  The spin-torque nano-oscillator in the presence of thermal fluctuation is
described by the normal form of the Hopf bifurcation with an additive white
noise. By the application of the reduction method, the amplitude-phase coupling
factor, which has a significant effect on the power spectrum of the spin-torque
nano-oscillator, is calculated from the Landau-Lifshitz-Gilbert-Slonczewski
equation with the nonlinear Gilbert damping. The amplitude-phase coupling
factor exhibits a large variation depending on in-plane anisotropy under the
practical external fields.
