Next, we take into account the discrete system where { ψ$_{n}$ } is an unpredictable solution of (4.6) with λ = 3 . 91 . Theorem 3.2 implies that the sequence { ψ$_{n}$ } , n ∈ Z , defined as ψ$_{n}$ = (3 ψ 3 $_{n}$, 4 ψ$_{n}$ ) ∈ R 2 is also unpredictable. In order to demonstrate the chaotic behavior of (4.7), we consider the system where { u$_{n}$ } is a solution of (4.6), again with λ = 3 . 91 . One can numerically verify that for each { u$_{n}$ } , the bounded solutions of (4.8) take place inside the compact region Therefore, the conditions ( C 1) − ( C 3) are satisfied for system (4.7), and there exists a unique unpre- dictable solution of (4.7) in accordance with Theorem 3.1. Figure 1 shows the first and second coordinates of the solution of system (4.8) with the initial data u$_{0}$ = 0 . 4 , x$_{0}$ = 0 . 58 , and y$_{0}$ = 1 . 95 . The utilized value of the parameter λ = 3 . 91 and the initial point u$_{0}$ = 0 . 4 were considered for shadowing in paper [20]. Moreover, we represent in Figure 2 the two dimensional trajectory of the same solution. Both Figures 1 and 2 support the result of Theorem 3.1 such that an unpredictable sequence takes place in the dynamics of the discrete system (4.7) and the behaviour of the system is chaotic. x$_{n}$$_{+1}$ = x$_{n}$ 2 − y$_{n}$ 7 + 3 ψ 3 n y$_{n}$$_{+1}$ = − x$_{n}$ 8 + y$_{n}$ 3 + x 2 / 3 n 12 + 4 ψ$_{n}$, (4.7) x$_{n}$$_{+1}$ = x$_{n}$ 2 − y$_{n}$ 7 + 3 u 3 n y$_{n}$$_{+1}$ = − x$_{n}$ 8 + y$_{n}$ 3 + x 2 / 3 n 12 + 4 u$_{n}$, (4.8) D = { ( x, y ) ∈ R 2 : 0 . 1 ≤ x ≤ 2 . 7 , 1 . 7 ≤ y ≤ 5 . 1 } . 0 50 100 150 0 1 2 3 n x n 0 50 100 150 2 3 4 5 n y n Figure 1: The solution of (4.8) with the initial data u$_{0}$ = 0 . 4 , x$_{0}$ = 0 . 58 , and y$_{0}$ = 1 . 95 . The figure supports the result of Theorem 3.1 such that (4.7) possesses an unpredictable solution. 7