Asymptotic Analysis and Singular Perturbation Theory

5.3 The method of averaging
Consider a system of ODEs for x(t) ∈ R n which can be written in the following
standard form
x ′ = εf(x, t, ε). (5.1)
Here, f : R n × R × R → R n is a smooth function that is periodic in t. We assume
the period is 2π for definiteness, so that
f(x, t + 2π, ε) = f(x, t, ε).
Many problems can be reduced to this standard form by an appropriate change of
variables.
Example 5.3 Consider a perturbed simple harmonic oscillator
y ′′ + y = εh(y, y ′ , ε).
We rewrite this equation as a first-order system and remove the unperturbed dynamics
by introducing new dependent variables x = (x1, x2) defined by
y
y ′
cos t sin t x1
=
.
− sin t cos t
We find, after some calculations, that (x1, x2) satisfy the system
x ′ 1 = −εh (x1 cos t + x2 sin t, −x1 sin t + x2 cos t, ε) sin t,
x ′ 2 = εh (x1 cos t + x2 sin t, −x1 sin t + x2 cos t, ε) cos t,
which is in standard periodic form.
Using the method of multiple scales, we seek an asymptotic solution of (5.1)
depending on a ‘fast’ time variable t and a ‘slow’ time variable τ = εt:
x = x(t, εt, ε).
We require that x(t, τ, ε) is a 2π-periodic function of t:
Then x(t, τ, ε) satisfies the PDE
We expand
At leading order, we find that
x(t + 2π, τ, ε) = x(t, τ, ε).
xt + εxτ = f(x, t, ε).
x2
x(t, τ, ε) = x0(t, τ) + εx1(t, τ) + O(ε 2 ).
x0t = 0.
79