Moving equilibrium theorem

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 {displaystyle qquad {dot {Y}}=g({bar {x}}(Y),Y)=:G(Y).} Here {displaystyle y} has been replaced by {displaystyle Y} to indicate that the solution {displaystyle Y} to (3) differs from the solution for {displaystyle y} obtainable from the system (1), (2).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/ Categories: Economics theorems