Non-Birkhoff Orbits with 2n Turning Points in the Standard Map

Abstract

One of the two fixed points of the standard map gives rise to a period-doubling bifurcation and becomes a saddle with reflection beyond a certain parameter value. In association with this bifurcation, symmetric non-Birkhoff periodic orbits (SNBOs) with 2n (n ≥ 2) turning points appear and exhibit complicated behavior. We first analyze the structure of stable and unstable manifolds of this saddle and then derive dynamical order relations for these SNBOs and show that a period-3 SNBO implies the existence of SNBOs with all possible numbers of turning points.