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FAST VARYING VARIABLE

Assume that we have two differential equations for two variables x and y, that

           = AF(x,y),
           = Q(x,y),     where

A >> 1 is a larger value. It means that the product AF(x,y) is a larger value, and, consequently, the rate of change is also the larger one. Hence it follows that x is a fast varying variable. Designate and divide the right and left parts to A. We obtain

          ε = F(x,y),
           = Q(x,y),     where

ε << 1. (ε = 1/A).

It can be seen that if ε → 0    ε = F(x,y) = 0.

It means that the differential equation for the variable x can be replaced by algebraic one F(x,y) = 0, in which x takes the stationary value, which depends on y as a parameter, that is x = x(y).

In this sense the slow variable y is the governing parameter, and one can influence the coordinates of the stationary point x(y). by varying the governing parameter.