Dimensional reduction of system of ODEs

[kalish1]

Given a nonlinear system of eight autonomous differential equations with all variables and parameters living in the positive octant of real numbers:

$$dX_1/dt = \ldots\\
dX_2/dt = \ldots \\
\ldots \\
dX_8/dt = \ldots$$

and given that $\lim\limits_{t \to \infty} K(t) \to 0$ for $K(t) = X_3(t) + X_4(t) +X_5(t) + X_6(t) + X_7(t) + X_8(t),$

is it true that the long-term behavior of the original system will be identical to its behavior on the $X_1-X_2$ plane? And why or why not?

This question has been crossposted here: http://math.stackexchange.com/questions/1371952/dimensional-reduction-of-system-of-odes

 

[gtoguy97]

No, the long-term behavior of the original system will not be identical to its behavior on the $X_1-X_2$ plane. This is because the other variables, $X_3$ to $X_8$, do have an effect on the system dynamics even if their values approach 0 asymptotically in the long-term. As such, they can influence the behavior of the system even if their values are small, and so the behavior of the system on the $X_1-X_2$ plane may be different from the behavior of the full system.