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A recently developed numerical method [1] is employed to study the existence of periodic orbits in the well-known Rössler system [2]. This system was originally conceived as a simple prototype for studying chaos and it continues to provide new insights into chaotic and other nonlinear phenomena. To within the uncertainty of the numerical method, it is found that the three (real) parameters of the Rössler system can always be chosen so that there exists at least one periodic orbit through any point in the space R3-{z=0}. After a discussion of the result it is concluded that the construction of its analytical proof poses an interesting challenge to theoreticians in the field of nonlinear dynamics. A Python implementation of the numerical method for finding the periodic orbits is provided as an Appendix.
[1] Dednam W and Botha A E 2014 Communicated to Engineering with Computers
[2] Rössler O E 1976 Phys. Lett. A 57 397
