Moving equilibrium theorem

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Consider a dynamical system



with the state variables and . Assume that is fast and is slow. Assume that the system (1) gives, for any fixed , an asymptotically stable solution . Substituting this for in (2) yields


Here has been replaced by to indicate that the solution to (3) differs from the solution for obtainable from the system (1), (2).

The Moving Equilibrium Theorem suggested by Lotka states that the solutions obtainable from (3) approximate the solutions obtainable from (1), (2) provided the partial system (1) is asymptotically stable in for any given and heavily damped (fast).

The theorem has been proved for linear systems comprising real vectors and . It permits reducing high-dimensional dynamical problems to lower dimensions and underlies Alfred Marshall's temporary equilibrium method.


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