Tikhonov's theorem (dynamical systems) explained

In applied mathematics, Tikhonov's theorem on dynamical systems is a result on stability of solutions of systems of differential equations. It has applications to chemical kinetics.^{[1] [2]} The theorem is named after Andrey Nikolayevich Tikhonov.

Statement

Consider this system of differential equations:

\begin{align} (d x)/(dt) & = f ( x , z ,t), \\ \mu (d z)/(dt) & = g ( x , z ,t). \end{align}

Taking the limit as

\mu\to 0 , this becomes the "degenerate system":\begin{align} (d x)/(dt) & = f ( x , z ,t), \\ z & = \varphi( x ,t), \end{align}

where the second equation is the solution of the algebraic equation

g ( x , z ,t) = 0 .

Note that there may be more than one such function

\varphi .

Tikhonov's theorem states that as

\mu\to 0 , the solution of the system of two differential equations above approaches the solution of the degenerate system if z = \varphi( x ,t) is a stable root of the "adjoined system"(d z)/(dt) = g ( x , z ,t).

Notes and References

1. Wlodzimierz . Klonowski . Wlodzimierz Klonowski . . Biophysical Chemistry . 18 . 2 . 1983 . 73–87 . 10.1016/0301-4622(83)85001-7 . 6626688 .
2. Marc R. . Roussel . Singular perturbation theory . October 19, 2005 . Lecture Notes .