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"

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

References

1. Klonowski, Wlodzimierz (1983). "Simplifying Principles for Chemical and Enzyme Reaction Kinetics". Biophysical Chemistry. 18 (2): 73–87. doi:10.1016/0301-4622(83)85001-7.
2. Roussel, Marc R. (October 19, 2005). "Singular perturbation theory" (PDF). Lecture notes.