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

TL;DR

This paper uses a numerical shooting method to computationally verify the existence of periodic orbits in the Rössler system for any initial condition, suggesting a new mathematical challenge for proof.

Contribution

It introduces a computational approach to demonstrate the universal existence of periodic orbits in the Rössler system, highlighting a potential avenue for rigorous proof.

Findings

For any initial condition, a set of parameters yields a periodic orbit.

Numerical optimization confirms the existence of these orbits.

The result invites further analytical mathematical investigation.

Abstract

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 $(x_0,y_0,z_0) \in \Re^3$, 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.