Dynamic properties of cyclic cosmologies

Our first goal in this work is to study general and model-independent properties of cyclic cosmologies. The large number studies bouncing cosmologies different scenarios published recently calls for a proper understanding the universal models. We thus review further elaborate common physical geometrical various classes models then discuss how Universe can be treated as dynamic system. two theorems from systems analysis used ensure existence cosmological solutions under certain conditions on field equations. After we proceed towards our second which application obtained results frameworks modified gravity theories: $f(R)$ gravity, dark energy $f(T)$ gravity. requirements these theories also obtain examples cosmologies, while discussing their basic properties.