Chaoticity of time-varying discrete dynamic systems via equicontinuity

This paper studies the preservation of chaotic properties in time-varying discrete dynamic systems, including various forms of expansiveness, sensitivity, and $\mathcal{M}^\alpha $-shadowing properties, and explores the interrelations among these properties. Specifically, it is first proved that positive ep-expansiveness and positive continuum-wise expansiveness are preserved under product operators, topological conjugacy, or composite operators. Subsequently, it is shown that a system characterized by positive ep-expansiveness has only a finite number of periodic points. Furthermore, an equivalence relation is established between positive expansiveness (or positive ep-expansiveness, or positive continuum-wise expansiveness) and sensitivity. Finally, it is noted that the $\mathcal{M}^\alpha$-shadowing property is incompatible with equicontinuity. An equivalent condition is presented to determine whether the limit map of a given system retains the $\mathcal{M}^\alpha$-shadowing property.