Tikhonov's theorem (dynamical systems)
# 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. The theorem is named after Andrey Nikolayevich Tikhonov.

## Statement

Consider this system of differential equations:

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

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

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

where the second equation is the solution of the algebraic equation

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

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 $\mathbf {z} =\varphi (\mathbf {x} ,t)$ is a stable root of the "adjoined system"

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