Renormalization invariance of a Hamiltonian system near the saddle point

The rescaling property of a one degree of freedom Hamiltonian system near the saddle point is analytically studied as regards the transformations of time-periodic perturbations. Two kinds of perturbation functions are considered: (a) linear functions and (b) homogeneous polynomial functions of canonical variables with time-periodic coefficients. The simple rescaling law of the phase-space of the Hamiltonian system near the hyperbolic fixed point with respect to a transformation of the amplitude and phase of the time-periodic perturbation is derived.