Based on the general discussion of bifurcation theory and stability analysis, we will classify various kinds of smooth local bifurcations of codimension-1 that our system model shows as one of the system parameters is changed. The grazing bifurcation caused by the grazing impact of an orbit is discussed here. In support of our claim about various kinds of smooth local bifurcations which the period 1 solutions go through, we numerically calculate the eigenvalues of the linearized Poincaré map taken at constant drive phase near the period 1 orbits and examine the change in the eigenvalues as a function of the drive amplitude. The eigenvalues are calculated using the Poincaré map linearized in the neighborhood of the period-one orbits. The Jacobian is composed of P matrix between impacts and C matrix at impact. Thus, the Poincaré mapping of the constant phase type thus locally consists of series of P and C matrices. .

It should be noted that the calculation of the eigenvalues is not enough to distinguish among different types of local bifurcations. However, the eigenvalue calculation does establish the distinction between period doubling bifurcation and the other kinds of local bifurcations. To establish the saddle-node bifurcation point we numerically locate the unstable saddle at the Poincaré section and track it through the parameter space until the bifurcation point. For the case of pitchfork (symmetry breaking) bifurcation phase plane projection of the two asymmetric orbits and tracking of unstable symmetric orbit was necessary to establish the nature of the bifurcation.

The result of the eigenvalue calculation as a function of A is shown in Fig. 1. The solid line curve is for the real part of the eigenvalues and dashed line curve is for the imaginary part of the eigenvalues. From A = 0.0 to 0.4 the figure shows the eigenvalues of the nonimpacting orbit which is a stable focus with eigenvalues given by e^-b±ip/w. At A = 0.4 (point labeled SN₁), the impacting symmetric orbit is created as a result of saddle-node bifurcation. The eigenvalues, shown in the figure, crosses the unit circle in the complex plane, along the real axis at 1 and two periodic orbits are created - a saddle (unstable) and a node (stable). At A = 2.95 (point labeled PF), the symmetric orbit loses it stability by pitchfork bifurcation because an eigenvalue crosses the unit circle along the real axis at 1 and this bifurcation creates two similar asymmetric orbits. The fact that it is a pitchfork bifurcation was established by observing the phase plane projection of the two asymmetric orbits in the vicinity of the bifurcation point. As the amplitude is increased through the symmetry breaking (pitchfork) bifurcation, the system could go to either of the two stable asymmetric orbits. At A = 3.78 (point labeled SN₂), we again have a saddle-node bifurcation, but this time it creates stable and unstable asymmetric orbits P₁₂₁ and P₁₁₂. The stable orbit, P₁₁₂, eventually loses its stability as a result of period-doubling bifurcation at A = 3.95 (point labeled PD), and spawns a period 2 orbit, since this time the eigenvalue crosses the unit circle at -1.

The symmetric orbit, after it becomes unstable because of symmetry breaking bifurcation and the asymmetric P₁₂₁ orbit which becomes unstable as a result of the period doubling bifurcation were numerically tracked using the numerical method of following an unstable orbit. The Jacobian at the constant phase Poincaré map was obtained numerically and then the eigenvalues of this Jacobian were calculated.

To establish the saddle-node bifurcation event, we first locate the unstable orbit (saddle) in the constant phase Poincaré section, and track the unstable orbit, in the parameter space, numerically using the follow orbit algorithm to the bifurcation point. Figures 2 and 3 show the creation of stable and unstable periodic orbits at A = 0.4 and A = 3.78.

Figure 2 shows the movement of symmetric P₁₁₁ orbit, as a function of A, in the phase plane projection. The arrow indicates the direction of the increasing A. For the plot shown, the position and the velocity at zero drive phase is recorded as A is varied from 0.4 to 2.61. As described earlier, the position and velocity of the symmetric P₁₁₁ orbits are determined using the analytic results derived in here. The stability of the solutions is determined using P and C matrices. The calculations show that one of the solutions is stable and the other is unstable.

Figure 3 shows the creation of stable and unstable periodic orbits, P₁₂₁ (Fig.(a)) and P₁₁₂ (Fig.(b)), at A = 3.78. The value of A is increased from 3.78 to 4.21 in the direction of the arrows. The position and velocity for P₁₂₁ and P₁₁₂ was calculated numerically by finding the Jacobian at the surface defined by constant drive phase. Using the multi-variable Newton-Raphson root finding algorithm, we find the location of the unstable orbit at the Poincaré. With this numerical algorithm we were able to track the unstable orbit in the parameter space. The details of the algorithm and results of tracking different kinds of unstable orbits will be discussed later. In both the figures  2 and  3 the `cross' marked points are for stable orbit and `circle' marked points are for unstable orbits.