Moving equilibrium theorem について

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Moving equilibrium theorem ：ウィキペディア英語版

Moving equilibrium theorem
Consider a dynamical system
(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has been replaced by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's temporary equilibrium method.
==References==

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