Zero-Hopf bifurcation, periodic orbits and $C^{1}$ non-integrability of the classical Rössler system

RESUMO

We investigate the zero-Hopf bifurcation and the local $C^{1}$-non-integrability of the original Rössler system. First, we characterize the unique zero-Hopf equilibrium of the system and study its perturbation by means of the first-order averaging theory. We prove that, under suitable assumptions on the bifurcation parameters, three distinct families of periodic orbits bifurcate from the zero-Hopf equilibrium, substantially extending the previously known result in which only one bifurcating periodic orbit was obtained. In the second part of the paper, these bifurcating periodic solutions are used to investigate the integrability properties of the Rössler system. By combining the first-order averaging theory with the classical Poincaré criterion for first integrals, we prove that the bifurcating periodic orbits constitute a generic obstruction to the existence of local first integrals of class $C^{1}$. This demonstrates that averaging theory provides not only an effective tool for detecting periodic solutions but also a powerful method for the integrability analysis.