# Control of Near-Grazing Dynamics in the Two-Degree-of-Freedom Vibroimpact System with Symmetrical Constraints

Authors: Zihan Wang; Jieqiong Xu; Shuai Wu; Quan Yuan
Journal: Complexity
 (2020)

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## Abstract

The stability of grazing bifurcation is lost in three ways through the local analysis of the near-grazing dynamics using the classical concept of discontinuity mappings in the two-degree-of-freedom vibroimpact system with symmetrical constraints. For this instability problem, a control strategy for the stability of grazing bifurcation is presented by controlling the persistence of local attractors near the grazing trajectory in this vibroimpact system with symmetrical constraints. Discrete-in-time feedback controllers designed on two Poincare sections are employed to retain the existence of an attractor near the grazing trajectory. The implementation relies on the stability criterion under which a local attractor persists near a grazing trajectory. Based on the stability criterion, the control region of the two parameters is obtained and the control strategy for the persistence of near-grazing attractors is designed accordingly. Especially, the chaos near codimension-two grazing bifurcation points was controlled by the control strategy. In the end, the results of numerical simulation are used to verify the feasibility of the control method.


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## Body

## 1. Introduction

Grazing bifurcation, one type of discontinuity-induced bifurcations, has been extensively studied in vibroimpact system as it has complex dynamics and is widely encountered in many engineering examples. On analysis of the dynamics near grazing in a general class of impact oscillator systems, a classical concept of analysis is the so-called discontinuity-mapping approach initially conceived of by Nordmark [1, 2] (see [3, 4] for an overview). The analysis is usually carried out by finding an appropriate local map describing the system dynamics in neighborhood of the grazing event. The local map can then be combined with an analytic Poincaré map to give the so-called grazing normal form whose dynamics can be shown to be topologically equivalent to those of the underlying flow. The grazing normal form derived by the discontinuity-mapping approach is used to analyze the local dynamics in the vicinity of a grazing trajectory. As shown in [5–8], the normal form map of the rigid impact oscillator contains a square-root term causing a singularity in the first derivative, which results in an abrupt loss of the stability. And different bifurcation scenarios associated with switching between impacting motions and nonimpacting motions near grazing were also described. In particular, the discontinuity-mapping approach was used by Fredriksson and Nordmark [9] to establish conditions for the persistence or disappearance of a local attractor in the vicinity of a grazing periodic trajectory. In addition, some conditions for the persistence of a local attractor in the immediate vicinity of quasiperiodic grazing trajectories in an impacting dynamical system were formulated by Thota and Dankowicz [10]. When the stability conditions of grazing bifurcation are degenerate, the codimension-two would occur, which is always a hot topic. Kowalczyk et al. [11] proposed a strategy for classification of codimension-two discontinuity-induced bifurcations of limit cycles in piecewise smooth systems and studied their nonsmooth transitions. Foale [12] analyzed the results of a special codimension-two grazing bifurcation in a single-degree-of-freedom impact oscillator by using the impact surface as a Poincaré section. In addition, the study also shows that the bifurcation of the saddle node bifurcation and the flip bifurcation can meet at a certain codimension-two grazing points in the parameter plane. Thota et al. [13] studied the distribution of codimension-two grazing bifurcation point according to the discontinuous mapping in the single-degree-of-freedom collision oscillator and discussed the possible dynamic characteristics of the system response near this bifurcation point. Xu et al. [14] studied the codimension-two grazing bifurcation of n-degree-of-freedom vibrators with bilateral constraints and obtained and simplified the existence conditions of codimension-two grazing bifurcation. Similar phenomena about codimension-two grazing bifurcation can also be found in Refs. [15, 16]. Yin et al. [17] discussed the important role of some degenerated grazing bifurcation points in the transition between saddle node bifurcation and period doubling bifurcation. Dankowicz and Zhao [18, 19] studied the grazing bifurcations of the codimension-one and the codimension-two of a class of impact microactuators.The loss of local attractors near grazing bifurcation may arise catastrophic changes of system response and lead to codimension-two or more complicated bifurcation; therefore, controlling near-grazing dynamics becomes necessary and significant. Dankowicz and Jerrelind [20] used the linear feedback control method to control the grazing bifurcation of the piece smooth dynamic system, so that the system has local attractors near the grazing orbit. Dankowicz and Svahn [21] presented for the existence of event-driven control strategies that guarantee the local persistence of system attractors with at most low-velocity contact in vibro impacting oscillators. Misra and Dankowicz [22] developed a rigorous control paradigm for regulating the near-grazing bifurcation behavior of limit cycles in piecewise-smooth dynamical systems. Yin et al. [23] analyzed the stability for near-grazing period-one impact motion to suppress grazing-induced instabilities. The bounded eigenvalues are further confined to the unit circle, and the continuous transition between the nonimpact motion and the controlled impact motion is obtained. Xu et al. [24] discussed the control problem of near-grazing dynamics in a two-degree-of-freedom vibroimpact system with a clearance.Based on the concept of controlling the persistence of local attractors near the grazing trajectory in impact oscillator with unilateral constraints mentioned in [20–24], this paper aims to control the stability of grazing bifurcation or control the persistence of local attractors near the grazing trajectory in this vibroimpact system with symmetrical constraints. Compared with impact oscillator with unilateral constraint, the instability problems near grazing trajectory become more complex for impact oscillator with symmetrical constraints as mentioned in [17]. The stability of double grazing motion bifurcation in the system is lost in three ways, and the existence conditions of the codimension-two grazing bifurcation occur in four different cases accordingly. For this complex unstable problem, analytic expressions of stability criterion are obtained in this paper. Based on the stability criterion, the stability control strategy of the persistence of near-grazing attractors is proposed. Furthermore, the chaos near codimension-two grazing bifurcation points was controlled by the control strategy. This paper is organized as follows. In Section 2, a two-degree-of-freedom vibroimpact system with symmetrical constraints is introduced. In Section 3, near-grazing bifurcation dynamics are analyzed. In Section 4, the discrete-in-time feedback control method is designed to maintain the persistence of the local attractor of double grazing period motion. In Section 5, numerical simulation is used to verify the feasibility of the control method. Finally, the conclusion is given in Section 6.


## 2. Mechanical Model and Double Grazing Periodic Motion

Figure1 shows the schematic model of a two-degree-of-freedom impact oscillator with a clearance. Masses M1 and M2 are connected to linear viscous dampers C1 and C2 by linear springs with stiffness K1 and K2, respectively. The harmonic forces of the amplitude P1 and P2 are applied to the masses M1 and M2, respectively, and the harmonic force is applied only to the mass in the horizontal direction. The mass M1 moves between the symmetrical rigid stops A and C. When the mass M1 strikes rigid stop A or C, the motion becomes a nonlinear motion and the impact is described by the recovery factor R. Assuming that the damping in the mechanical model is the Rayleigh type proportional damping, it can be known that C1/K1=C2/K2.Figure 1
Schematic of the two-degree-of-freedom impact oscillator with symmetrical constraints.The governing equation is described by(1)M100M2X¨1X¨2+C1−C1−C1C1+C2X˙1X˙2+K1−K1−K1K1+K2X1X2=P1P2sinΩT+τ,X1<D.WhenX1=D, the collision occurs. At this time, the collision equation is as follows:(2)X˙1+=−RX˙1−,X1=D,where X˙ and X¨ represent the first and second derivatives of X with respect to time T, respectively. X˙1+ indicates the instantaneous speed at which the mass M1 approaches the rigid stop A or C. X˙1− indicates the instantaneous speed at which the mass M1 leaves the rigid stop A or C.Introduce the nondimensional quantities as follows:(3)μm=M2M1,μk=K2K1,μc=C2C1,μc=μk,p=P2P1+P2,ω=ΩM1K1,t=TK1M1,ζ=C12K1M1,d=DK1P1+P2,x1=X1K1P1+P2,x2=X2K1P1+P2.According to equations (1)–(3), the system can be transformed into nondimensional forms.(4)100μmx¨1x¨2+2ζ−2ζ−2ζ2ζ1+μcx˙1x˙2+1−1−11+μkx1x2=1−ppsinωt+τ,x1<d,(5)x˙1+=−Rx˙1−,x1=d,where x˙ and x¨ represent the first and second derivatives of x with respect to the nondimensional time t, respectively.SupposeΨ be the canonical modal matrix of equation (4), and coordinate transformation of equation (4). Let x=Ψξ 4.(4)(6)Iξ¨+Cξ˙+Λξ=Psinωt+τ,where ω1 and ω2 represent the eigenfrequencies of the system, x=x1,x2T, ξ=ξ1,ξ2T, I is the unit matrix, Λ and C are diagonal matrices, Λ=diagω12,ω22, C=2ζΛ=diag2ζω12,2ζω22, and P=ΨT1−p,pT. The general solution of equation (4) is given by(7)xi=∑j=12Ψije−ηjt−t0ajcosωdjt+bjcosωdjt+Ajsinωt+τ+Bjcosωt+τ,x˙i=∑j=12Ψije−ηjt−t0bjωdj−ajηjcosωdjt−ajωdj+bjηjsinωdjt+Ajωcosωt+τ−Bjωsinωt+τ,where i=1,2, t0 represents the time at which the mass M1 collides with the rigid stop A or C; we set the mass M1 to collide with the rigid stop A when t0=0. Ψij is an element of the canonical modal matrix Ψ, ηj=ζωj2, and ωdj=ωj2−ηj2. The initial conditions and modal parameters of the system determine the integral constants aj and bj. Aj and Bj are amplitude parameters, and the expression is given by(8)Aj=12ωdjω+ωdjω+ωdj2+ηj2−ω−ωdjω−ωdj2+ηj2f¯j,Bj=ηj2ωdj1ω+ωdj2+ηj2−1ω−ωdj2+ηj2f¯j.According to the initial conditions and periodic conditions of the double grazing periodic-n motion, the existence conditions of two-degree-of-freedom impact oscillator double grazing periodic-n motion is as follows:(9)d=d12+d22,τ=arctand1d2,where d1=Ψ11A1+Ψ12A2 and d2=Ψ11B1+Ψ12B2.In addition, in order to ensure that the impacting cycle of the massM1 does not adhere to rigid stop, the acceleration a∗ of the mass M1 satisfies a∗<0 and the acceleration a∗ of the mass M2 satisfies a∗>0.


## 3. Near-Grazing Dynamics

### 3.1. The Stability Criterion of Double Grazing Bifurcation

Letp be a state vector, such that x=x1,v1,x2,v2,pT∈R5, and it follows that(10)dxdt=x˙=fx=v1,a1,v2,a2,0T,where ai represents the acceleration of the oscillator as a function of xi,vi. Define ϕx,t as the local flow function associated with fx. Suppose that the movement of the oscillator is limited by symmetrical rigid constraints placed at x1=d corresponding to state space discontinuity surfaces D1 and D2, where(11)D1=hD1x,d=d−x1=0,D2=hD2x,d=−d−x1=0.The oscillator moves between two rigid stops whenhD1x,d>0 and hD2x,d<0. When hD1x,d=0 or hD2x,d=0, the oscillator collides with the rigid stop. In addition, let hP1x=hxD1x,dfx=−v1, hxP2x=hD2x,dfx=−v1.When the oscillator collides with the constraint, we establish a functionRx with a characteristic restitution coefficient R to represent the jump map, i.e., Rx represents the instantaneous state after the collision and before the collision, where(12)Rx=x1,−Rv1,x2,v2,pT.The state space trajectory and the grazing contact points ofD1 and D2 are, respectively, corresponding to points x∗1 and x∗2, so that(13)hD1x∗1,d∗=0,hP1x∗1=−v1∗1=0,ddthP1xx=x∗1=hxP1x∗1fx∗1=−a1∗1>0,hD2x∗2,d∗=0,hP2x∗2=−v1∗2=0,ddthP2xx=x∗2=hxP2x∗2fx∗2=−a1∗2<0.Choose a constant phase angleθ∗ as a Poincaré section, and it has the following form: Π1=x^∈R5×S1θ=θ∗, where x^=x1,x˙1,x2,x˙2,p,θT, θ=ωt+τ. The modulus of θ is 2π, θ∗=τ∗. Since the periodic trajectory of the system is symmetrical, we set another Poincaré section as Π2=x^∈R5×S1θ=θ∗+π.Withd as the bifurcation parameter, we define the flow maps P1x and P2x by the evolution of the smooth flow Φ⋅,T on the Poincaré sections Π1 and Π2; the expressions of P1x and P2x are as follows:(14)P1x=x∗2+N1x−x∗1+M1d−d∗+h.o.t,where N1=∂/∂xP1xx=x∗1,d=d∗ and M1=∂/∂dP1xx=x∗1,d=d∗.(15)P2x=x∗1+N2x−x∗2+M2d−d∗+h.o.t,where N2=∂/∂xP2xx=x∗2,d=d∗ and M2=∂/∂dP2xx=x∗2,d=d∗.The critical bifurcation valued∗ can be obtained from expression (9). Since the vector field fx and the smooth flow function Φ⋅,T do not contain the bifurcation parameter d, the values of the matrices M1=M2=0.We use the discontinuous mapping method introduced by Nordmark to analyze the system; the two discontinuous mapsDM1 and DM2 are introduced into the neighborhood of points x∗1 and x∗2 such that the surface P1x is invariant under DM1,i.e., x∈P1x,DM1x∈P1x. The same surface P2x is invariant under DM2, i.e., x∈P2x,DM2x∈P2x. According to the discontinuous mapping method, the discontinuous mapping of DM1 and DM2 is expressed by(16)DM1=Id,hD1x,d≥0,x∗1+β12hxP1x∗1f∗1−hD1x,d,hD1x,d<0,where(17)β1=fx∗1−gxx∗1fx∗1,DM2=Id,hD2x,d≤0,x∗2+β22hxP2x∗2f∗2−hD2x,d,hD2x,d>0,where(18)β2=fx∗2−gxx∗2fx∗2.According to (13)–(17), the composite maps P˜1=P1x∘DM1 and P˜2=P2x∘DM2 are written as(19)P˜1=x∗2+N1x−x∗1+h.o.t,hD1x,d≥0,x∗2+N1β12hxP1x∗1fx∗1−hD1x,d+h.o.t,hD1x,d<0,(20)P˜2=x∗1+N2x−x∗2+h.o.t,hD2x,d≤0,x∗1+N2β22hxP2x∗2fx∗2−hD2x,d+h.o.t,hD2x,d>0.We will discuss the stability of double grazing bifurcation using the Poincaré mapP˜=P˜2∘P˜1. Starting from the vicinity of the grazing point, whether it is at the impact side or the nonimpact side, if it is still close to the grazing point after the iterative mapping P˜, then the local attractor near the grazing trajectory exists; we refer to such scenario as a continuous grazing bifurcation or we say that the grazing bifurcation is stable. Otherwise, we refer to scenario as a discontinuous grazing bifurcation.When pointx starts from the impact side near the grazing point x∗1, hD1x,d<0. Here, we can think of the hD1x,d approximation as hxD1x∗1x−x∗1. Similarly, we consider the hD2x,d approximation as hxD2x∗2x−x∗2.If(21)hxD2x∗2P˜1x−x∗2=hxD2x∗2N1β12hxP1x∗1fx∗1−hD1x,d≤0,it means P˜1x is located at the nonimpact side near the grazing point x∗2.After the iteration of the mappingP˜2, if(22)hxD1x∗1P˜2P˜1x−x∗1=hxD1x∗1N2N1β12hxP1x∗1fx∗1−hD1x,d<0,then,(23)hxD1x∗1N2N1β1<0.It means the impact point impacts discontinuity surfaceD1 again and the impact will be perpetuated, which results in a large stretching in a direction given by the image of the vector β1 under the Jacobian N1 and N2. Therefore, the local attractor near the grazing trajectory does not exist. According to the analysis, the stability criterion of grazing bifurcation under which a local attractor persists near a grazing trajectory is formulated as follows:(24)hxD2x∗2N1N2N1n−1β1≤0,and(25)hxD1x∗1N2N1nβ1≥0,for all positive integer n.According the same method, another stability criterion of grazing bifurcation under which a local attractor persists near a grazing trajectory is formulated as follows:(26)hxD1x∗1N2N1N2n−1β2≥0,and(27)hxD2x∗2N1N2nβ2≤0,for all positive integer n.


### 3.2. Codimension-Two Grazing Bifurcation

The local attractor near grazing trajectory is lost in there ways.Case 1.
If(28)hD2x∗2P˜1x,d=hxD2x∗2N1β12hxP1x∗1fx∗1−hD1x,d<0,i.e., hxD2x∗2N1β1<0, and(29)hD1x∗1P˜2P˜1x,d=hxD1x∗1N2N1β12hxP1x∗1fx∗1−hD1x,d<0,i.e., hxD1x∗1N2N1β1<0, then it means that the point from the impact side near the grazing point x∗1 will impact the discontinuous surface D1 again after the iteration of the mapping P˜ and the impact will continue, which results in a large stretching in a direction given by the image of the vector β1 under the Jacobian N1 and N2; therefore, the local near-grazing attractor will lose, where discontinuous grazing bifurcation occurs.
If the impact point is followed by nonimpacting for some iterations but eventually impacts discontinuous surfaceD1, the impact will be perpetuated which lead to the instability of grazing bifurcation or the loss of near-grazing attrators.
Therefore,(30)hxD2x∗2N1N2N1iβ1<0,hxD1x∗1N2N1j+1β1<0,for any 0≤i≤j, the stability of grazing bifurcation will be lost, where i and j are positive integers.Case 2.
When the pointx satisfying hxD1x−x∗1<0 is from the impact side near the grazing point x∗1, if hxD2x∗2N1β1>0 and hxD2x∗2N1N2β2>0, it means that the impact point will impact with the discontinuous surface D2 again and the impact will continue, and the grazing bifurcation is discontinuous.
If the impact point is followed by nonimpacting for some iterations but eventually impacts discontinuous surfaceD2, the impact will be perpetuated. Finally, the stability of grazing bifurcation will also be lost.
That is,(31)hxD2x∗2N1N2N1iβ1>0,hxD2x∗2N1N2j+1β2>0,for any j≥i≥0, the stability of grazing bifurcation will be lost.Case 3.
When the pointx satisfying hxD1x−x∗1<0 is from the impact side near the grazing point x∗1, if hxD2x∗2N1β1>0 and hxD1x∗1N2β2<0, this means that the impact point will impact with the discontinuous surfaces D1 and D2 again and the impact will continue, and the grazing bifurcation is discontinuous. If the impact point is followed by non-impacting for some iterations but eventually impacts discontinuous surface D1 and D2 the impact will be perpetuated. Finally, the stability of grazing bifurcation will also be lost.
That is,(32)hxD2x∗2N1N2N1iβ1>0,hxD1x∗1N2N1N2jβ2<0,for any j≥i≥0, the stability of grazing bifurcation will be lost.
According to the above analysis, the conditions of codimension-two grazing bifurcation are obtained as follows (the definition of such points is seen in Ref. [10]):(33)hxD1x∗1N2N1nβ1=0,hxD1x∗1N2N1N2nβ2=0,hxD2x∗2N1N2N1nβ1=0,hxD2x∗2N1N2nβ2=0,where n=0,1,2,….


## 3.1. The Stability Criterion of Double Grazing Bifurcation

Letp be a state vector, such that x=x1,v1,x2,v2,pT∈R5, and it follows that(10)dxdt=x˙=fx=v1,a1,v2,a2,0T,where ai represents the acceleration of the oscillator as a function of xi,vi. Define ϕx,t as the local flow function associated with fx. Suppose that the movement of the oscillator is limited by symmetrical rigid constraints placed at x1=d corresponding to state space discontinuity surfaces D1 and D2, where(11)D1=hD1x,d=d−x1=0,D2=hD2x,d=−d−x1=0.The oscillator moves between two rigid stops whenhD1x,d>0 and hD2x,d<0. When hD1x,d=0 or hD2x,d=0, the oscillator collides with the rigid stop. In addition, let hP1x=hxD1x,dfx=−v1, hxP2x=hD2x,dfx=−v1.When the oscillator collides with the constraint, we establish a functionRx with a characteristic restitution coefficient R to represent the jump map, i.e., Rx represents the instantaneous state after the collision and before the collision, where(12)Rx=x1,−Rv1,x2,v2,pT.The state space trajectory and the grazing contact points ofD1 and D2 are, respectively, corresponding to points x∗1 and x∗2, so that(13)hD1x∗1,d∗=0,hP1x∗1=−v1∗1=0,ddthP1xx=x∗1=hxP1x∗1fx∗1=−a1∗1>0,hD2x∗2,d∗=0,hP2x∗2=−v1∗2=0,ddthP2xx=x∗2=hxP2x∗2fx∗2=−a1∗2<0.Choose a constant phase angleθ∗ as a Poincaré section, and it has the following form: Π1=x^∈R5×S1θ=θ∗, where x^=x1,x˙1,x2,x˙2,p,θT, θ=ωt+τ. The modulus of θ is 2π, θ∗=τ∗. Since the periodic trajectory of the system is symmetrical, we set another Poincaré section as Π2=x^∈R5×S1θ=θ∗+π.Withd as the bifurcation parameter, we define the flow maps P1x and P2x by the evolution of the smooth flow Φ⋅,T on the Poincaré sections Π1 and Π2; the expressions of P1x and P2x are as follows:(14)P1x=x∗2+N1x−x∗1+M1d−d∗+h.o.t,where N1=∂/∂xP1xx=x∗1,d=d∗ and M1=∂/∂dP1xx=x∗1,d=d∗.(15)P2x=x∗1+N2x−x∗2+M2d−d∗+h.o.t,where N2=∂/∂xP2xx=x∗2,d=d∗ and M2=∂/∂dP2xx=x∗2,d=d∗.The critical bifurcation valued∗ can be obtained from expression (9). Since the vector field fx and the smooth flow function Φ⋅,T do not contain the bifurcation parameter d, the values of the matrices M1=M2=0.We use the discontinuous mapping method introduced by Nordmark to analyze the system; the two discontinuous mapsDM1 and DM2 are introduced into the neighborhood of points x∗1 and x∗2 such that the surface P1x is invariant under DM1,i.e., x∈P1x,DM1x∈P1x. The same surface P2x is invariant under DM2, i.e., x∈P2x,DM2x∈P2x. According to the discontinuous mapping method, the discontinuous mapping of DM1 and DM2 is expressed by(16)DM1=Id,hD1x,d≥0,x∗1+β12hxP1x∗1f∗1−hD1x,d,hD1x,d<0,where(17)β1=fx∗1−gxx∗1fx∗1,DM2=Id,hD2x,d≤0,x∗2+β22hxP2x∗2f∗2−hD2x,d,hD2x,d>0,where(18)β2=fx∗2−gxx∗2fx∗2.According to (13)–(17), the composite maps P˜1=P1x∘DM1 and P˜2=P2x∘DM2 are written as(19)P˜1=x∗2+N1x−x∗1+h.o.t,hD1x,d≥0,x∗2+N1β12hxP1x∗1fx∗1−hD1x,d+h.o.t,hD1x,d<0,(20)P˜2=x∗1+N2x−x∗2+h.o.t,hD2x,d≤0,x∗1+N2β22hxP2x∗2fx∗2−hD2x,d+h.o.t,hD2x,d>0.We will discuss the stability of double grazing bifurcation using the Poincaré mapP˜=P˜2∘P˜1. Starting from the vicinity of the grazing point, whether it is at the impact side or the nonimpact side, if it is still close to the grazing point after the iterative mapping P˜, then the local attractor near the grazing trajectory exists; we refer to such scenario as a continuous grazing bifurcation or we say that the grazing bifurcation is stable. Otherwise, we refer to scenario as a discontinuous grazing bifurcation.When pointx starts from the impact side near the grazing point x∗1, hD1x,d<0. Here, we can think of the hD1x,d approximation as hxD1x∗1x−x∗1. Similarly, we consider the hD2x,d approximation as hxD2x∗2x−x∗2.If(21)hxD2x∗2P˜1x−x∗2=hxD2x∗2N1β12hxP1x∗1fx∗1−hD1x,d≤0,it means P˜1x is located at the nonimpact side near the grazing point x∗2.After the iteration of the mappingP˜2, if(22)hxD1x∗1P˜2P˜1x−x∗1=hxD1x∗1N2N1β12hxP1x∗1fx∗1−hD1x,d<0,then,(23)hxD1x∗1N2N1β1<0.It means the impact point impacts discontinuity surfaceD1 again and the impact will be perpetuated, which results in a large stretching in a direction given by the image of the vector β1 under the Jacobian N1 and N2. Therefore, the local attractor near the grazing trajectory does not exist. According to the analysis, the stability criterion of grazing bifurcation under which a local attractor persists near a grazing trajectory is formulated as follows:(24)hxD2x∗2N1N2N1n−1β1≤0,and(25)hxD1x∗1N2N1nβ1≥0,for all positive integer n.According the same method, another stability criterion of grazing bifurcation under which a local attractor persists near a grazing trajectory is formulated as follows:(26)hxD1x∗1N2N1N2n−1β2≥0,and(27)hxD2x∗2N1N2nβ2≤0,for all positive integer n.


## 3.2. Codimension-Two Grazing Bifurcation

The local attractor near grazing trajectory is lost in there ways.Case 1.
If(28)hD2x∗2P˜1x,d=hxD2x∗2N1β12hxP1x∗1fx∗1−hD1x,d<0,i.e., hxD2x∗2N1β1<0, and(29)hD1x∗1P˜2P˜1x,d=hxD1x∗1N2N1β12hxP1x∗1fx∗1−hD1x,d<0,i.e., hxD1x∗1N2N1β1<0, then it means that the point from the impact side near the grazing point x∗1 will impact the discontinuous surface D1 again after the iteration of the mapping P˜ and the impact will continue, which results in a large stretching in a direction given by the image of the vector β1 under the Jacobian N1 and N2; therefore, the local near-grazing attractor will lose, where discontinuous grazing bifurcation occurs.
If the impact point is followed by nonimpacting for some iterations but eventually impacts discontinuous surfaceD1, the impact will be perpetuated which lead to the instability of grazing bifurcation or the loss of near-grazing attrators.
Therefore,(30)hxD2x∗2N1N2N1iβ1<0,hxD1x∗1N2N1j+1β1<0,for any 0≤i≤j, the stability of grazing bifurcation will be lost, where i and j are positive integers.Case 2.
When the pointx satisfying hxD1x−x∗1<0 is from the impact side near the grazing point x∗1, if hxD2x∗2N1β1>0 and hxD2x∗2N1N2β2>0, it means that the impact point will impact with the discontinuous surface D2 again and the impact will continue, and the grazing bifurcation is discontinuous.
If the impact point is followed by nonimpacting for some iterations but eventually impacts discontinuous surfaceD2, the impact will be perpetuated. Finally, the stability of grazing bifurcation will also be lost.
That is,(31)hxD2x∗2N1N2N1iβ1>0,hxD2x∗2N1N2j+1β2>0,for any j≥i≥0, the stability of grazing bifurcation will be lost.Case 3.
When the pointx satisfying hxD1x−x∗1<0 is from the impact side near the grazing point x∗1, if hxD2x∗2N1β1>0 and hxD1x∗1N2β2<0, this means that the impact point will impact with the discontinuous surfaces D1 and D2 again and the impact will continue, and the grazing bifurcation is discontinuous. If the impact point is followed by non-impacting for some iterations but eventually impacts discontinuous surface D1 and D2 the impact will be perpetuated. Finally, the stability of grazing bifurcation will also be lost.
That is,(32)hxD2x∗2N1N2N1iβ1>0,hxD1x∗1N2N1N2jβ2<0,for any j≥i≥0, the stability of grazing bifurcation will be lost.
According to the above analysis, the conditions of codimension-two grazing bifurcation are obtained as follows (the definition of such points is seen in Ref. [10]):(33)hxD1x∗1N2N1nβ1=0,hxD1x∗1N2N1N2nβ2=0,hxD2x∗2N1N2N1nβ1=0,hxD2x∗2N1N2nβ2=0,where n=0,1,2,….


## 4. Controlling the Persistence of Near-Grazing Attractors

As the lose of stability for grazing bifurcation may arise catastrophic changes of system response and codimension-two even more complicated bifurcation, it is necessary to control the stability of grazing bifurcation by controlling the persistence of the local near-grazing attractor. We use discrete-in-time feedback control to stabilize the grazing bifurcation. Two constant phase angles are defined as Poincaré sections, and two discrete-in-time feedback controllers on Poincaré sections are designed. After that, we obtain a new compound map and then control the stability of the grazing bifurcation by controlling the parameters on the controller.We define the other two constant phase angles as Poincaré sectionsΠ3 and Π4, where Π3=x^∈R5×S1θ=θ∗+π/2 and Π4x^∈R5×S1θ=θ∗+3π/2.According toP1x and Π3, we define the mappings P3x:Π1⟶Π3 and P4x:Π3⟶Π2, and the resulting expansion is as follows:(34)P3x=x∗3+N3x−x∗1+h.o.t,P4x=x∗2+N4x−x∗3+h.o.t,where x∗3 is the fixed point of the grazing orbit on the Poincaré section Π3, N3=∂/∂xP3xx=x∗1, and N4=∂/∂xP4xx=x∗3.In the same way, according toP2x and Π4, we define the mappings P5x:Π2⟶Π4 and P6x:Π4⟶Π1, and the expansion is as follows:(35)P5x=x∗4+N5x−x∗2+h.o.t,P6x=x∗1+N6x−x∗4+h.o.t,where x∗4 is the fixed point of the grazing orbit on the Poincaré section Π4, N5=∂/∂xP5xx=x∗2, and N6=∂/∂xP6xx=x∗4.Discrete-in-time feedback controllers are designed on the Poincaré sectionsΠ3 and Π4, respectively, as follows:(36)G1x=x1v1x2v2k1x1−x1∗3+k2v1−v1∗3+k3p−p∗+p∗,G2x=x1v1x2v2k4x1−x1∗4+k5v1−v1∗4+k6p−p∗+p∗.We can calculate the expressions of mapsG1x and G2x which have the following forms:(37)G1x=x1∗3+G1xx−x1∗3,G2x=x1∗4+G2xx−x1∗4,where(38)G1x=10000010000010010010k1k200k3,G2x=10000010000010010010k4k500k6.The combining the mapG1x and P3x,P4x, obtain a controlled map as Pg1x=P4x∘G1x∘P3x, expansion as follows:(39)Pg1x=x∗2+N4G1xN3x−x∗1+h.o.t,where G1x=∂/∂xG1xx=x∗3.Similarly, combining the mapG2x with the map P5x,P6x, the map under control Pg2x=P6x∘G2x∘P5x, is obtained and its expansion has the following forms:(40)Pg2x=x∗1+N6G2xN5x−x∗2+h.o.t,where G2x=∂/∂xG2xx=x∗4.We can calculate the expressions ofNi as follows:(41)Ni=△△△△△△△△△△△△△△△△△△△△00001,where i=3,4,5,6 and the symbol Δ refers to a nontrivial coefficient.Combining (39) and (15), we construct a composite map of P˜g1=Pg1x∘DM1; the expressions are as follows:(42)P˜g1=x∗2+Ng1x−x∗1+h.o.t,hD1x,d≥0,x∗2+Ng1β12hxP1x∗1fx∗1−hD1x,d+h.o.t,hD1x,d<0,where Ng1=N4G1xN3.Similarly, we can combine (40) and (17) to construct a composite map of P˜g2=Pg2x∘DM2; the expression is as follows:(43)P˜g2=x∗1+Ng2x−x∗2h.o.t,hD2x,d≤0,x∗1+Ng2β22hxP2x∗2fx∗2−hD2x,d+h.o.t,hD2x,d>0,where Ng2=N6G2xN2.According to the analysis in Section3, there are three cases in which the grazing bifurcation is unstable, where discontinuous grazing bifurcation occurs. For Case 1, the impact point will impact with the discontinuous surface D1 again after some iterations and the impact will be perpetuated. Based on the stability criteria (24) and (25), we control k1−k6 on the controller to ensure that hxD2x∗2Ng1Ng2Ng1n−1β1≤0 and hxD1x∗1Ng2Ng1nβ1≥0, for positive integer n, such that the grazing bifurcation will be continuous.For Case2, the impact point will impact with the discontinuous surface D2 again after some iterations and the impact will be perpetuated. Based on the stability criteria (26) and (27), we control k1−k6 on the controller so that hxD2x∗2Ng1Ng2nβ2≤0 and hxD1x∗1Ng2Ng1Ng2n−1β2≥0 for positive integer n. The grazing bifurcation will be continuous.For Case3, the impact point will impact with the discontinuous surfaces D1 and D2 again, and after some iterations, the impact will be perpetuated. Based on the stability criteria (24) and (25) or (26) and (27), the grazing bifurcation will be continuous by controlling parameters k1−k6 to ensure the local attractor near grazing bifurcation persists.For the codimension-two grazing bifurcation point, the persistence of near-grazing attractors is also controlled by controllingk1−k6 based on the stability criteria (24) and (25) or (26) and (27). For example, if hxD1x∗1N2N1nβ1=0, we might let hxD2x∗2Ng1Ng2Ng1n−1β1≤0 and hxD1x∗1Ng2Ng1nβ1≥0 by controlling k1−k6.IfhxD1x∗1N2N1N2nβ2=0, we might let hxD2x∗2Ng1Ng2nβ2≤0 and hxD1x∗1Ng2Ng1Ng2n−1β2≥0 by controlling k1−k6.


## 5. Numerical Experiments

For this system, we take suitable parametersμm=6.0,μk=3.0,ω=0.63,R=0.82,ζ=0.2,andp=2.513 as an example. d is supposed as the bifurcation parameter; the critical value d∗=1.61139 for grazing bifurcation is obtained from formula (9). According to the analysis of near-grazing dynamics in Section 3, under the set of fixed parameters, hxD2x∗2N1β1=−0.128027<0 and hxD1x∗1N2N1β1=−0.078865<0. The impact point will impact with the discontinuous surfaces D1 again and the impact will be perpetuated, which satisfies the instability conditions in Case 1. The bifurcation diagram as shown in Figure 2(a) is obtained through numerical simulation by discontinuity maps (19) and (20). It is clear that the grazing bifurcation is discontinuous because the discontinuous transition occurs at the bifurcation point d−d∗=0. In addition, we take Π2 as the Poincaré section; the bifurcation diagram as shown in Figure 2(b) is obtained through direct numerical simulation by (4) and (5), and it is clear that the discontinuous jump also occurs at the grazing bifurcation point d−d∗=0.Figure 2
Bifurcation diagram from the grazing bifurcation. (a)1/1 impact periodic motions based on the numerical simulation obtained by the discontinuity maps (19) and (20). (b) Direct numerical simulation on Poincaré section Π2 of 1/1 impact periodic motions. (c)1/2 impact periodic motions based on the numerical simulation obtained by the discontinuity maps (19) and (20). (d) Direct numerical simulation on Poincaré section Π2 of 1/2 impact periodic motions. (e) The codimension-two grazing bifurcation points based on the numerical simulation obtained by the discontinuity maps (19) and (20). (f) Direct numerical simulation on Poincaré section Π2 of the codimension-two grazing bifurcation points.
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(f)We use the control method in Section4 to stabilize the unstable grazing bifurcation phenomena. When the parameters of k1,k3 and k4,k6are fixed, the ranges of parameters k2 and k5 are calculated according to control strategy, which is named two-dimensional control region of k2 and k5. Here, we select k1=−0.5,k3=−1,k4=−0.5,andk6=−1 as the fixed parameters; a two-dimensional control region graph of k2 and k5 is obtained as shown in Figure 3(a), where k2 is x−axis, k5 is y−axis, and the blue region is the region of control parameters for ensuring the stability of the grazing bifurcation.Figure 3
The parametersk2 and k5 control region. (a) The grazing bifurcation of 1/1 impact periodic motions. (b) The grazing bifurcation of 1/2 impact periodic motions. (c) The codimension-two grazing bifurcation points.
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(c)We selectk2=−5.8andk5=−5.8 to control the system and draw Figure 4(a) through the discontinuity maps (40) and (41). At the same time, we take Π2 as the Poincaré section and obtain Figure 4(b) by direct numerical simulation. As shown in Figures 4(a) and 4(b), there exists the local attractor near grazing bifurcation.Figure 4
Bifurcation diagram of a stable grazing bifurcation after control. (a)1/1 impact periodic motions based on the numerical simulation obtained by the discontinuity maps (38) and (39). (b) Direct numerical simulation on Poincaré section Π2 of 1/1 impact periodic motions. (c)1/2 impact periodic motions based on the numerical simulation obtained by the discontinuity maps (38) and (39). (d) Direct numerical simulation on Poincaré section Π2 of 1/2 impact periodic motions. (e) The codimension-two grazing bifurcation points based on the numerical simulation obtained by the discontinuity maps (38) and (39). (f) Direct numerical simulation on Poincaré section Π2 of the codimension-two grazing bifurcation points.
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(f)Take the set of parametersμm=6.0,μk=3.0,ω=0.48,R=0.82,ζ=0.2,p=2.013, as an example. d∗=0.65183, is obtained from equation (9). Without control, hxD2x∗2N1N2N1β1=−0.00868998<0 and hxD1x∗1N2N12β1=−0.00822255<0. are obtained under the above fixed parameters. It satisfies the instability conditions in case 1 according to expression (30). The bifurcation diagram as shown in Figure 2(c) is obtained through numerical simulation by discontinuity maps (19) and (20). It is clear that the grazing bifurcation is discontinuous because the discontinuous transition occurs at the bifurcation point d−d∗=0. In addition, we take Π2 as the Poincaré section, and the bifurcation diagram as shown in Figure 2(d) is obtained through direct numerical simulation by (4) and (5); it is clear that the discontinuous jump also occurs at the grazing bifurcation point d−d∗=0.Based on the stability criterion mentioned in Section4, we select k1=−0.6,k3=−1,k4=−0.6,andk6=−1 as the fixed parameters; a two-dimensional control region graph of k2 and k5 is obtained as shown in Figure 3(b), where k2 is x−axis, k5 is y−axis, and the blue region is the region of control parameters for ensuring the stability of the grazing bifurcation. Then, we select k2=−5.8andk5=−5.8 to control the system, and the bifurcation diagram is obtained as shown in Figure 4(c) by the discontinuity maps (40) and (41). In addition, we take Π2 as the Poincaré section and obtain Figure 4(d) by direct numerical simulation. As shown in Figures 4(c) and 4(d), there exists the local attractor near grazing bifurcation.Fixed parametersμm=6.0,μk=3.0,ω=0.5649677,R=0.82,ζ=0.2,p=0.813. the critical valued∗=1.821554, such that hxD1x∗1N2N15β1=0. It corresponds to the codimension-two grazing bifurcation. We obtain the bifurcation diagram Figure 2(e) through numerical simulation by the discontinuity maps (19) and (20). As shown in Figure 2(e), the system is in a complex chaotic state near the grazing point. Then, we take Π2 as the Poincaré section and obtain Figure 2(f) by direct numerical simulation.Based on the control strategy mentioned in Section4, we select k1=−0.6,k3=−1,k4=−0.6,andk6=−1 as the fixed parameters; a two-dimensional control region graph of k2 and k5 is obtained as shown in Figure 3(c), where k2 is x−axis, k5 is y−axis, and the blue region is the region of control parameters for suppressing the chaos. Then, we select k2=−5.8andk5=−5.8 to control the system, and the bifurcation diagram is obtained as shown in Figure 4(e) by the discontinuity maps (40) and (41). In addition, we take Π2 as the Poincaré section and obtain Figure 4(f) by direct numerical simulation.Comparing Figure2 with Figure 4, it is shown that the results obtained by mappings are in good agreement with the direct numerical simulation, and the control effect is further verified.


## 6. Conclusions

The discontinuous grazing bifurcation is often accompanied by a jump phenomenon. In addition, complex chaotic regions occur near the codimension-two bifurcation points. How to avoid the dramatic change of system response caused by jump phenomenon of grazing bifurcation or the codimension-two bifurcation point is an urgent requirement for the control mechanism of discontinuous grazing bifurcation. In this paper, a discrete-in-time feedback control strategy is used to control the near-grazing dynamics of double grazing period motion in a two-degree-of-freedom vibroimpact system with symmetrical constraints. Compared to unilateral constrained systems, the stability criterion of double grazing period motion becomes more complex. Based on the stability criterion, the control strategy is designed to control the near-grazing dynamics and the two parameters’ control region is obtained. For the case of discontinuous grazing bifurcation, we take two jumping phenomena (jumping from nonimpact periodic motion to1/1 and 1/2 impact periodic motion) as examples to stabilize grazing bifurcation by controlling the parameters on the controller. In addition, the chaos dynamics near codimension-two grazing bifurcation point is controlled by using this strategy. Finally, the feasibility of the control strategy is illustrated by comparing the numerical simulation of composite mapping with the direct numerical simulation of Poincare section.



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