content_image/326119.jpg content_image/326127.jpg content_image/326128.jpg
## 5 Poincaré chaos near periodic orbits 

<img src='content_image/326119.jpg'>

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 

<img src='content_image/326127.jpg'>

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). 

<img src='content_image/326128.jpg'>

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). 

