,v20=v20,v30=v30,x12nπω=x10=-b,x22nπω=x20,x32nπω=x30,v12nπω=v10=0,v22nπω=v20,v32nπω=v30.Substituting the above condition into the general solutions of (6), we can obtain the expression of aj and bj as(11)aj=ejbjsj1-ejcj,j=1,2,3,where(12)ej=e-ηj(2nπ/ω),sj=sin2nπωdjω,cj=cos⁡2nπωdjω,j=1,2,3.Substituting (11) into the initial conditions and the periodic conditions yields(13)φ11l¯1b1+φ12l¯2b2+φ13l¯3b3=0,φ21l¯1b1+φ22l¯2b2+φ23l¯3b3=0,φ31l¯1b1+φ32l¯2b2+φ33l¯3b3=0,where(14)l¯j=ωdj1-ej(cj-ej)1-ejcj,j=1,2,3.Thus, if(15)φ11l¯1φ12l¯2φ13l¯3φ21l¯1φ22l¯2φ23l¯3φ31l¯φ32l¯2φ33l¯3≠0,we have b1=0, b2=0, and b3=0. Hence a1=0, a2=0, and a3=0. For simplicity, assume that the parameters are chosen such that the integral constants b1, b2, b3, a1, a2, and a3 are vanishing. Inserting integral constants into (6) gives(16)d1sinτ+d2cos⁡τ=b,-d2sinτ+d1cos⁡τ=0,as the grazing periodic motion sets off from the grazing point on constraint A or(17)d1sinτ+d2cos⁡τ=-b,-d2sinτ+d1cos⁡τ=0as the grazing periodic motion sets off from the grazing point on constraint C. Then it follows that(18)b2=d12+d22,τ=arctand1d2,where  d1=A1φ11+A2φ12+A3φ13 and d2=B1φ11+B2φ12+B3φ13. Denote the acceleration of oscillator M1 as Ag1(orAg2) for the case in which the periodic grazing motion begins from the grazing point on constraint A(orC) with Ag1=x¨(t)t=0=-ω2b and Ag2=x¨(t)t=0=ω2b. If b>0, we have Ag1<0 and Ag2>0.Based on the analysis above, if there exists a double grazing periodic trajectory in the system with initial condition and periodic conditions, then system parameters must satisfy the following condition:(19)φ11l¯1φ12l¯2φ13l¯3φ21l¯1φ22l¯2φ23l¯3φ31l¯φ32l¯2φ33l¯3≠0,b=d12+d22,d12+d22≠0,τ=arctand1d2.Using the condition for existence of double grazing periodic motion for the three-degree-of-freedom impact system, the curve of points in the(ω,b) parameters space corresponding to the existence of a double grazing periodic trajectory with k2=k3=5, m2=m3=10, ζi=0.05, and R=0.8 is shown in Figure 2, where i=1,2,3.Figure 2
Grazing curve.In order to verify the existence condition obtained, numerical simulation of the original system equations (3)-(4) will be given. For fixed k2=k3=5, m2=m3=10, ζi=0.05, R=0.8, f10=1, f20=0, and f30=0, a double grazing period-1 motion is obtained with ω=0.1, b=1.44159 as shown in Figure 3. Figures 3(a) and 3(b) are the phase portrait and time history of oscillator M1. Figure 3(a) shows that oscillator M1 collides with constraints A and C with zero velocity. It illustrates the validity of the condition for existence of grazing periodic motion.Figure 3
Double grazing period-1 motions of the system: (a) the phase portraits of oscillatorM1; (b) the time response of oscillator M1.
(a)
(b)


## 3. Poincaré Mapping of Double Grazing Periodic Motion in Three-Degree-of-Freedom Impact Oscillator

### 3.1. Definitions in Geometric Structure

In terms of the state vectorx=(x1,v1,x2,v2,x3,v3,θ)T∈R7, it follows that(20)dxdt=x˙=fx=v1,a1,v2,a2,v3,a3,ωT,where θ=ωt mod 2π denotes the phase of the excitation and ai equals the acceleration of the oscillator as a function of xi, vi, and θ. Denote the corresponding flow function by Φ(x,t).Suppose that the movement of the oscillator is limited by symmetrical rigid constraints placed at|x1|=b corresponding to state-space discontinuity surfaces D1 and D2 (see Figure 4):(21)D1=x∣hD1(x)=b-x1=0,D2=x∣hD2(x)=-b-x1=0,such that hD1(x)>0 and hD2(x)<0 during the motion of the oscillator and hxD1(x)=hxD2(x)=(-1,0,0,0,0,0,0). Let hP1(x)=hD1(x)f(x)=-v1 and hP2(x)=hD2(x)f(x)=-v1; it follows that hxP1(x)=hxP2(x)=(0,-1,0,0,0,0,0).Figure 4
A schematic of the flow with impacting.The transversal intersections of a state-space trajectory withD1 and D2 are at points x∗1 and x∗2, respectively, such that hD1(x∗1)=hD2(x∗2)=0 and hP1(x∗1)≠0, hP2(x∗2)≠0.Suppose that trajectoryx(t) with x(t∗1)=x∗1 at t∗1 and x(t∗2)=x∗2 at t∗2; it follows that(22)D1dthD1xtt=t∗1=hxD1(x∗1)f(x∗1)=hP1(x∗1),D2dthD2xtt=t∗2=hxD2(x∗2)f(x∗2)=hP2(x∗2).Indeed, ifhP1(x∗1)<0, a collision occurs between the oscillator and right constraint D1, and if hP2(x∗2)>0, a collision occurs between the oscillator and left constraint D2. We model the collision as an instantaneous impact with a characteristic coefficient of restitution R; that is, the state immediately after impact relates to that immediately before impact according to the jump map:(23)g1(x)=x1,-Rv1,x2,v2,x3,v3,θT,g2(x)=x1,-Rv1,x2,v2,x3,v3,θT.In contrast, points of grazing contact of a state-space trajectory withD1 and D2 correspond to points x∗1 and x∗2, respectively, such that(24)hD1x∗1=0,hP1x∗1=-v1∗1=0,dhP1(x)dtx=x∗1=hxP1(x∗1)f(x∗1)=-a1∗1>0,hD2(x∗2)=0,hP2(x∗2)=-v1∗2=0,dhP2(x)dtx=x∗2=hxP2(x∗2)f(x∗2)=-a1∗2<0.Introduce Poincaré surfacesP1 and P2:(25)P1=x∣hP1(x)=0,P2=x∣hP2(x)=0such that x∗1 and x∗2 correspond to a transversal intersection of a state-space trajectory with surfaces P1 and P2, respectively. Since (d2/dt2)hD1(x(t))t=t∗1=hxP1(x∗1)f(x∗1)=-a1∗1>0, it follows that x∗1 is a local minimum in the value of hD1(x) (i.e., the distance to D1) along state-space trajectories of the system. By transversality, it follows that nearby trajectories achieve locally unique points of intersection with P1 corresponding to local minima in the value of hD1(x). By the same as above, since -a1∗2<0, the trajectories nearby the grazing one achieve locally unique points of intersection with P2 corresponding to local maxima in the value of hD2(x).Now suppose thatx∗1 and x∗2 are symmetrical grazing contact points of a periodic trajectory of the system with D1 and D2, such that Φ(x∗1,t+T/2)=Φ(x∗2,t) and Φ(x∗2,t+T/2)=Φ(x∗1,t). Ignoring the effects of the jump map and using the transversality, it is possible to define local Poincaré maps P1smooth:P1→P2 and P2smooth:P2→P1, such that P1smooth(x∗1)=x∗2 and P2smooth(x∗2)=x∗1.


### 3.2. Discontinuity-Mapping

When considering the effects of the jump map associated with the impact surface, the dynamics of the impact oscillator under perturbations in initial conditions away from the grazing periodic trajectory may be analyzed using the discontinuity-mapping approach originally introduced by Nordmark [9]. Here, two discontinuity-mappings DM1 and DM2 are introduced on a neighborhood of points x∗1 and x∗2, such that surface P1 is invariant under DM1 (i.e., x∈P1,DM1(x)∈P1) and surface P2 is invariant under DM2 (i.e., x∈P2,DM2(x)∈P2); therefore, Poincaré mapping P associated with surface P1 for the flow near the grazing trajectory including the effects of the jump map can be written as(26)P=P2smooth∘DM2∘P1smooth∘DM1.According to the discontinuity-mapping approach, the discontinuity-mappings ofDM1 and DM2 are obtained as follows:(27)DM1=Id,hD1(x)≥0,x∗1+β12hxP1(x∗1)f(x∗1)·-hxD1(x∗1)(x-x∗1),hD1(x)<0,DM2=Id,hD2(x)≤0,x∗2+β22hxP2(x∗2)f(x∗2)·-hxD2(x∗2)(x-x∗2),hD2(x)>0,where β1=(f(x∗1)hxP1(x∗1)g1,x(x∗1)/hxP1(x∗1)f(x∗1)-g1,x(x∗1))f(x∗1) and β2=(f(x∗2)hxP2(x∗2)g2,x(x∗2)/hxP2(x∗2)f(x∗2)-g2,x(x∗2))f(x∗2).


### 3.3. The Poincaré Mapping

Expanding the smooth mappingP1smooth(x) near x∗1, it follows that(28)P1smooth(x)=P1smooth(x∗1)+P1smooth,x(x∗1)(x-x∗1)+h.o.t=x∗2+P1smooth,x(x∗1)(x-x∗1)+h.o.t.Expanding event functionhD1(x) near x∗1, it follows that(29)hD1x=hD1x∗1+hxD1x∗1x-x∗1+h.o.t.Sincex∗1 satisfies hD1(x∗1)=0, such that (29) is written as(30)hD1x=hxD1x∗1x-x∗1+h.o.t,consequently,(31)P1=P1smooth(x)∘DM1(x)=x∗2+P1smooth,x(x∗1)·(x-x∗1)+h.o.t,hxD1(x∗1)(x-x∗1)≥0,x∗2+P1smooth,x(x∗1)β1·2hxP1(x∗1)f(x∗1)·-hxD1(x∗1)(x-x∗1)+h.o.t,hxD1(x∗1)(x-x∗1)<0.From the above analysis, the mappingP2smooth(x)∘DM2(x) is written as(32)P2=P2smooth(x)∘DM2(x)=x∗1+P2smooth,x(x∗2)·(x-x∗2)+h.o.t,hxD2(x∗2)(x-x∗2)≤0,x∗1+P2smooth,x(x∗2)β2·2hxP2(x∗2)f(x∗2)·-hxD2(x∗2)(x-x∗2)+h.o.t,hxD2(x∗2)(x-x∗2)>0.


## 3.1. Definitions in Geometric Structure

In terms of the state vectorx=(x1,v1,x2,v2,x3,v3,θ)T∈R7, it follows that(20)dxdt=x˙=fx=v1,a1,v2,a2,v3,a3,ωT,where θ=ωt mod 2π denotes the phase of the excitation and ai equals the acceleration of the oscillator as a function of xi, vi, and θ. Denote the corresponding flow function by Φ(x,t).Suppose that the movement of the oscillator is limited by symmetrical rigid constraints placed at|x1|=b corresponding to state-space discontinuity surfaces D1 and D2 (see Figure 4):(21)D1=x∣hD1(x)=b-x1=0,D2=x∣hD2(x)=-b-x1=0,such that hD1(x)>0 and hD2(x)<0 during the motion of the oscillator and hxD1(x)=hxD2(x)=(-1,0,0,0,0,0,0). Let hP1(x)=hD1(x)f(x)=-v1 and hP2(x)=hD2(x)f(x)=-v1; it follows that hxP1(x)=hxP2(x)=(0,-1,0,0,0,0,0).Figure 4
A schematic of the flow with impacting.The transversal intersections of a state-space trajectory withD1 and D2 are at points x∗1 and x∗2, respectively, such that hD1(x∗1)=hD2(x∗2)=0 and hP1(x∗1)≠0, hP2(x∗2)≠0.Suppose that trajectoryx(t) with x(t∗1)=x∗1 at t∗1 and x(t∗2)=x∗2 at t∗2; it follows that(22)D1dthD1xtt=t∗1=hxD1(x∗1)f(x∗1)=hP1(x∗1),D2dthD2xtt=t∗2=hxD2(x∗2)f(x∗2)=hP2(x∗2).Indeed, ifhP1(x∗1)<0, a collision occurs between the oscillator and right constraint D1, and if hP2(x∗2)>0, a collision occurs between the oscillator and left constraint D2. We model the collision as an instantaneous impact with a characteristic coefficient of restitution R; that is, the state immediately after impact relates to that immediately before impact according to the jump map:(23)g1(x)=x1,-Rv1,x2,v2,x3,v3,θT,g2(x)=x1,-Rv1,x2,v2,x3,v3,θT.In contrast, points of grazing contact of a state-space trajectory withD1 and D2 correspond to points x∗1 and x∗2, respectively, such that(24)hD1x∗1=0,hP1x∗1=-v1∗1=0,dhP1(x)dtx=x∗1=hxP1(x∗1)f(x∗1)=-a1∗1>0,hD2(x∗2)=0,hP2(x∗2)=-v1∗2=0,dhP2(x)dtx=x∗2=hxP2(x∗2)f(x∗2)=-a1∗2<0.Introduce Poincaré surfacesP1 and P2:(25)P1=x∣hP1(x)=0,P2=x∣hP2(x)=0such that x∗1 and x∗2 correspond to a transversal intersection of a state-space trajectory with surfaces P1 and P2, respectively. Since (d2/dt2)hD1(x(t))t=t∗1=hxP1(x∗1)f(x∗1)=-a1∗1>0, it follows that x∗1 is a local minimum in the value of hD1(x) (i.e., the distance to D1) along state-space trajectories of the system. By transversality, it follows that nearby trajectories achieve locally unique points of intersection with P1 corresponding to local minima in the value of hD1(x). By the same as above, since -a1∗2<0, the trajectories nearby the grazing one achieve locally unique points of intersection with P2 corresponding to local maxima in the value of hD2(x).Now suppose thatx∗1 and x∗2 are symmetrical grazing contact points of a periodic trajectory of the system with D1 and D2, such that Φ(x∗1,t+T/2)=Φ(x∗2,t) and Φ(x∗2,t+T/2)=Φ(x∗1,t). Ignoring the effects of the jump map and using the transversality, it is possible to define local Poincaré maps P1smooth:P1→P2 and P2smooth:P2→P1, such that P1smooth(x∗1)=x∗2 and P2smooth(x∗2)=x∗1.


## 3.2. Discontinuity-Mapping

When considering the effects of the jump map associated with the impact surface, the dynamics of the impact oscillator under perturbations in initial conditions away from the grazing periodic trajectory may be analyzed using the discontinuity-mapping approach originally introduced by Nordmark [9]. Here, two discontinuity-mappings DM1 and DM2 are introduced on a neighborhood of points x∗1 and x∗2, such that surface P1 is invariant under DM1 (i.e., x∈P1,DM1(x)∈P1) and surface P2 is invariant under DM2 (i.e., x∈P2,DM2(x)∈P2); therefore, Poincaré mapping P associated with surface P1 for the flow near the grazing trajectory including the effects of the jump map can be written as(26)P=P2smooth∘DM2∘P1smooth∘DM1.According to the discontinuity-mapping approach, the discontinuity-mappings ofDM1 and DM2 are obtained as follows:(27)DM1=Id,hD1(x)≥0,x∗1+β12hxP1(x∗1)f(x∗1)·-hxD1(x∗1)(x-x∗1),hD1(x)<0,DM2=Id,hD2(x)≤0,x∗2+β22hxP2(x∗2)f(x∗2)·-hxD2(x∗2)(x-x∗2),hD2(x)>0,where β1=(f(x∗1)hxP1(x∗1)g1,x(x∗1)/hxP1(x∗1)f(x∗1)-g1,x(x∗1))f(x∗1) and β2=(f(x∗2)hxP2(x∗2)g2,x(x∗2)/hxP2(x∗2)f(x∗2)-g2,x(x∗2))f(x∗2).


## 3.3. The Poincaré Mapping

Expanding the smooth mappingP1smooth(x) near x∗1, it follows that(28)P1smooth(x)=P1smooth(x∗1)+P1smooth,x(x∗1)(x-x∗1)+h.o.t=x∗2+P1smooth,x(x∗1)(x-x∗1)+h.o.t.Expanding event functionhD1(x) near x∗1, it follows that(29)hD1x=hD1x∗1+hxD1x∗1x-x∗1+h.o.t.Sincex∗1 satisfies hD1(x∗1)=0, such that (29) is written as(30)hD1x=hxD1x∗1x-x∗1+h.o.t,consequently,(31)P1=P1smooth(x)∘DM1(x)=x∗2+P1smooth,x(x∗1)·(x-x∗1)+h.o.t,hxD1(x∗1)(x-x∗1)≥0,x∗2+P1smooth,x(x∗1)β1·2hxP1(x∗1)f(x∗1)·-hxD1(x∗1)(x-x∗1)+h.o.t,hxD1(x∗1)(x-x∗1)<0.From the above analysis, the mappingP2smooth(x)∘DM2(x) is written as(32)P2=P2smooth(x)∘DM2(x)=x∗1+P2smooth,x(x∗2)·(x-x∗2)+h.o.t,hxD2(x∗2)(x-x∗2)≤0,x∗1+P2smooth,x(x∗2)β2·2hxP2(x∗2)f(x∗2)·-hxD2(x∗2)(x-x∗2)+h.o.t,hxD2(x∗2)(x-x∗2)>0.


## 4. Stability at Grazing and Codimension-Two Grazing Bifurcation

The stability of the grazing periodic trajectory when ignoring the effects of the constraint is determined by the eigenvalues of its Jacobian matrix. In contrast, the Jacobian matrix of the grazing periodic trajectory in the absence of the constraint is discontinuous and becomes singular; the stability properties of the grazing periodic trajectory in the presence of the bilateral constraint are determined by Poincaré mappingP. If the points near the grazing point which start from either the impact side or nonimpacting side are trapped close to the grazing point after iterating the mapping equations (31)-(32), the grazing periodic trajectory is stability.  For an impact point x in the vicinity of the grazing point x∗1, which satisfies hxD1(x∗1)(x-x∗1)<0, if(33)hxD2x∗2P1x-x∗2=hxD2(x∗2)P1smooth,x(x∗1)β1·2hxPx∗1fx∗1-hxD1x∗1x-x∗1<0,that is,(34)hxD2(x∗2)P1smooth,x(x∗1)β1<0,hxD1x∗1P2x-x∗1=hxD1x∗1P2smooth,xx∗2P1smooth,xx∗1β1·2hxPx∗1fx∗1·-hxD1x∗1x-x∗1<0,that is,(35)hxD1(x∗1)P2smooth,x(x∗2)P1smooth,x(x∗1)β1<0,it means an impact point impacts discontinuity surface D1 again and the impact will be perpetuated, which results in a large stretching in a direction given by the image of vector β1 under Jacobians P1smooth,x and P2smooth,x, and the trajectory is unstable.According to the above analysis(36)hxD2(x∗2)P1smooth,x(x∗1)β1<0,hxD1(x∗1)P2smooth,x(x∗2)P1smooth,x(x∗1)β1<0,meaning an impact point impacts discontinuity surface D1 again and the impact will be perpetuated; the grazing periodic trajectory is unstable.By just changing(37)hxD2(x∗2)P1smooth,x(x∗1)β1<0,hxD1(x∗1)hxD2(x∗2)P1smooth,x(x∗1)β1<0to(38)hxD2(x∗2)P1smooth,x(x∗1)·P2smooth,xx∗2P1smooth,xx∗1j-1β1<0,hxD1(x∗1)P2smooth,xx∗2P1smooth,xx∗1iβ1<0,for j≤i, it happens that an impact is followed by nonimpacting for some iterations but eventually impacts discontinuity surface D1 again and the impact will be perpetuated, and the grazing periodic trajectory is unstable. Thus, if hxD2(x∗2)P1smooth,x(x∗1)(P2smooth,x(x∗2)P1smooth,x(x∗1))(i-1)β1<0 and hxD1(x∗1)(P2smooth,x(x∗2)P1smooth,x(x∗1))iβ1<0 for any i≥1, 1≤j≤i, stability is lost.In the same way, if(39)hxD2(x∗2)P1smooth,x(x∗1)·P2smooth,xx∗2P1smooth,xx∗1iβ1>0,hxD2(x∗2)P1smooth,xx∗1P2smooth,xx∗2j+1β2>0,for any i≥0, j≥i, it happens that an impact is followed by nonimpacting for some iterations but eventually impacts discontinuity surface D2 again and the impact will be perpetuated, and the grazing periodic trajectory is unstable.Moreover, if(40)hxD2x∗2P1smooth,xx∗1·P2smooth,xx∗2P1smooth,xx∗1iβ1>0,hxD1(x∗1)P2smooth,xx∗2·P1smooth,xx∗1P2smooth,xx∗2jβ2<0,for any i≥0, j≥i, it happens that an impact is followed by nonimpacting for some iterations but eventually impacts discontinuity surfaces D1 and D2 again and the impact will be perpetuated, and the grazing periodic trajectory is unstable.According to above analysis, the codimension-two grazing bifurcation points (the definition of such points is seen in [17]) correspond to(41)hxD1(x∗1)P2smooth,xx∗2P1smooth,xx∗1iβ1=0,hxD1(x∗1)P2smooth,x(x∗2)·P1smooth,xx∗1P2smooth,xx∗2jβ2=0,hxD2(x∗2)P1smooth,x(x∗1)·P2smooth,xx∗2P1smooth,xx∗1iβ1=0,hxD2(x∗2)P1smooth,xx∗1P2smooth,xx∗2j+1β2=0,for all i≥0 and j≥0, respectively. Let ξn express the codimension-two grazing bifurcation points; ξn can be written in four cases; that is,(42)ξn=hxD1(x∗1)P2smooth,xx∗2P1smooth,xx∗1nβ1=0,ξn=hxD1(x∗1)P2smooth,x(x∗2)·P1smooth,xx∗1P2smooth,xx∗2nβ2=0,ξn=hxD2(x∗2)P1smooth,x(x∗1)·P2smooth,xx∗2P1smooth,xx∗1nβ1=0,ξn=hxD2x∗2P1smooth,xx∗1P2smooth,xx∗2n+1β2,=0for n=0,1,2,….In the following, take the third case, for example, the codimension-two grazing bifurcation criterionξn is simplified as far as possible and the more analytic expressions are obtained.Let(43)P1smooth,x(x∗1)P2smooth,xx∗2P1smooth,xx∗1n=P11n+1P12n+1P13n+1⋯P16n+1P17n+1000⋯00P31n+1P32n+1P33n+1⋯P36n+1P37n+1⋮⋮⋮⋮⋮P61n+1P62n+1P63n+1⋯P66n+1P67n+1P71n+1P72n+1P73n+1⋯P76n+1P77n+1.SinceP1smooth,x(x∗1)f(x∗1)=0, it follows that(44)P1smooth,x(x∗1)P2smooth,xx∗2P1smooth,xx∗1nf(x∗1)=0.Thus,(45)ξn=hxD2(x∗2)P1smooth,x(x∗1)·P2smooth,xx∗2P1smooth,xx∗1nβ1=-R+1a1∗1P12n+1.According to the relevant definitions in Section3 and the implicit function theorem, it is straightforward to show that(46)P1smooth,xx∗1=Id-fx∗2hxp2x∗2hxp2x∗2fx∗2Φxx∗1,T2,P2smooth,xx∗2=Id-fx∗1hxp1x∗1hxp1x∗1fx∗1Φxx∗2,T2,where Id is unit matrix.From the semigroup property of the smooth flow(47)ΦΦx,t,s=Φx,t+s,it follows by differentiation with respect to t and evaluation at t=0, s=T/2, and x=x∗1 that(48)Φxx∗1,T2fx∗1=fx∗2.Consequently,(49)P1smooth,x(x∗1)f(x∗1)=0,0,0,0,0,0,0T.In the same way, we obtain that(50)P2smooth,x(x∗2)f(x∗2)=0,0,0,0,0,0,0T.Moreover, it is straightforward to show that(51)P1smooth,xx∗1P2smooth,xx∗2n=Id-fx∗2hxp2x∗2hxp2x∗2fx∗2·Φxx∗1,T2Φxx∗2,T2n,P2smooth,x(x∗2)P1smooth,xx∗1P2smooth,xx∗2n=Id-fx∗1hxp1x∗1hxp1x∗1fx∗1Φxx∗2,T2·Φxx∗1,T2Φxx∗2,T2n.In the same way, we obtain that(52)P2smooth,xx∗2P1smooth,xx∗1n=Id-fx∗1hxp1x∗1hxp1x∗1fx∗1·Φxx∗2,T2Φxx∗1,T2n,(53)P1smooth,x(x∗1)P2smooth,xx∗2P1smooth,xx∗1n=Id-fx∗2hxp2x∗2hxp2x∗2fx∗2·Φxx∗1,T2Φxx∗2,T2Φxx∗1,T2n,where integer superscripts denote matrix powers.SincehxD2(x∗2)f(x∗2)=-v1(x∗2)=0, and according to (53), it follows that(54)hxD2(x∗2)P1smooth,x(x∗1)P2smooth,xx∗2P1smooth,xx∗1n=hxD2x∗2Φxx∗1,T2Φxx∗2,T2Φxx∗1,T2n.Both sides of (54) are multiplied by f(x∗1), and since(55)hxD2(x∗2)P1smooth,x(x∗1)·P2smooth,xx∗2P1smooth,xx∗1nf(x∗1)=0,it follows that(56)hxD2x∗2Φxx∗1,T2Φxx∗2,T2Φxx∗1,T2n·fx∗1=0.Thus, (45) can be written as(57)ξn=hxD2x∗2Φxx∗1,T2·Φxx∗2,T2Φxx∗1,T2nβ1=-R+1a1∗1Φ12n+1,where Φ12 denotes the (1,2) element of Φx(x∗1,T/2)[Φx(x∗2,T/2)Φx(x∗1,T/2)]n.In the same way, we can obtain the simplified expressions of the other three cases as follows:(58)ξn=hxD1(x∗1)P2smooth,xx∗2P1smooth,xx∗1nβ1=-R+1a1∗1Φ12n,ξn=hxD1(x∗1)P2smooth,x(x∗2)·P1smooth,xx∗1P2smooth,xx∗2nβ2=-R+1a1∗2Φ12n+1,ξn=hxD2(x∗2)P1smooth,xx∗1P2smooth,xx∗2n+1β2=-R+1a1∗2Φ12n+1.


## 5. Numerical Simulations

Taking the sencond case derived in the above section as an example, the codimension-two grazing bifurcation points are corresponding toξn=-(R+1)a1∗1Φ12n=0. The curve of points in the (ω,b) parameters space corresponding to the existence of a grazing periodic trajectory with k2=k3=5.0, m2=m3=10.0, ζ1=ζ2=ζ3=0.05, and R=0.8 is shown in Figure 5. Here, points for which ξ1=0 are indicated by the asterisk (∗). As shown in Figure 5, with ω decreases, codimension-two grazing bifurcation points appear to have an increased tangency with the grazing curve.Figure 5
Schematic of codimension-two grazing bifurcation points forn=1.For example, under case off10=1, f20=0, and f30=0, taking ω=0.28109, b=1.92036 along the grazing curve shown in Figure 5, then existence condition of double grazing periodic motion and ξ1=0 are both satisfied using the formulae derived from previous sections, which means that the point is a codimension-two grazing bifurcation point. Simulating with the above parameters, a grazing periodic trajectory is obtained as shown in Figure 6, which may cause complicated dynamical behaviors with the change of some parameters.Figure 6
A grazing periodic trajectory of oscillatorM1.Whenn=2, using ξn=-(R+1)a1∗1Φ12n=0 with the same parameters k2=k3=5.0, m2=m3=10.0, ζ1=ζ2=ζ3=0.05, and R=0.8, the curve of points in the (ω,b) parameters space corresponding to the existence of a grazing periodic trajectory is shown in Figure 7. Here, points for which ξ2=0 are indicated by the asterisk (∗).Figure 7
Schematic of codimension-two grazing bifurcation points forn=2.In order to make the bifurcation behavior in the vicinity of the codimension-two grazing bifurcation point clear, the unfolding diagram is presented in the neighborhood of codimension-two grazing bifurcation point corresponding toξ1=0 by a large number of numerical simulations. For fixed k2=k3=5.0, m2=m3=10.0, ζ1=ζ2=ζ3=0.05, R=0.8, f10=1, f20=0, and f30=0, it is known from above analysis that a codimension-two grazing bifurcation point corresponds to ω=0.28109 and b=1.92036. Changing values of parameters b and ω near ω=0.28109, b=1.92036. Existence regions of different types of periodic-impact motions of the system, that is, the unfolding diagram, are presented in the (ω,b) parameter plane plotted in Figure 8, where the dash curve represents double grazing motion. As shown in Figure 8, the dynamical behavior in the vicinity of the codimension-two grazing bifurcation point is complex. The (ω,b) parameter space is divided into regions characterized by different types of motions.Figure 8
Unfolding diagram near a codimension-two grazing bifurcation point.For fixedω=0.29, sample phase portraits of each region obtained through perturbing b are shown in Figures 9(a)–9(k). Conveniently, we introduce the symbol n-pr-ql to describe different types of periodic-impact motion of the system, where n represents the number of the exciting force period in the motion period and pr and ql mean the number of impacts occurring at right A and left C constraints, respectively, when p=0 or q=0; that is, 0r and 0l mean the number of grazings occurring at right A and left C constraints, respectively, while p=-1 or q=-1; that is, -1r and -1l symbolize no impact occurring at the right and left constraints. Decreasing b from 2.0 to 1.494, the phase portraits are 1-(-1)r-(-1)l in region (a), multi-0r-0l in region (b), 1-1r-(-1)l in region (c), 1-(-1)r-1l in region (d), 1-1r-1l in region (e), 2-1r-1l in region (f), multi-multir-multil in region (g), 2-2r-2l in region (h), 3-2r-2l in region (i), 3-3r-3l in region (j), and 4-3r-3l in region (k), respectively.Figure 9
Phase diagrams ((a)–(k)) of periodic motions (ω=0.29) and phase diagrams ((l)–(v)) of periodic motions (ω=0.27).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)
(q)
(r)
(s)
(t)
(u)
(v)For fixedω=0.27, sample phase portraits of each region obtained through perturbing b are shown in Figures 9(l)–9(v). Decreasing b from 1.86 to 1.34, the phase portraits are 1-1r-(-1)l in region (l), 3-(-1)r-3l in region (m), 1-1r-1l in region (n), 2-1r-1l in region (o), 6-4r-4l in region (p), 10-8r-8l in region (q), multi-multir-multil in region (r), 2-2r-2l in region (s), 3-2r-2l in region (t), 3-3r-3l in region (u), and 4-4r-4l in region (v), respectively.


## 6. Conclusions

It is difficult to analyze codimension-two bifurcation of grazing periodic motion for multidegree-of-freedom vibroimpact system theoretically, especially with symmetrical constraints. So far, there are some researches that explored the codimension-two bifurcation of grazing periodic motion for dynamical system with unilateral constraint. As far as we know, the work on codimension-two bifurcation of grazing periodic motion of vibratory system with bilateral constraints is not seen. Focusing on the above unsolved problem, this paper presents the existence condition of double grazing motion and obtains its Poincaré mapping using the classical discontinuity-mapping approach. Based on the results, the condition of codimension-two grazing bifurcation is established and simplified. Therefore, it is convenient to find the codimension-two bifurcation points by using the obtained formulae. By numerical simulation, the complex and rich dynamical behaviors near the codimension-two grazing bifurcation point are presented. For example, there exist period-10 impact-8 motion, period-6 impact-4 motion, period-4 impact-4 motion, and so forth in the vicinity of the codimension-two grazing bifurcation point. The detailed dynamical features near critical point of grazing codimension-two bifurcation will be analyzed further in the future works.



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