In the study of dynamical systems, the method of averaging is used to study certain time-varying systems by analyzing easier, time-invariant systems obtained by averaging the original system.
Consider a general, nonlinear dynamical system
x
˙
=
ϵ
f
(
t
,
x
,
ϵ
)
where
f
(
t
,
x
)
is periodic in
t
with period
T
. The evolution of this system is said to occur in two timescales: a fast oscillatory one associated with the presence of
t
in
f
and a slow one associated with the presence of
ϵ
in front of
f
. The corresponding (leading order in
ϵ
) averaged system is
x
˙
a
=
ϵ
1
T
∫
0
T
f
(
τ
,
x
,
0
)
d
τ
=
f
~
(
x
a
)
.
Averaging mods out the fast oscillatory dynamics by averaging their effect (through time integration - see the formula above). In this way, the mean (or long-term) behavior of the system is retained in the form of the dynamical equation for the evolution for
x
a
. Standard methods for time-invariant (autonomous) systems may then be employed to analyze the equilibria (and their stability) as well as other dynamical objects of interest present in the phase space of the averaged system.
Consider a damped pendulum whose point of suspension is vibrated vertically by a small amplitude, high frequency signal (this is usually known as dithering). The equation of motion for such a pendulum is given by
m
(
l
θ
¨
−
a
k
ω
2
sin
ω
t
sin
θ
)
=
−
m
g
sin
θ
−
k
(
l
θ
˙
+
a
ω
cos
ω
t
sin
θ
)
where
a
sin
ω
t
describes the motion of the suspension point,
k
describes the damping of the pendulum, and
θ
is the angle made by the pendulum with the vertical.
The phase space form of this equation is given by
t
˙
=
1
θ
˙
=
p
p
˙
=
1
m
l
(
m
a
k
ω
2
sin
ω
t
sin
θ
−
m
g
sin
θ
−
k
(
l
p
+
a
ω
cos
ω
t
sin
θ
)
)
where we have introduced the variable
p
and written the system as an autonomous, first-order system in
(
t
,
θ
,
p
)
-space.
Suppose that the angular frequency of the vertical vibrations,
ω
, is much greater than the natural frequency of the pendulum,
g
/
l
. Suppose also that the amplitude of the vertical vibrations,
a
, is much less than the length
l
of the pendulum. The pendulum's trajectory in phase space will trace out a spiral around a curve
C
, moving along
C
at the slow rate
g
/
l
but moving around it at the fast rate
ω
. The radius of the spiral around
C
will be small and proportional to
a
. The average behaviour of the trajectory, over a timescale much larger than
2
π
/
ω
, will be to follow the curve
C
.
