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In order to show that Π ψ ( t ) is almost periodic, let us fix an arbitrary positive number ϵ and set 

Since Π ψ is uniformly continuous, there exists a positive number η satisfying η < ζ 5 and η ≤ ϵ 3 H$_{0}$ such that ‖ Π ψ ( t$_{1}$ ) − Π ψ ( t$_{2}$ ) ‖ < ϵ 3 whenever | t$_{1}$ − t$_{2}$ | < 4 η. j 

According to Lemma A.3 [19], { ζ i } , j ∈ Z , is a family of equipotentially almost periodic sequences, since { τ j i } , j ∈ Z , is a family of equipotentially almost periodic sequences and the sequence { κ$_{i}$ } , i ∈ Z , is periodic. Let θ be a number with 0 < θ < η, and consider the numbers ω ∈ Ω and q ∈ Q as in Lemma 3.1 such that (i) ‖ ψ ( t + ω ) − ψ ( t ) ‖ < η, t ∈ R ; (ii) ‖ f ( z, t + ω ) − f ( z, t ) ‖ < η, z ∈ R $^{n}$, t ∈ Z , (iii) ‖ x ( t + ω ) − x ( t ) ‖ < η, t ∈ R ; (iv) | ζ q i − ω | < θ, i ∈ Z . 

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For any k ∈ Z , it can be verified that if s ∈ ( ζ$_{k}$ + θ, ζ$_{k}$$_{+1}$ − θ ) , then ν ( s + ω, ζ ) − ν ( s, ζ ) = 0 . Therefore, for each t from the intervals ( ζ$_{i}$ + η, ζ$_{i}$$_{+1}$ − η ) , i ∈ Z , one can confirm that 

<img src='content_image/307168.jpg'>

Suppose that t ∈ ( ζ$_{˜}$ i − η, ζ$_{˜}$ $_{i}$+ η ) for some ˜ i ∈ Z . Because η is sufficiently small such that 5 η < ζ, t + 3 η belongs to the interval ( ζ$_{˜}$ i + η, ζ$_{˜}$ i +1 − η ) so that 

Thus, we have that 

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Therefore, ‖ Π ψ ( t + ω ) − Π ψ ( t ) ‖ < ϵ for all t ∈ R . Accordingly Π ψ ( t ) is almost periodic and Π( C $_{0}$) ⊆ C $_{0}$. Now, let ψ$_{1}$ and ψ$_{2}$ be elements of C $_{0}$. Then, 

<img src='content_image/307165.jpg'>

Hence, ‖ Π ψ$_{1}$ − Π ψ$_{2}$ ‖ 0 ≤ KL$_{f}$ α ‖ ψ$_{1}$ − ψ$_{2}$ ‖ 0 . Since KL$_{f}$ α < 1 , the operator Π : C 0 → C 0 is contractive according to condition ( C 2) . Consequently, the bounded solution φ$_{x,ζ}$ ( t ) is the unique almost periodic solution of system (1.6). □ 

We will consider the chaotic dynamics of system (1.6) in the next section. 

