New Dynamics in the Rössler System

We study the well-known Rössler system ẋ=-y-z, ẏ=x+a y, ż=b-c z+x z. First, we give a global qualitative description of the flow of the completely degenerate case a=b=c=0 restricted to each invariant surface H=h of its first integral, including the behaviour at infinity via Poincaré compactification. Second, we use first-order averaging to prove the existence of periodic orbits for sufficiently small parameters (in a perturbation of the integrable case) and provide leading-order approximations of their initial conditions.