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Here $y$ has been replaced by $Y$ to indicate that the solution $Y$ to (3) differs from the solution for $y$ obtainable from the system (1), (2).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

• Schlicht, E. (1985). Isolation and Aggregation in Economics. Springer Verlag. ISBN 0-387-15254-7.
• 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/