Homoclinical Structure of Dynamic Equations on Time Scales Mehmet Onur Fen Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey Department of Mathematics and Computer Science, Çankaya University, 06790, Ankara, Turkey E-mail: monur.fen@gmail.com Homoclinic and heteroclinic motions in dynamics equations on time scales is investigated. The utilized time scale is a specific one such that it is a union of disjoint compact intervals. A numerical example that supports the theoretical results is presented. Keywords: Dynamic equations on time scales; Homoclinic and heteroclinic motions; Stable and unstable sets Dynamic equations on time scales has become one of the attractive subjects among mathematicians beginning with the studies of S. Hilger (1988). The theory of time scales has many applications in various disciplines such as mechanics, electronics, neural networks, population models and economics (Bohner & Peterson, 2001; Chen & Song, 2013; Tisdell & Zaidi, 2008; Zhang et al., 2010). Homoclinic and heteroclinic motions play an important role in the theory of dynamical systems since they are concerned with the presence of chaos (Bertozzi, 1988; Chacon & Bejarano, 1995; Gonchenko et al., 1996; Shil’nikov, 1967; Wiggins, 1988). The existence of a homoclinic solution was obtained as a limit of periodic solutions in a second order non-autonomous Hamiltonian system on time scales in the paper (Su & Feng, 2014) by means of the mountain pass theorem and the standard minimizing argument. In the present paper, we consider differential equations on a specific time scale and show that perturbations through discrete equations give rise to the existence of homoclinic and heteroclinic motions in the system. Let us denote by R and Z the sets of real numbers and integers, respectively. The basic concepts about differential equations on time scales are as follows (Bohner & Peterson, 2001; Lakshmikantham et al. 1996; Lakshmikantham & Vatsala, 2002; Lakshmikantham & Devi, 2006). A time scale is a nonempty closed subset of R . On a time scale T , the forward and backward jump operators are defined as σ ( t ) = inf { s ∈ T : s > t } and ρ ( t ) = sup { s ∈ T : s < t } , respectively. We say that a point t ∈ T is right-scattered if σ ( t ) > t and right-dense if σ ( t ) = $_{t}$. Similarly, if ρ ( t ) < t, then t ∈ T is called left- scattered, and otherwise it is called left-dense. Besides, a function h : T × R n → R n is called rd-continuous 1 arXiv:1601.04917v1 [nlin.CD] 19 Jan 2016 1 Introduction Abstract