Stabilities and bifurcations of sine dynamic equations on time scale

Abstract

A time scale is a nonempty closed subset of the real numbers R. Recently, the dynamic equations on time scale have received much attention, which have the generalized forms of differential and differential dynamic equations. In this paper, we study the stabilities of fixed points and bifurcations of the sine dynamic equations on time scale. The results show that the solutions of the sine dynamic equations become different with the time scale parameter changing. And n-period-doubling bifurcations and splits of fixed points are observed. Moreover, the chaotic parameter spaces of the dynamic equations are expanded by the increase of complexity of time scale but without increasing the system parameter, thus providing a potential advantage for chaos encryption, radar waveform design and other application areas.