Multiple time scales and averaging

Write the fast variable as $t$ and the slow variable as $T=\varepsilon t$, treating them as independent while constructing an approximation. The total time derivative becomes $\partial_t+\varepsilon\partial_T$. For the weakly damped oscillator $u''+2\varepsilon u'+u=0$, the leading fast solution can have slowly varying cosine and sine amplitudes. Substituting this form at the next order produces resonant forcing unless the slow amplitudes satisfy first-order decay equations. The derivative rule is essential; differentiating only with respect to the fast variable would miss those solvability conditions.

Eliminating the resonant sine and cosine terms gives $A_T=-A$ and $B_T=-B$. Leading initial displacement one and velocity zero select $A=e^{-T}$ and $B=0$, hence $e^{-\varepsilon t}\cos t$. This leading expression has an initial velocity discrepancy of order the parameter; a higher-order correction can repair it. The construction identifies the envelope, but a claimed uniform error on the slow time interval must be proved separately, for example by comparing with this linear oscillator's exact damped solution.

For a finite-dimensional system $x'=\varepsilon f(t,x)$, suppose $f$ is periodic in time with a fixed positive period, continuously differentiable, and has bounded value and state derivative on a neighborhood of a compact state set. Assume the averaged solution with the same initial state exists for slow time from zero to a fixed $L$ and stays a positive distance inside that neighborhood. Then for sufficiently small positive parameter the original solution exists on the corresponding interval and differs from the averaged one by at most a constant times the parameter, uniformly for $0\le t\le L/\varepsilon$. The constants depend on the fixed interval and neighborhood, not on the parameter.

The averaged vector field is the time-period mean with the state held fixed during that integration. A proof integrates the zero-mean periodic part to a bounded periodic primitive, uses it in a near-identity change of variables, and applies a stability estimate over the slow interval. The compact-interior condition prevents an unmodeled exit before the estimate closes. This theorem does not automatically prove accuracy for all future time, stability of every periodic orbit, or averaging for a nonperiodic forcing with no additional assumptions.

For the scalar equation $x'=\varepsilon(-x+\sin t)$ with initial value one, averaging removes the sine and gives $\overline x=e^{-\varepsilon t}$. Solving the full linear equation yields an explicit difference involving a decaying transient plus a small periodic response. Its amplitude can be bounded directly for every nonnegative time, giving a stronger statement than the finite slow-time theorem. The global result belongs to this stable linear example, not to every averaged system.

Averaging does not require the original state to be constant during a physical period; it uses that its change per period is small in the declared regime. Near resonance, a suitable rotating-frame or amplitude coordinate may be required before the system has slow standard form. Near an unstable or escaping trajectory, an order-one change in the slow variable can amplify approximation error. Monitor the actual state range and error norm, rather than assuming that removing visible oscillations always improves prediction at every time.