Computer Assisted 'Proof' of the Global Existence of Periodic Orbits in the R\"ossler System

The numerical optimized shooting method for finding periodic orbits in nonlinear dynamical systems was employed to determine the existence of periodic orbits in the well-known R"ossler system. By optimizing the period $T$ and the three system parameters, $a$, $b$ and $c$, simultaneously, it was found that, for any initial condition $(x_0,y_0,z_0)\in {\mathrm{\Re }}^3$, there exists at least one set of optimized parameters corresponding to a periodic orbit passing through $(x_0,y_0,z_0)$. After a discussion of this result it was concluded that its analytical proof may present an interesting new mathematical challenge.