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Here y has been replaced by Y to indicate that the solution Y to (3) differs from the solution for y obtainable from the system (1), (2).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.
