Moving equilibrium theorem explained

Consider a dynamical system

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

•
x

=f(x,y)

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

  

•
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)..........

  

•
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has been replaced by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.

References

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Moving equilibrium theorem".

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