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homoclinic and heteroclinic motions on time scales are given on the basis of functional spaces. Simulations that support the theoretical results are also provided. 

## 2 Preliminaries 

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. 

(C1) det( I + δ$_{k}$A ) ̸ = 0 for all k ∈ Z , where I is the n × n identity matrix; 

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(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 . 

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 

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