Various notions of shadowing in triangular system and its component systems

In this article, we investigate various forms of shadowing for a general triangular system. In particular, we relate various notions of shadowing for a triangular system with various notions of shadowing in the component systems. We prove that if the base f for T is transitive then shadowing in the base map and the non-autonomous system generated by a transitive point ensures shadowing of the triangular system. We prove that if the base map for T is expansive then shadowing in the triangular system ensures shadowing in the component systems. We prove that if the non-autonomous component systems form a synchronized family and the base map possesses a globally attracting fixed point $x_0$ then eventual shadowing in system generated by $x_0$ ensures eventual shadowing in each of the non-autonomous component systems. We also investigate chain transitivity and chain mixing for a general triangular system.