Phantom crossing and oscillating dark energy with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> gravity

In this work, we 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 an 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 simultaneously exhibit phantom crossing behavior and oscillating dark energy. We consider these models by numerically solving 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.