Moving equilibrium theorem

Consider a dynamical system (1)..........[math]\displaystyle{ \dot{x}=f(x,y) }[/math]

(2)..........[math]\displaystyle{ \qquad \dot{y}=g(x,y) }[/math]

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

(3)..........[math]\displaystyle{ \qquad \dot{Y}=g(\bar{x}(Y),Y)=:G(Y). }[/math]

Here [math]\displaystyle{ y }[/math] has been replaced by [math]\displaystyle{ Y }[/math] to indicate that the solution [math]\displaystyle{ Y }[/math] to (3) differs from the solution for [math]\displaystyle{ y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

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

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