# Li-Yorke chaos in hybrid systems on a time scale 

Marat Akhmet $^{a,}$∗ , Mehmet Onur Fen b 

$^{a}$Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey 

$^{b}$Neuroscience Institute, Georgia State University, Atlanta, Georgia 30303, USA 

## Abstract 

By using the reduction technique to impulsive differential equations [1], we rigorously prove the presence of chaos in dynamic equations on time scales (DETS). The results of the present study are based on the Li-Yorke definition of chaos. This is the first time in the literature that chaos is obtained for DETS. An illustrative example is presented by means of a Duffing equation on a time scale. 

Keywords: Li-Yorke chaos; Dynamic equations on time scales; Proximality; Frequent separation; Duff- ing equation; Hybrid systems 

## 1 Introduction 

The concept of chaos has been one of the attractive topics among scientists since the studies of Poincaré [12], Cartwright and Littlewood [21], Levinson [34], Lorenz [38] and Ueda [47]. Another subject that is also popular is the theory of time scales, which is first presented by Hilger [26]. Both concepts have many applications in various disciplines such as mechanics, electronics, neural networks, population models and economics. See, for instance, [14, 16, 22, 23, 39, 41, 45, 46, 48] and the references therein. 

Dynamic equations on time scales (DETS) have been extensively investigated in the literature [16, 31]. However, to the best of our knowledge, the presence of chaos has never been achieved in DETS. Motivated by the deficiency of mathematical methods for the investigation of chaos in such equations, we suggest the results of the present study. 

The first mathematical definition of chaos was introduced by Li and Yorke [35] for discrete dynamical systems in a compact interval of the real line. The presence of an uncountable scrambled set is one of the main features of the Li-Yorke chaos. The original definition of Li and Yorke was extended to dimensions greater than one by Marotto [40]. According to Marotto [40], a multidimensional continuously differentiable map possesses generalized Li-Yorke chaos if it has a snap-back repeller. The existence of Li-Yorke chaos in a spatiotemporal chaotic system was proved in [37] by means of Marotto’s Theorem, and generalizations of Li-Yorke chaos to mappings in Banach spaces and complete metric spaces were provided in [28, 43, 44]. It was shown by Kuchta [30] that if a map on a compact interval has a two point scrambled set, then it possesses an uncountable scrambled set. Blanchard [15] proved that the 

