Moving equilibrium theorem

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Here y has been replaced by Y to indicate that the solution Y to (3) differs from the solution for y obtainable from the system (1), (2).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

## References

* Schlicht, E. (1985). Isolation and Aggregation in Economics. Springer Verlag. ISBN 0-387-15254-7. https://archive.org/details/isolationaggrega0000schl.
* Schlicht, E. (1997). "The Moving Equilibrium Theorem again". Economic Modelling 14 (2): 271–278. doi:10.1016/S0264-9993(96)01034-6. https://epub.ub.uni-muenchen.de/39121/. https://epub.ub.uni-muenchen.de/39121/

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