concerning Duffing equations on a time scale is presented in Section 5 to support the theoretical results. Finally, some concluding remarks are given in Section 6. The basic concepts that are needed in the present paper about differential equations on time scales are as follows [16, 31, 32, 33]. 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 ) = $_{t}$. In a similar way, 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 if it is continuous at each ( t, u ) ∈ T × R n with right- dense t, and the limits lim ( r,ν $^{)}$→( t − ,u $_{)}$h ( r, ν ) = h ( t − , u ) and lim $^{ν}$→$_{u}$h ( t, ν ) = h ( t, u ) exist at each ( t, u ) with left-dense t. At a right-scattered point t ∈ T , the $_{Δ}$-derivative of a continuous function ϑ is defined as ϑ $^{Δ}$( t ) = ϑ ( σ ( t )) − ϑ ( t ) σ ( t ) t . On the other hand, at a right-dense point t, we have ϑ $^{Δ}$( t ) = lim $^{r}$→t ϑ ( t ) − ϑ ( r ) t r provided that the limit exists. It is worth noting that on the time scale T 0 used in system (1.1) the points $^{θ$_{2}$$_{k}$}$− $_{1}$, k ∈ Z , are left-scattered and right-dense, and the points $_{θ$_{2}$$_{k}$}$, k ∈ Z , are right-scattered and left-dense. Moreover, σ ( θ$_{2}$$_{k}$ ) = $_{θ$_{2}$$_{k}$$_{+1}$}$, ρ ( θ$_{2}$$_{k}$$_{+1}$ ) = $_{θ$_{2}$$_{k}$}$, k ∈ Z , and σ ( t ) = ρ ( t ) = t for any t ∈ T 0 except at the points θ$_{k}$, k ∈ Z . Suppose that the time scale T 0 used in the description of equation (1.1) satisfies the ω − property. That is, there exists a number ω > 0 such that t + ω ∈ T 0 whenever t ∈ T $_{0}$. In this case, there exists a natural number p such that δ$_{k}$$_{+}$$_{p}$ = δ$_{k}$ for all k ∈ Z , where δ$_{k}$ = θ$_{2}$$_{k}$$_{+1}$ − θ$_{2}$$_{k}$ [1]. Suppose that p is the minimal among those numbers. We assume without loss of generality that $^{θ}$− 1 < 0 < θ$_{0}$. Define on the set T ′ 0 = T 0 \ ⋃ ∞ k $^{=}$−∞ { $^{θ$_{2}$$_{k}$}$− 1 } the ψ − substitution [1] as The function ψ ( t ) is one-to-one, ψ (0) = 0 , ψ ( T ′ $_{0}$) = R and lim $^{t}$→∞$^{, t}$∈ T ′ 0 ψ ( t ) = ∞ . According to the results of the paper [1], dψ ( t ) /dt = 1 , t ∈ T ′ $_{0}$, and dψ − $^{1}$( s ) /ds = 1 provided that s = s$_{k}$, k ∈ Z , where and the sequence { s$_{k}$ } , k ∈ Z , is defined through the equation s$_{k}$ = ψ ( θ$_{2}$$_{k}$ ) . The function ψ − 1 is piecewise ψ ( t ) =      t − ∑ 0 <θ$_{2}$$_{k}$<t δ$_{k}$, t ≥ 0 , t + $^{t}$≤θ$_{2}$$_{k}$< 0 δ$_{k}$, t < 0 , . (2.3) $_{ψ}$− $^{1}$( s ) =      s + ∑ 0 <s$_{k}$<s δ$_{k}$, s ≥ 0 , s − $^{s}$≤s$_{k}$< 0 δ$_{k}$, s < 0 , (2.4) 2 Preliminaries 3