Moving equilibrium theorem

Consider a dynamical system

(1)..........\dot{x}=f(x,y)

(2)..........\qquad \dot{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 \bar{x}(y). Substituting this for x in (2) yields

(3)..........\qquad \dot{Y}=g(\bar{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Here y has been replaced by Y to indicate that the solution Y to (3) differs from the solution for y obtainable from the system (1), (2).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.

References

• https://epub.ub.uni-muenchen.de/39121/