Talk:Moving equilibrium theorem

Suggestions

 * Maybe the concept of "fast" and "slow" could be explained (or a link could be added).


 * Also, I think it's not obvious what "approximates" means.


 * I was wondering who proved the theorem? Are there other versions for nonlinear systems?


 * I suggest reorganizing the article like this:

The Moving Equilibrium Theorem suggested by Lotka states that for a system of linear differential equations in two real variables dependent on time, where one changes comparatively slow and the other fast, the difference in time scale allows you to approximate solutions of the system by solving the equations separately.

The theorem permits reducing high-dimensional dynamical problems to lower dimensions and underlies Alfred Marshall's temporary equilibrium method.

For a proper statement of the theorem, consider a dynamical system

$$(1) \qquad \dot{x}=f(x,y)$$

$$(2) \quad \quad \dot{y}=g(x,y)$$

with state variables $$x$$ and $$y$$. Assume that $$x$$ is fast and $$y$$ is slow. Assume that for any fixed $$y$$, equation (1) has an asymptotically stable solution $$\bar{x}(y)$$. Substituting this for $$x$$ in (2) yields

$$(3) \qquad \dot{Y}=g(\bar{x}(Y),Y)=:G(Y)$$.

Here $$y$$ has been replaced by $$Y$$ to indicate that the solution $$Y$$ obtained from (3) differs from the solution for $$y$$ obtainable from the system (1), (2). The theorem asserts that the solution for $$Y$$ approximates the solution for $$y$$, provided the partial system (1) is heavily damped (fast) for any given $$y$$.