Persistence of Li-Yorke chaos in systems with relay $^{1}$Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey Marat Akhmet $^{1}$, Mehmet Onur Fen2 , $^{∗}$, Ardak Kashkynbayev 1 $^{2}$Basic Sciences Unit, TED University, 06420 Ankara, Turkey It is rigorously proved that the chaotic dynamics of the non-smooth system with relay function is persistent even if a chaotic perturbation is applied. We consider chaos in a modified Li-Yorke sense such that infinitely many almost periodic motions take place in its basis. It is demonstrated that the system under investigation possesses countable infinity of chaotic sets of solutions. Coupled Duffing oscillators are used to show the effectiveness of our technique, and simulations that support the theoretical results are represented. Moreover, a chaos control procedure based on the Ott-Grebogi-Yorke algorithm is proposed to stabilize the unstable almost periodic motions embedded in the chaotic attractor. Keywords: Persistence of chaos; Li-Yorke chaos; Almost periodic motions; Relay system; Chaos control; Duffing equation The word “persistence” is not popular for differential equations since it is not usual to say about per- sistence of periodic solutions against periodic perturbations as well as other forms of regular motions such as quasi-periodic and almost periodic solutions in a similar way. In the literature, these types of problems have been investigated as a part of synchronization [1]-[3]. Chaos is not an exceptional term in this row, and those results which we recognize as synchronization of chaos can be interpreted as a specific type of persistence of chaos [4]-[9]. The specification is characterized through an asymptotic relation between solutions of coupled systems. In other words, persistence has not been considered by researchers explicitly, except under the mask of synchronization or entrainment [1]-[14]. In this study, we consider synchronization in its ultimately generalized form, without any additional asymptotic con- ditions, considering only ingredients of Li-Yorke chaos [15]. This is the main theoretical novelty of the present paper. An extension of the original definition of Li and Yorke [15] to dimensions greater than one was per- formed by Marotto [16]. It was demonstrated in [16] that a multidimensional continuously differentiable map possesses Li-Yorke chaos if it has a snap-back repeller. Moreover, generalizations of Li-Yorke chaos ∗ Corresponding Author. E-mail: monur.fen@gmail.com, Tel: +90 312 585 02 17 1 1 Introduction Abstract arXiv:1701.08385v1 [nlin.CD] 29 Jan 2017