Coexistence of symmetric periodic points in the standard map

Abstract

The problem of the coexistence of symmetric periodic points of the standard map has been investigated. First, it is shown that there is another reversibility in addition to the one already known. The dominant axis of this reversibility is the remaining coordinate axis. Using these reversibilities, simple recursion formulas that determine the positions of symmetric periodic points have been derived. Second, bifurcation structures of symmetric periodic points from the fixed point at the origin have been investigated. It is shown that periodic points bifurcate along both the dominant axes of two reversibilities every time the rotation number of the fixed point passes through a rational number. Third, the problem of the coexistence of symmetric periodic points has been numerically investigated with the aid of the recursion formulas. A devil’s staircase form of the evolution of the outermost Kolmogorov–Arnold–Moser (KAM) curve of the fixed point with the change of the perturbation parameter is observed. This indicates that bifurcated periodic points do not disappear by inverse bifurcation as long as they are in the invariant region of the fixed point.