dynamics behavior such as chaos and bifurcation [7]-[10], [13], [15], [18], [36]-[39]. However, there is still no sufficient conditions for the discontinuous motion to admit main features of dynamical systems : the group property, continuous and differentiable dependence on initial data and continuation of motions, which are useful for both local and global analysis. Variational systems for grazing solutions have not been considered in general as well as orbital stability theorem and regular perturbation theory around cycles, despite, particular cases can be found in specialized papers. See, for example, [12]. To investigate these problems in the present paper, we have applied the method of B − equivalence and results on discontinuous dynamics developed and summarized in [1]. In our analysis the grazing singularity is observed through the gradient of the time function τ ( x ) , since some of its coordinates are infinite. We have found the components of the discontinuous dynamical system that is the vector field, surfaces of discontinuity and the equations of jump such that interacting they neutralize the effect of singularity. Then, we linearize the system at the grazing moments and this brings the dynamics to regular analysis and make suitable for the application. By means of the linearization, the theory can be understood as a part of the general theory of discontinuous dynamical system. Thus, we have considered grazing phenomena as a subject of the general theory of discontinuous dynamical systems [1], discovered a partition of set of solutions near grazing solution such that we determine linearization around a grazing solution is a collection of several linear impulsive systems with fixed moments of impulses. This constitutes the main novelty of the present paper. To linearize a solution around the grazing one, a system from the collection is to be utilized. This result has been applied to prove the orbital stability theorem. The way of analysis in [1]-[3] continues in the present paper and it admits all attributes which are proper for continuous dynamics [23]. That is why, we believe that the method can be extended for introduction and research of graziness in other types of dynamics such as partial and functional differential equations and others. Next, we plan to apply the present results and the method of investigation for problems initiated in [7]-[10], [37]-[39], [41]. References [1] M. Akhmet. Principles of Discontinuous Dynamical Systems . Springer-Verlag, New York, 2010. [2] M.U. Akhmet. Perturbations and Hopf bifurcation of the planar discontinuous dynamical system. Nonlinear Analysis , 60:163–178, 2005. [3] M.U. Akhmet. On the smoothness of solutions of impulsive autonomous systems, Nonlinear Anal.: TMA, 60:311–324, 2005. [4] M. U. Akhmet and A. Kıvılcım. The Models with Impact Deformations, Discontinuity, Nonlinearity, and Complexity, 4(1):49–78, 2015. 41