Moving equilibrium theorem

Consider a dynamical system

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

(2)..........${\displaystyle \qquad {\dot {y}}=g(x,y)}$

with the state variables ${\displaystyle x}$ and ${\displaystyle y}$. Assume that ${\displaystyle x}$ is fast and ${\displaystyle y}$ is slow. Assume that the system (1) gives, for any fixed ${\displaystyle y}$, an asymptotically stable solution ${\displaystyle {\bar {x}}(y)}$. Substituting this for ${\displaystyle x}$ in (2) yields

(3)..........${\displaystyle \qquad {\dot {Y}}=g({\bar {x}}(Y),Y)=:G(Y).}$

Here ${\displaystyle y}$ has been replaced by ${\displaystyle Y}$ to indicate that the solution ${\displaystyle Y}$ to (3) differs from the solution for ${\displaystyle y}$ obtainable from the system (1),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

The Moving Equilibrium Theorem suggested by Lotka states that the solutions ${\displaystyle Y}$ obtainable from (3) approximate the solutions ${\displaystyle y}$ obtainable from (1), (2) provided the partial system (1) is asymptotically stable in ${\displaystyle x}$ for any given ${\displaystyle y}$ and heavily damped (fast).

The theorem has been proved for linear systems comprising real vectors ${\displaystyle x}$ and ${\displaystyle y}$. 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/