for hybrid systems. Applying this, the normal-form coefficients are evaluated, which in this case imply the occurrences a jump to chaos and period-adding cascade. The necessary and sufficient conditions of the general discontinuous boundary are expounded in [28]. In [27], by means of non-stick mapping, the necessary and sufficient conditions for the grazing in periodically forced linear oscillator with dry friction are obtained. By constructing special maps such as zero time discontinuity mapping [10]-[12] and Nordmark map [36]-[39], the existence of periodic solution and their stability were investigated in mechanical systems. In this paper, we model the dynamics with grazing impacts by utilizing differential equations with impulses at variable moments and applying the methods of [1]-[3]. As a consequence of such methods, the role of the mappings [36]-[39] is diminished. One can observe that a trajectory at a grazing point may have tangency to the surface of discontinuity, which is parallel to one or several coordinate axises. Particularly it means the velocity approaches to zero [36]-[39]. Then, we will say about the axial grazing. Otherwise, grazing is non-axial. This research contains the analysis of both axial and non-axial grazing. In [33], it is observed through simulations and experiments that the coefficient of restitution depends on the impact velocity of the particle by considering both the viscoelastic and the plastic deformations of particles occurring at low and high velocities, respectively. It has been proposed in [32] that at low impact velocities and for most materials with linear elastic range, the coefficient of restitution is of the form R ( v ) = 1 − av, where v is the velocity before collision and a is a constant. Also, for low impact velocities the restitution law can be considered quadratic [16]. This is why, we will use non-constant restitution coefficients in models with impacts in this study. For investigation of autonomous differential equations, it is convenient to utilize properties of dynam- ical systems. They are the group property, continuation of solutions in both time directions, continuity and differentiability in parameters. The studies of discontinuous dynamical systems with transversal intersections of orbits and surfaces, B − smooth discontinuous flows, can be found in [1, 2]. In this research, the dynamics is approved for systems with grazing orbits. Moreover, the definitions of orbital stability and asymptotic phase are adapted to grazing cycles. The orbital stability theorem is proved, which can not be underestimated for theory of impact mechanisms. The remaining part of the paper is organized as follows. In the next section, we will introduce necessary notations, definitions and theorems to specify discontinuous dynamical systems. In Section 3, it is shown how the dynamics can be linearized around grazing orbits. In Section 4, the theorem of orbital stability is adapted for grazing cycles. In Section 5, the small parameter analysis is applied near grazing orbits and bifurcation of cycles is observed. In last three sections, examples are presented to actualize the theoretical results numerically and analytically. Finally, Conclusion covers a summary of our study. 2