chaos replication method introduced and developed in our studies [27]-[38] allows to superpose solutions with incommensurate periods and to apply chaotic perturbation to the system with purely almost periodic motion to create infinitely many unstable almost periodic motions embedded in the chaotic attractor. 

The paper [28] deals with the general technique of dynamical synthesis, which was developed in [39]- [42]. The study [32] was concerned with the extension of chaos in the sense of Devaney [1], Li-Yorke [2] and the one obtained through period-doubling cascade [3, 21, 22] for unidirectionally coupled systems. The appearance of infinitely many quasi-periodic motions in a chaotic attractor was revealed in [32]. For that purpose, a system with an asymptotically stable equilibrium point was influenced by two chaotic systems that possess infinitely many periodic motions with incommensurate periods giving rise to quasi- periodicity. However, in the present study, we make use of only a single source of perturbations to obtain chaos with infinitely many almost periodic motions , and the perturbed system has a different structure. 

The idea that unstable periodic motions embedded in a chaotic attractor can be replaced by more general types of regular motions stems from the investigations of Poincaré and Birkhoff [43]. In any neighborhood of a structurally stable Poincaré homoclinic orbit, there exist nontrivial hyperbolic sets containing a countable number of saddle periodic orbits and continuum of non-periodic Poisson stable orbits [4]-[6]. According to Seifert [8], there exists discrete chaos with infinitely many almost periodic motions. It was certified in the paper [7] that, in general, in place of a countable set of periodic solutions to form chaos, one can take an uncountable collection of Poisson stable motions which are dense in a quasiminimal set. This can also be observed in the Horseshoe attractor [44]. Motivated by these studies, continuous chaotic attractors with infinitely many almost periodic motions are considered in the present paper. 

The concept of snap-back repellers for high dimensional maps was introduced in [45]. According to Marotto [45], if a multidimensional continuously differentiable map has a snap-back repeller, then it is Li-Yorke chaotic. Marotto’s Theorem was used in [46] to prove the existence of Li-Yorke chaos in a spatiotemporal chaotic system. Li-Yorke sensitivity, which links the Li-Yorke chaos with the notion of sensitivity, was studied in [47]. Moreover, generalizations of Li-Yorke chaos to mappings in Banach spaces and complete metric spaces were provided in [48]-[50]. In the present paper, we develop the concept of Li-Yorke chaos with infinitely many almost periodic motions, which are separated from the motions of a scrambled set, and provide a method for its formation. 

Almost periodic and in particular quasi-periodic motions have an important place in the theory of neural networks. In the book [51], the dynamics of the brain activity is considered as a system of many coupled oscillators with different incommensurable periods. According to Pasemann et al. [52], quasi-periodic solutions are noteworthy in biological and artificial systems since they are associated with various kinds of central pattern generators. Watanabe et al. [53] demonstrated that chaotic dynamics works as means to learn new patterns and increases the memory capacity of neural networks. The 

