Shearless curve breakup in the biquadratic nontwist map

Nontwist area-preserving maps violate the twist condition along shearless invariant curves, which act as transport barriers in phase space. Recently, some plasma models have presented multiple shearless curves in phase space and these curves can break up independently. In this paper, we describe the different shearless curve breakup scenarios of the so-called biquadratic nontwist map, a recently proposed area-preserving map derived from a plasma model, that captures the essential behavior of systems with multiple shearless curves. Three different scenarios are found and their dependence on the system parameters is analyzed. The results indicate a relation between shearless curve breakup and periodic orbit reconnection-collision sequences. In addition, even after a shearless curve breakup, the remaining curves inhibit global transport.Comment: 12 pages, 7 figure

Renormalization and destruction of 1/γ21/\gamma^2 tori in the standard nontwist map

Extending the work of del-Castillo-Negrete, Greene, and Morrison, Physica D {\bf 91}, 1 (1996) and {\bf 100}, 311 (1997) on the standard nontwist map, the breakup of an invariant torus with winding number equal to the inverse golden mean squared is studied. Improved numerical techniques provide the greater accuracy that is needed for this case. The new results are interpreted within the renormalization group framework by constructing a renormalization operator on the space of commuting map pairs, and by studying the fixed points of the so constructed operator.Comment: To be Submitted to Chao

Normal Forms for Symplectic Maps with Twist Singularities

We derive a normal form for a near-integrable, four-dimensional symplectic map with a fold or cusp singularity in its frequency mapping. The normal form is obtained for when the frequency is near a resonance and the mapping is approximately given by the time-$T$ mapping of a two-degree-of freedom Hamiltonian flow. Consequently there is an energy-like invariant. The fold Hamiltonian is similar to the well-studied, one-degree-of freedom case but is essentially nonintegrable when the direction of the singular curve in action does not coincide with curves of the resonance module. We show that many familiar features, such as multiple island chains and reconnecting invariant manifolds, are retained even in this case. The cusp Hamiltonian has an essential coupling between its two degrees of freedom even when the singular set is aligned with the resonance module. Using averaging, we approximately reduced this case to one degree of freedom as well. The resulting Hamiltonian and its perturbation with small cusp-angle is analyzed in detail.Comment: LaTex, 27 pages, 21 figure