Consider a

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

(1)..........

\dot=f(x,y)

(2)..........

\qquad \dot=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(y)

. Substituting this for

x

in (2) yields (3)..........

\qquad \dot=g(\bar(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has been replaced by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 Alfred Marshall (26 July 1842 – 13 July 1924) was an English economist and one of the most influential economists of his time. His book ''Principles of Economics (Marshall), Principles of Economics'' (1890) was the dominant economic textboo ...

's temporary equilibrium method.

References

* *{{cite journal , author=Schlicht, E. , title=The Moving Equilibrium Theorem again , journal=Economic Modelling , year=1997 , volume=14 , pages=271–278 , doi=10.1016/S0264-9993(96)01034-6 , issue=2, url=https://epub.ub.uni-muenchen.de/39121/ https://epub.ub.uni-muenchen.de/39121/ Economics theorems