Under the condition (C2), it is known that q ( t ) is a transversal homoclinic orbit, i.e., taking the $_{1}$-time map G : R m → R m of the system (2.4), it has a hyperbolic fixed point p (0) with a transversal homoclinic orbit { q ( n ) $_{}$_{n}$$_{∈}$$_{Z}$}$. Following Sections 3 and 4 of [41], especially Theorem 4.8 of [41], we get a collection of bounded solutions { ν$_{β}$ ( t ) }$_{β}$$_{∈}$$_{S}$$_{m}$ of system (2.4) orbitally near q ( t $_{)}$, where the index set S$_{m}$, m ≥ 2 , is the set of doubly infinite sequences a = ( . . . , a$_{−}$$_{1}$, a$_{0}$, a$_{1}$, . . . ) with a$_{i}$ ∈ { 1 , . . . , m } for all i ∈ Z , i.e., S$_{m}$ = { 1 , 2 , . . . , m } Z such that each linear system has an exponential dichotomy on R with uniform positive constants K and α and projections $_{P$_{β}$}$: where V$_{β}$ is the fundamental solution of the system (2.7). By Theorem 4.8 of [41], an iterative G l , for some fixed ¯ l ∈ N , is conjugate to the Bernoulli shift on an invariant compact subset I ⊂ R n , G ¯ l : I → I . So G ¯ l has $_{i}$-periodic orbits in I for any natural number $_{i}$. This gives that the original map G has periodic orbits with periods i ¯ l starting in I . Since by definition ν$_{β}$ (0) = ς$_{β}$ for some ς$_{β}$ ∈ I and then G $^{k}$( ς$_{β}$ ) = ν$_{β}$ ( k ) for any k ∈ Z , we see that among these ν$_{β}$ ( t ) there are i ¯ $_{l}$-periodic solutions for any i ∈ N . Introducing the new variable y through u = y + ν$_{β}$, system (2.5) can be written in the following form: Since f is C 2 function, there exist positive numbers P$_{1}$ and P$_{2}$ such that sup t ∈ R ,β ∈ S m ‖ D$_{u}$f ( ν$_{β}$ ( t ) , t ) ‖ ≤ P$_{1}$ and sup t ∈ R ,β ∈ S m ‖ D$_{uu}$f ( ν$_{β}$ ( t ) , t ) ‖ ≤ P$_{2}$ for each bounded solution ν$_{β}$ ( t ) of (2.4). Our main assumption is the existence of a nonempty set A$_{x}$ of all solutions of system (2.3), uniformly bounded on R . That is, there exists a positive real number H such that sup t ∈ R ‖ x ( t ) ‖ ≤ H for all x ( t ) ∈ A$_{x}$ . The following conditions are also needed: Using the dichotomy theory [20], one can verify that for a given solution x ( t ) of system (2.3), a bounded on R function φ β x ( t $_{)}$( t ) = y ( t ) is a solution of system (2.9) if and only if the following integral equation is 5 (C6) M$_{h}$ < α 2 16 K $^{2}$P 2 , where M$_{h}$ = sup ‖ x ‖≤ H ‖ h ( x ) ‖ . (C5) There exists a positive number M$_{f}$ such that sup u ∈ R $^{n}$,t ∈ R ‖ f ( u, t ) ‖ ≤ M$_{f}$ ; y ′ = D$_{u}$f ( ν$_{β}$, t ) y + f ( y + ν$_{β}$, t ) − f ( ν$_{β}$, t ) − D$_{u}$f ( ν$_{β}$, t ) y + h ( x ) . (2.9) ∥ V$_{β}$ ( t ) P$_{β}$V$_{β}$ ( s ) − $^{1}$∥ ∥ ≤ Ke − α ( t − s ) ∀ t, s, t ≥ s, ∥ V$_{β}$ ( t )( I − P$_{β}$ ) V$_{β}$ ( s ) − $^{1}$∥ ∥ ≤ Ke α ( t − s ) ∀ t, s, t ≤ s, (2.8) v ′ = D$_{u}$f ( ν$_{β}$ ( t ) , t ) v (2.7)