by using the values ε = 0 . 085 and C = 2 . 6 . The OGY control switched off at t = ζ$_{350}$ and the Pyragas control is switched off at t = ζ$_{400}$. In other words, we take λ$_{j}$ = 3 . 9 after t = ζ$_{350}$, and take C = 0 after t = ζ$_{400}$. Figure 4 shows the simulation results for the w$_{3}$ and w$_{4}$ coordinates of system (4.28) corresponding to the initial data w$_{1}$ (0 . 41) = 0 . 8 , w$_{2}$ (0 . 41) = 0 . 7 , w$_{3}$ (0 . 41) = 0 . 12 , w$_{4}$ (0 . 41) = 0 . 013 , w$_{5}$ (0 . 41) = 0 . 41 . It is seen in the figure that the quasi-periodic solution of (4.26) is stabilized. To present a better visuality, the stabilized quasi-periodic solution of (4.26) is shown in Figure 5 for 200 ≤ t ≤ 250 . 5 Appendix For the reader convenience, we present and prove a Growall-Coppel type inequality (see [14]) result used in this paper. Theorem 5.1 Let a, b, c be nonnegative constants, γ > 0 and u ∈ C ([ p, q ] , R ) be nonnegative on an interval [ p, q ] , p < q , satisfying on [ p, q ] . If 2 c < γ then for any t ∈ [ p, q ] , where δ = √ γ 2 − 2 cγ . Proof. By [14, Theorems 2.3, 2.4], functions satisfy the equations respectively. Since all u$_{1}$ ( t ) , u$_{2}$ ( t ) , u$_{3}$ ( t ) are nonnegative, the function 19 u$_{4}$ ( t ) = u$_{1}$ ( t ) + u$_{2}$ ( t ) + u$_{3}$ ( t ) u$_{1}$ ( t ) = be − γ ( t − p ) + c ∫ ∞ p e − γ | t − s $^{|}$u$_{1}$ ( s ) ds, u$_{2}$ ( t ) = be − γ ( q − t ) + c ∫ q −∞ e − γ | t − s $^{|}$u$_{2}$ ( s ) ds, u$_{3}$ ( t ) = a + c ∫ ∞ −∞ e − γ | t − s $^{|}$u$_{3}$ ( s ) ds, u$_{1}$ ( t ) = b c ( γ − δ ) e − δ ( t − p $^{)}$, t ≥ p, u$_{2}$ ( t ) = b c ( γ − δ ) e − δ ( q − t $^{)}$, t ≤ q, u$_{3}$ ( t ) = aγ γ − 2 c u ( t ) ≤ aγ γ − 2 c + b c ( γ − δ ) ( e − δ ( t − p ) + e − δ ( q − t ) ) u ( t ) ≤ a + b ( e − γ ( t − p ) + e − γ ( q − t ) ) + c ∫ q p e − γ | t − s $^{|}$u ( s ) ds