New Dynamics in the Rössler System

We study the well-known Rössler system \( \dot{x}=-y-z,\qquad \dot{y}=x+a y,\qquad \dot{z}=b-c z+x z. \) $$\dot{x}=-y-z,\dot{y}=x+ay,\dot{z}=b-cz+xz.$$ First, we give a global qualitative description of the flow of the completely degenerate case \(a=b=c=0\) $a=b=c=0$ restricted to each invariant surface \(H=h\) $H=h$ 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.