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

• DOI: 10.48550/arxiv.1408.3397
• Publication Date: 2014-01-01
• Authors: Botha, A. E.; Dednam, W.
• Keywords: FOS: Physical sciences; Chaotic Dynamics (nlin.CD); Nonlinear Sciences - Chaotic Dynamics

6 pages, 1 figure<br/>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��ssler system. By optimizing the period $T$ and the three system parameters, $a$, $b$ and $c$, simultaneously, it was found that, for any initial condition $(x_0,y_0,z_0) \in \Re^3$, 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.<br/>