Periodic orbits and their stability in the Rössler prototype-4 system

For the Rössler prototype-4 system x.=−y−z, y.=x, z.=αy(1−y)−βz we prove the existence of periodic orbits and study their stability or instability. The main tool for proving these results is the averaging theory. Recently the existence of some of these periodic orbits were detected numerically. -- Highlights: ► We deal with the Rössler prototype-4 system x.=−y−z, y.=x, z.=αy(1−y)−βz. ► It is one of the simplest autonomous differential equations exhibiting chaos. ► Recently some periodic orbits for this system has been detected numerically. ► We provide an analytical proof of these orbits and study their stability. ► Also we prove the existence of periodic orbits not detected numerically.