Li-Yorke chaos in hybrid systems on a time scale

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.Comment: 16 pages, 2 figure

Existence of Unpredictable Solutions and Chaos

In paper [1] unpredictable points were introduced based on Poisson stability, and this gives rise to the existence of chaos in the quasi-minimal set. This time, an unpredictable function is determined as an unpredictable point in the Bebutov dynamical system. The existence of an unpredictable solution and consequently chaos of a quasi-linear system of ordinary differential equations are verified. This is the first time that the description of chaos is initiated from a single function, but not on a collection of them. The results can be easily extended to different types of differential equations. An application of the main theorem for Duffing equations is provided.Comment: 15 pages, 4 figure