Chaotic solutions and black hole shadow in f(R) gravity

We discuss the emergence of black hole shadow and photon-sphere in the context of $f\left(R\right)$ gravity. It is shown that the shadow is exponentially sensitive to linear instabilities of metric coming from some $f\left(R\right)$ solutions. Thus, the instabilities of photon circular trajectories, delimiting the black hole photon-sphere, are double exponentialized. Specifically we individuate two Lyapunov exponents, rather than only one, related to two different sources of chaos in geodesic orbits as a sort of butterfly effect. Such a result violates the black hole chaos bound proposed by Maldacena, Shenker and Stanford for General Relativity. We also explore the impact of the black hole metric instabilities in $f\left(R\right)$ gravity on the quasi-normal modes. In the framework of Extended Theories of Gravity, our analysis suggests a new paradigm to deal with black hole shadow and gravitational waves observations coming from black hole merging in the ringdown phase.