0 1 2 3 1.5 2 2.5 3 3.5 4 4.5 5 5.5 x$_{n}$ y n Figure 2: The trajectory of the discrete system (4.8) corresponding to the initial data u$_{0}$ = 0 . 4 , x$_{0}$ = 0 . 58 , and y$_{0}$ = 1 . 95 . The figure reveals the chaotic behavior of system (4.7). 5 Poincaré chaos near periodic orbits In this section, we will demonstrate the appearance of irregular behavior near periodic orbits of discrete systems. For that purpose, let us consider the system It is shown in the book [19] that the system (5.9) admits a stable periodic orbit whenever the value ω/ 2 π is rational. Taking λ = 3 . 86 in the logistic map (4.6), and perturbing system (5.9) with solutions (4.6) we set up the system where { u$_{n}$ } is a solution of (4.6). Let us use the value of ω = π/ 7 so that the non-perturbed system (5.9) possesses a one parameter family of stable 14 -periodic orbits. We depict in Figure 3 the trajectory of (5.10) corresponding to the initial data u$_{0}$ = 0 . 4 , x$_{0}$ = 1 , and y$_{0}$ = 1 . The total number of iterations used in the simulation is 65 × 10 $^{6}$. The choice for the parameter value λ = 3 . 86 and the initial value u$_{0}$ = 0 . 4 is analyzed for shadowing in paper [20]. It is seen in Figure 3 that the applied perturbation makes the system (5.10) behave chaotically near the 14 -periodic orbit of (5.9). It is worth noting that Figure 3 represents a single orbit. The fractal structure of the orbit is also observable in the simulation. Figure 3 manifests the appearance of Poincaré chaos near the periodic orbit of (5.9). 8 x$_{n}$$_{+1}$ = cos( ω ) x$_{n}$ + sin( ω ) y$_{n}$ y$_{n}$$_{+1}$ = − sin( ω ) x$_{n}$ + cos( ω ) y$_{n}$. (5.9) x$_{n}$$_{+1}$ = cos( ω ) x$_{n}$ + sin( ω ) y$_{n}$ + 0 . 001 u$_{n}$ y$_{n}$$_{+1}$ = − sin( ω ) x$_{n}$ + cos( ω ) y$_{n}$ + 0 . 001 u$_{n}$ (5.10)