Tikhonov's theorem (dynamical systems)

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:

${\displaystyle {\begin{aligned}{\frac {d\mathbf {x} }{dt}}&=\mathbf {f} (\mathbf {x} ,\mathbf {z} ,t),\\\mu {\frac {d\mathbf {z} }{dt}}&=\mathbf {g} (\mathbf {x} ,\mathbf {z} ,t).\end{aligned}}}$

Taking the limit as ${\displaystyle \mu \to 0}$, this becomes the "degenerate system":

${\displaystyle {\begin{aligned}{\frac {d\mathbf {x} }{dt}}&=\mathbf {f} (\mathbf {x} ,\mathbf {z} ,t),\\\mathbf {z} &=\varphi (\mathbf {x} ,t),\end{aligned}}}$

where the second equation is the solution of the algebraic equation

${\displaystyle \mathbf {g} (\mathbf {x} ,\mathbf {z} ,t)=0.}$

Note that there may be more than one such function ${\displaystyle \varphi }$.

Tikhonov's theorem states that as ${\displaystyle \mu \to 0,}$ the solution of the system of two differential equations above approaches the solution of the degenerate system if ${\displaystyle \mathbf {z} =\varphi (\mathbf {x} ,t)}$ is a stable root of the "adjoined system"

${\displaystyle {\frac {d\mathbf {z} }{dt}}=\mathbf {g} (\mathbf {x} ,\mathbf {z} ,t).}$