and the eigenvalues of the matrix e 4 $^{A}$( I + 2 A ) are inside the unit circle, where I is the 2 × 2 identity matrix. Due to the fact that the coefficient of the nonlinear term y 3 $_{1}$( t ) in (5.12) is sufficiently small, it can be numerically verified for ζ$_{0}$ ∈ [0 , 1] that the bounded solutions of system (5.12) lie inside the region D = { ( y$_{1}$, y$_{2}$ ) ∈ R 2 : − 0 . 01 ≤ y$_{1}$ ≤ 0 . 07 , − 0 . 12 ≤ y$_{2}$ ≤ 0 . 07 } . Therefore, it is reasonable to consider the dynamics of (5.12) inside D . The conditions ( C 5) and ( C 6) hold for (5.12) with N = 193 , λ = 1 . 6 , p = 1 , ¯ δ = 2 and L$_{f}$ = 0 . 000294 . In accordance with Theorem 4.1, system (5.12) is Li-Yorke chaotic. It is worth noting that the chaoticity of the logistic map (5.11) gives rise to the presence of chaos in (5.12). Moreover, Lemma 3.1 implies that for a fixed solution ζ = { ζ$_{k}$ } of (5.11) the unique bounded solution of (5.12) attracts all other solutions of the system. Let us use the solution ζ = { ζ$_{k}$ } of (5.11) with ζ$_{0}$ = 0 . $_{19}$in system (5.12). We depict in Figure 1 the$_{y}$$_{1}$ - coordinate of the solution of (5.12) corresponding to the initial data y$_{1}$ (0) = 0 . 019 and y$_{2}$ (0) = − 0 . 004 . Figure 1 supports the result of Theorem 4.1 such that system (5.12) possesses chaos. Moreover, the trajectory of the same solution in the y$_{1}$ − y$_{2}$ plane is represented in Figure 2, which reveals the existence of a chaotic attractor in the dynamics of (5.12). We rigorously prove the existence of chaos in dynamic equations on time scales, where the right hand side of the equations depends on a chaotic map. The reduction technique to impulsive differential equations presented in the paper [1] is used in our investigations. A mathematical description of chaos in the sense Figure 1: The chaotic behavior in the solution of system (5.12). One can show that where 0 20 40 60 80 100 120 140 160 180 200 0 0.02 0.04 0.06 t y 1 Q =    0 1 $_{3}$√$_{5}$ 2 − 5 2    , e At = e − 5 $_{2}$t $_{Q}$   cos ( $_{3}$√$_{5}$ 2 t ) − sin ( $_{3}$√$_{5}$ 2 t ) sin $_{3}$√$_{5}$ 2 t cos $_{3}$√$_{5}$ 2 t    Q − $^{1}$, 12 6 Conclusion