The present study is devoted to the existence of homoclinic and heteroclinic motions in dynamic equations on time scales. Sufficient conditions that guarantee the existence of homoclinic and heteroclinic motions are provided, and the definitions of stable and unstable sets are given on the basis of functional spaces. The equation under investigation can be considered as a hybrid system since it combines the dynamics of the discrete map with the continuous dynamics of the equation on time scales. The presented example and simulations reveal the applicability of our theoretical results. This work is supported by the 2219 scholarship programme of TÜBİTAK, the Scientific and Technological Research Council of Turkey. 7 References Acknowledgments 5 Conclusion Figure 3: The heteroclinic solution of (4.3). It is observable that the bounded solution$_{ϕ}$$_{η}$$_{(}$$_{t}$$_{)}$ is heteroclinic to the $_{8}$-periodic solutions ϕ$_{η}$ ∗ ( t ) , ϕ$_{η}$ ∗∗ ( t ) . [1] M. U. Akhmet and M. Turan, The differential equations on time scales through impulsive differential equations , Nonlinear Analysis, 65 (2006) 2043-2060. [2] M. U. Akhmet, Hyperbolic sets of impact systems , Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15 (Suppl. S1) (2008) 1-2, in: Proceedings of the $_{5}$th International Conference on Impulsive and Hybrid Dynamical Systems and Applications, Beijing, Watan Press, 2008. [3] M. U. Akhmet, Devaney’s chaos of a relay system , Commun. Nonlinear Sci. Numer. Simulat., 14 (2009a), 1486-1493. [4] M. U. Akhmet, Li-Yorke chaos in the system with impacts , J. Math. Anal. Appl., 351 (2009b) 804-810. [5] M. U. Akhmet, Homoclinical structure of the chaotic attractor , Commun. Nonlinear Sci. Numer. Simulat., 15 (2010a) 819-822. −60 −40 −20 0 20 40 60 80 100 −2.5 −2 −1.5 −1 −0.5 0 t y 2 ϕ$_{η}$ ∗ ( t ) ϕ$_{η}$ ∗∗ ( t ) ϕ$_{η}$ ( t )