Symmetry structure of the hyperbolic bifurcation without reflection of periodic orbits in the standard map
For two-dimensional area preserving maps, the linearized tangent map determines the stability of a fixed point. When the trace of the tangent map is less than \(-2\), a fixed point is inversion hyperbolic, thus subsequent points of a map alternate across the destabilized fixed point (periodic doubling bifurcation). While for the trace of the tangent map is larger than \(+2\), a fixed point undergoes the hyperbolic bifurcation without reflection. In the paper, the processes of the hyperbolic bifurcation without reflection in the standard map \((q,p)\to (q+p+A\sin 2\pi q,p+A\sin 2\pi q)\), where \(A\) is a stochastic parameter, are examined in terms of the higher-order symmetry in the momentum inversion. The higher-order symmetry lines are shown to approach asymptotically to the separatrix of the hyperbolic fixed point. It is also shown that the existing symmetry lines cannot determine the structure of periodic islands born after the hyperbolic bifurcation without reflection. The analysis is undertaken for a period-4 orbit and a period-3 accelerator mode in the standard map.
hyperbolic bifurcation of periodic orbits
scientific article; zbMATH DE number 1884660