Cosmological bounce in Horndeski theory and beyond

We discuss the stability of the classical bouncing solutions in the general Horndeski theory and beyond Horndeski theory. We restate the no-go theorem, showing that in the general Horndeski theory there are no spatially flat non-singular cosmological solutions which are stable during entire evolution. We show the way to evade the no-go in beyond Horndeski theory and give two specific examples of bouncing solutions, whose asymptotic past and future or both are described by General Relativity (GR) with a conventional massless scalar field. Both solutions are free of any pathologies at all times