# Moving equilibrium theorem

Consider a dynamical system

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

Here ${\displaystyle y}$ has been replaced by ${\displaystyle Y}$ to indicate that the solution ${\displaystyle Y}$ to (3) differs from the solution for ${\displaystyle y}$ obtainable from the system (1),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

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