Heteroclinic tangles of the separatrix of double-null map

The double-null map is the simplest symplectic map that has the generic magnetic topology of double-null divertor tokamaks. The generating function of the double-null map is given by $S(x$,$y)=x^{2}$/2$+y^{2}$/2-$y^{4}$/4. $S=$1/4 gives the separatrix surface. The scaling of safety factor on the magnetic axis, $q_{0}$, with map parameter $k$ is used to calculate the number of iterations of the double-null map $N_{p}$ that is equivalent to a single toroidal circuit of the tokamak. The scaling of root mean square deviation of energy on the $q_{95}$ surface with map parameter $k$ is taken as the estimate of magnetic asymmetry to represent the magnetic perturbation from map parameter $k$. These data is used in the forward and backward double-null maps to calculate the heteroclinic tangles of the ideal separatrix of generic double-null divertor tokamaks from magnetic asymmetries. This work is supported by grants DE-FG02-01ER54624, DE-FG02-04ER54793, and DE-FG02-07ER54937.