Locating Cantori for Symmetric Tokamap and Symmetric Ergodic Magnetic Limiter Map Using Mean-Energy Error Criterion

Abstract

We use a method based on the conservation of energy, the mean-energy error criterion, to approximately locate the place of a cantorus by locating the series of its convergents. The mean-energy error curve has nearly stationary parts in the vicinity of elliptic (minimax) orbits, the so-called magnetic islands. Stable minimax orbits converge to orbits homoclinic to a cantorus. By tracing the island series, we limit the cantorus to a narrow region. A near-critical perturbation parameter is used so that, while the cantorus may be destabilized, its high-order minimax orbits remain intact. As illustrations, we consider two symplectic maps, systematically derived from the Hamilton–Jacobi equation and Jacobi’s theorem, in the context of the magnetically confined plasmas in a tokamak: a symmetric tokamap realistically reproduces the main features of a tokamak, and a symmetric ergodic magnetic limiter (EML) map is defined to describe the action of EML rings on the magnetic field lines in the tokamak.