SICNNs with Li-Yorke Chaotic Outputs on a Time Scale Mehmet Onur Fen a, $^{∗}$, Fatma Tokmak Fen b $^{a}$Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey $^{b}$Department of Mathematics, Gazi University, 06500, Teknikokullar, Ankara, Turkey In the present study, we investigate the existence of Li-Yorke chaos in the dynamics of shunting inhibitory cellular neural networks (SICNNs) on time scales. It is rigorously proved by taking advantage of external inputs that the outputs of SICNNs exhibit Li-Yorke chaos. The theoretical results are supported by simulations, and the controllability of chaos on the time scale is demonstrated by means of the Pyragas control technique. This is the first time in the literature that the existence as well as the control of chaos are provided for neural networks on time scales. Keywords: Shunting inhibitory cellular neural networks; Time scales; Li-Yorke chaos; Proximality; Frequent separation; Chaos control Cellular neural networks (CNNs) have been paid much attention starting with the studies of Chua and Yang [1, 2]. Exceptional role in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing has been played by shunting inhibitory cellular neural networks (SICNNs), which was introduced by Bouzerdoum and Pinter [3]. The reader is referred to [4]-[8] for the applications of SICNNs. Another subject that is also popular is the theory of time scales, which is first presented by Hilger [9]. This theory has many applications in various scientific fields, and neural networks are no exception [10]-[26]. The first mathematically rigorous definition of chaos was introduced by Li and Yorke [27]. It was shown in [27] that if a map on an interval has a point of period three, then it possesses chaos in the sense of Li-Yorke. The presence of an uncountable scrambled set is a distinguishing feature of the Li- Yorke chaos. It was indicated in [28] that if a map on a compact interval has a two point scrambled set, then it possesses an uncountable scrambled set. Li-Yorke sensitivity, which links the Li-Yorke chaos with the notion of sensitivity, was studied by Akin and Kolyada [29]. It was proved in [30] that the presence of positive topological entropy implies chaos in the sense of Li-Yorke. According to Marotto [31], a multidimensional continuously differentiable map is Li-Yorke chaotic provided that it has a snap- back repeller. On the other hand, Marotto’s Theorem was utilized by Li et al. [32] to demonstrate the ∗ Corresponding Author Tel.: +90 312 365 9276, E-mail: monur.fen@gmail.com 1 arXiv:1512.08750v1 [nlin.CD] 29 Dec 2015 1 Introduction Abstract