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Here $$y$$ has been replaced by $$Y$$ to indicate that the solution $$Y$$ to (3) differs from the solution for $$y$$ obtainable from the system (1), (2).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