Localization of periodic orbits of the Rössler system under variation of its parameters

The localization problem of compact invariant sets of the Rössler system is considered in this paper. The main interest is attracted to a localization of periodic orbits. We establish a number of algebraic conditions imposed on parameters under which the Rössler system has no compact invariant sets contained in half-spaces z > 0; z < 0 and in some others. We prove that if parameters (a, b, c) of the Rössler system are such that this system has no equilibrium points then it has no periodic orbits as well. In addition, we give localization conditions of compact invariant sets by using linear functions and one quadratic function.