Discontinuous dynamics with grazing points M. U. Akhmet $^{a,}$∗ , A. Kıvılcım a a Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey Discontinuous dynamical systems with grazing solutions are discussed. The group property, continu- ation of solutions, continuity and smoothness of motions are thoroughly analyzed. A variational system around a grazing solution which depends on near solutions is constructed. Orbital stability of grazing cycles is examined by linearization. Small parameter method is extended for analysis of neighborhoods of grazing orbits, and grazing bifurcation of cycles is observed in an example. Linearization around an equilibrium grazing point is discussed. The mathematical background of the study relies on the theory of discontinuous dynamical systems [1]. Our approach is analogous to that one of the continuous dy- namics analysis and results can be extended on functional differential, partial differential equations and others. Appropriate illustrations with grazing limit cycles and bifurcations are depicted to support the theoretical results. Keywords: Discontinuous dynamical systems; Grazing points and orbits; Axial and non-axial grazing; Variational system; Orbital stability; Small parameter; Bifurcation of cycles; Impact mechanisms Vibro-impacting systems provide examples of non-linear dynamical systems, exhibiting new levels of complicated dynamics due to their non-smoothness. Grazing phenomenon is one of the attractive features for these dynamics [13]-[36], [26]-[29]. There are two approaches for the definition of grazing in literature. One is presented in the studies of Bernardo, Budd and Champneys [7, 8], Bernardo and Hogan [9], and Luo [26]-[29]. In these studies, it is asserted that grazing occurs when a trajectory hits the surface of discontinuity tangentially. In [36]-[39], Nordmark defines grazing as the approach of the velocity to zero in the neighborhood of the surface of discontinuity which is the case of the studies conducted in [7]-[9], [26]-[29]. Our comprehension of grazing in this paper is close to that one in [8],[9],[26]. In the paper [15], grazing is considered as a bounding case which separates regions of quite different dynamic behaviors. It is understood that the system trajectory makes tangential contact with an event triggering hypersurface. The shooting, continuation and optimization methods are developed and illustrated for both transient and grazing phenomena. It is exemplified by utilizing power electronics and robotics. In [41], the grazing periodic orbit and its linearization are obtained by means of a numerical continuation method 1 Introduction Abstract ∗ Corresponding Author Tel.: +90 312 210 5355, Fax: +90 312 210 2972, E-mail: marat@metu.edu.tr 1 arXiv:1509.01678v1 [math.DS] 5 Sep 2015