Phantom Crossing and Oscillating Dark Energy with $F(R)$ Gravity

In this work, we shall consider how a dynamical oscillating and phantom crossing dark energy era can be realized in the context of $F(R)$ gravity. We approach the topic from a theoretical standpoint considering all the conditions that may lead to a consistent phantom crossing behavior and separately how the $F(R)$ gravity context may realize oscillating dark energy era. Apart from our qualitative considerations, we study in a quantitative way two $F(R)$ gravity dark energy models which are viable cosmologically and also exhibit simultaneously phantom crossing behavior and also oscillating dark energy. We consider these models by solving numerically the field equations using appropriate statefinder parameters engineered for dark energy studies. As we show, $F(R)$ provides a natural extension of Einstein's general relativity which can naturally realize a transition from a phantom era to a quintessential era, a feature supported by recent observational data, without resorting to phantom scalar fields to realize the phantom evolution.