The “hidden” dynamics of the Rössler attractor

The fast/slow dynamics of the Rössler model in the chaotic regime are compared to the dynamics of the reduced (slow) model governing the flow along the slow invariant manifold, on which the trajectory is restrained to evolve during the slow part of the attractor. It is shown that the dynamics of the reduced model incorporate the slow time scales of the full model. However, instead of the fast dissipative time scale of the full model that restrains the trajectory on the slow invariant manifold, the reduced model generates a new time scale that (i) relates to the curvature of the manifold and (ii) becomes faster as higher order corrections are incorporated in the approximation of the manifold and the reduced model. It is shown that this new time scale does not characterize the motion of the solution on the manifold but it can be employed as a signal for the strengthening or weakening of the manifold, depending on whether it relates to components of the reduced model that tend to lead its solution towards equilibrium or away from it. Finally, it is demonstrated that the new time scale introduced by the reduced model has a significant role in determining the maximum accuracy that can be delivered by the singular perturbation methodology.