homoclinic and heteroclinic motions on time scales are given on the basis of functional spaces. Simulations that support the theoretical results are also provided. Throughout the paper, the usual Euclidean norm for vectors and the norm induced by the Euclidean norm for square matrices (Horn & Johnson, 1992) will be utilized. Let us take into account the function ψ : T ′ 0 → R defined as (Akhmet & Turan, 2006), where δ$_{k}$ = θ$_{2}$$_{k}$$_{+1}$ − θ$_{2}$$_{k}$, k ∈ Z , and T ′ 0 = T$_{0}$ \ ⋃ ∞ k = −∞ { θ$_{2}$$_{k}$$_{−}$$_{1}$ } , and denote by X ( s, r ) the transition matrix of the linear homogeneous impulsive system where s$_{k}$ = ψ ( θ$_{2}$$_{k}$ ) , k ∈ Z , Δ x |$_{s}$$_{=}$$_{s}$$_{k}$ = x ( s$_{k}$ +) − x ( s$_{k}$ ) and x ( s$_{k}$ +) = lim s → s + k x ( s ) . The following conditions are required. If the conditions ( C 1) − ( C 5) are valid, then for a fixed solution ζ = { ζ$_{k}$ }$_{k}$$_{∈}$$_{Z}$ of (1.2) there exists a unique solution ϕ$_{ζ}$ ( t ) of (1.1) which is bounded on T$_{0}$ such that 3 sup t ∈ T$_{0}$ ‖ ϕ$_{ζ}$ ( t ) ‖ ≤ N ( M$_{f}$ + M$_{F}$ ) ( 1 λ + ¯ δ 1 − e − λ ¯ κ ) . x $^{′}$( s ) = Ax ( s ) , s = s$_{k}$, Δ x |$_{s}$$_{=}$$_{s}$$_{k}$ = δ$_{k}$Ax ( s$_{k}$ ) , ψ ( t ) =      t − ∑ 0 <θ$_{2}$ k <t δ$_{k}$, t ≥ 0 , t + t ≤ θ$_{2}$ k < 0 δ$_{k}$, t < 0 , 2 Preliminaries (C1) det( I + δ$_{k}$A ) = 0 for all k ∈ Z , where I is the n × n identity matrix; (C2) There exist positive numbers N and λ such that ‖ X ( s, r ) ‖ ≤ Ne − λ ( s − r ) for s ≥ r ; (C3) There exist positive numbers M$_{f}$ and M$_{F}$ such that sup t ∈ T$_{0}$ , y ∈ R n ‖ f ( t, y ) ‖ ≤ M$_{f}$ and sup υ ∈ Λ ‖ F ( υ ) ‖ ≤ M$_{F}$ ; (C4) There exists a positive number L$_{f}$ such that ‖ f ( t, y$_{1}$ ) − f ( t, y$_{2}$ ) ‖ ≤ L$_{f}$ ‖ y$_{1}$ − y$_{2}$ ‖ for all t ∈ T$_{0}$ , y$_{1}$, y$_{2}$ ∈ R $^{n}$. (C5) NL$_{f}$ ( 1 λ + ¯ δ 1 − e − λ ¯ κ ) < 1; (C6) − λ + NL$_{f}$ + 1 ¯ κ ln ( 1 + NL$_{f}$ ¯ δ ) < 0 .