On Slater’s criterion for the breakup of invariant curves

• We investigate Slater’s theorem in the context of area-preserving maps.
• The breakup diagram of the nontwist map was obtained using Slater’s criterion.
• Slater’s criterion can be implemented to determine the last invariant curve.
• To the standard map our heuristic Slater’s criterion was Kc=0.9716394Kc=0.9716394.
• Our result is very close to the widely accepted Greene’s result, Kc=0.971635Kc=0.971635.

We numerically explore Slater’s theorem in the context of dynamical systems to study the breakup of invariant curves. Slater’s theorem states that an irrational translation over a circle returns to an arbitrary interval in at most three different recurrence times expressible by the continued fraction expansion of the related irrational number. The hypothesis considered in this paper is that Slater’s theorem can be also verified in the dynamics of invariant curves. Hence, we use Slater’s theorem to develop a qualitative and quantitative numerical approach to determine the breakup of invariant curves in the phase space of area-preserving maps.