to mappings in Banach spaces and complete metric spaces can be found in [17, 18]. Besides, in the paper [19], the Li-Yorke definition of chaos was modified in such a way that infinitely many periodic motions separated from the motions of the scrambled set are replaced with almost periodic ones. In the present paper, we will also consider the Li-Yorke chaos in this modified sense. In paper [20], we have considered unpredictability as a global phenomenon in weather dynamics on the basis of connected Lorenz systems, and this extension of chaos was performed by means of perturbations of Lorenz systems by chaotic solutions of their counterparts. Our suggestion may be a key to explain why the weather unpredictability is observed everywhere. This is true also for unpredictability and lack of forecasting in economics [21, 22]. To complete the explanation of weather and economical unpredictability as global phenomena by the analysis of interconnected models, we need to argument persistence of chaos of a model against chaotic perturbations as solutions of another similar models. From this point of view, results of the present paper are very motivated. Another motivation of this study relies on the richness of a single chaotic model for motions, as a supply of infinitely many different periodic [15, 23], almost periodic [19] and even Poisson stable [24] motions. Of course, the diversity of motions is useless if one cannot control chaos [25, 26]. In other words, chaos persistence means extension of chaos controllability. In our former paper [27], persistence of chaos was considered in coupled Lorenz systems by tak- ing into account sensitivity and existence of infinitely many periodic motions embedded in the chaotic attractor. However, in the present paper, we consider chaos in the sense of Li-Yorke with countable infinity of almost periodic solutions in basis instead of periodic ones. In the present study, all results concerning the existence of almost periodic motions as well as Li-Yorke chaos are rigorously proved, and a more comprehensive theoretical discussion is performed compared to [27]. The demonstration of infinite number of Li-Yorke chaotic sets of solutions in the dynamics is another novelty of the present paper. Moreover, a numerical chaos control technique based on the Ott-Grebogi-Yorke (OGY) [25] algorithm is proposed for the stabilization of the unstable almost periodic motions. On the other hand, the paper [19] was concerned with the Li-Yorke chaotic dynamics of shunting inhibitory cellular neural networks with discontinuous external inputs. The concept of persistence of chaos was not considered in [19] at all. It was demonstrated in [19] that the chaotic structure of the discontinuity moments of the external inputs gives rise to the appearance of chaos, and chaos does not take place in the dynamics either in the case of regular discontinuity moments or in the absence of the discontinuous external inputs. On the contrary, in the present paper, a continuous chaotic perturbation is applied to a relay system which is already chaotic in the absence of perturbation, and it is proved that the chaotic structure is permanent in the dynamics regardless of the applied perturbation. As the source of chaotic perturbation we make use of solutions of another system of differential equations, but it is also possible to use any data which is known to be chaotic in the sense of Li-Yorke. 2