SHADOWING PROPERTIES AND CHAOTIC CHARACTERISTICS IN TIME-VARYING DISCRETE DYNAMICAL SYSTEMS

Let $ f_{1,\infty} $ be a map sequence on a compact metric space $ (X,d) $. First, this paper defines $ NK $-th iteration map sequence $ f_{1,\infty}^{n[k]} $, where $ f_{1,\infty}^{n[k]}=\{f^{nk}_{\frac{n(n-1)k}{2}+1}\}_{n=1}^{\infty} $ $ (k\in \mathbb{N}) $. It is proved that $ f_{1,\infty} $ is $ \mathcal{P} $-chaotic (or has pseudo-orbit shadowing property) if and only if $ f_{1,\infty}^{n[k]} $ is also $ \mathcal{P} $-chaotic (or has pseudo-orbit shadowing property). Where $ \mathcal{P} $-chaos denotes one of the nine properties: Li-Yorke chaos, distributional chaotic, dense chaos, dense $ \delta $-chaos, generic chaos, generic $ \delta $-chaos, Li-Yorke sensitivity, sensitivity, spatio-temporal chaos. Then, the preservation of persistence, expansive, linking, topological stability, and six types of shadowing properties in time-varying discrete dynamical systems (T-VDDSs) under product operator or topological conjugacy are shown. Moreover, a homeomorphism that is expansive and has the eventual shadowing property necessarily implies topological stability.