Consider a dynamical system
(1)..........
x
˙
=
f
(
x
,
y
)
(2)..........
y
˙
=
g
(
x
,
y
)
with the state variables
x
and
y
. Assume that
x
is fast and
y
is slow. Assume that the system (1) gives, for any fixed
y
, an asymptotically stable solution
x
¯
(
y
)
. Substituting this for
x
in (2) yields
(3)..........
Y
˙
=
g
(
x
¯
(
Y
)
,
Y
)
=:
G
(
Y
)
.
Here
y
has been replaced by
Y
to indicate that the solution
Y
to (3) differs from the solution for
y
obtainable from the system (1), (2).
The Moving Equilibrium Theorem suggested by Lotka states that the solutions
Y
obtainable from (3) approximate the solutions
y
obtainable from (1), (2) provided the partial system (1) is asymptotically stable in
x
for any given
y
and heavily damped (fast).
The theorem has been proved for linear systems comprising real vectors
x
and
y
. It permits reducing high-dimensional dynamical problems to lower dimensions and underlies Alfred Marshall's temporary equilibrium method.
