Amplitude-phase equations for a weakly perturbed oscillator

For $u^{\prime \prime }+u=ϵf(u,u^{\prime },ϵt)$, write $u_0=R(T)\cos (t+\varphi (T))$ with $T=ϵt$. Removing resonant sine and cosine forcing gives

For $x^{\prime \prime }+x=-2\epsilon y{x}^{\prime }$ and $y^{\prime }=\epsilon g(x)$, the method of multiple scales gives the displayed amplitude-phase equations, together with constant leading phase. For $g(x)=x$, $Y$ is constant and $R$ changes exponentially. For $g(x)=\log |x|$ and positive amplitude, the integrable phase average is $\log (R/2)$. Thus $L=\log (R/2)$ obeys $L_{TT}+L=0$ with $L_T=-Y$. The identically zero oscillator is inadmissible for logarithmic feedback, whose force is undefined there.

When the weak force depends only on velocity, the phase average is a periodic total derivative and vanishes. The leading phase is constant, while the amplitude evolves according to the remaining average. The sign of that drift determines damping or growth and must be checked from the force, rather than inferred from its velocity dependence alone.

For initial amplitude one, the method of multiple scales gives $x\simeq A(ϵt)\sin t$ with $A_T=-4A^2/(3\pi )$, hence $A=1/(1+4ϵt/(3\pi ))$. Positive damping causes algebraic decay. Negative damping yields a formal finite-time amplitude pole, but the weak-damping assumption fails before it because $|ϵ|A$ becomes order one. A leading approximation has order-$|ϵ|$ displacement error on fixed slow-time intervals away from that pole.

For $y^{\prime \prime }+y=\epsilon (y^{\prime }{)}^3$ with positive $\epsilon$, the oscillator energy increases at rate $\epsilon (y^{\prime }{)}^4$. The amplitude-phase equations give $R(T)=R_0(1-3R_0^{2}T/4{)}^{-1/2}$. The weakly nonlinear expansion fails when the growing amplitude makes $\epsilon R^2$ large, so its formal finite-time divergence does not itself control the exact late-time trajectory.

For a weak force depending only on position, the amplitude average is a periodic total derivative and vanishes. The leading harmonic amplitude is constant, while the phase may drift. The method of multiple scales agrees with the exact conservation of energy for bounded orbits in the effective potential. A cubic position force on the right side of the oscillator equation lowers the frequency at first order when its coefficient is positive.