Global periodic structure of integrable hyperbolic map

Integrable hyperbolic mappings are constructed within a scheme presented by Suris. The Cosh map is a singular map, of which fixed point is unstable. The global behavior of periodic orbits of the Sinh map is investigated referring to the Poincaré–Birkhoff resonance condition. Close to the fixed point, the periodicity is indeed determined from the Poincaré–Birkhoff resonance condition. Increasing the distance from the fixed point, the orbit is affected by the nonlinear effect and the average periodicity varies globally. The Fourier transformation of the individual orbits determines overall spectrum of global variation of the periodicity.