Tikhonov's theorem (dynamical systems)
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{{context|date=May 2025}}
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}}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.{{cite journal |first=Wlodzimierz |last=Klonowski |authorlink=Wlodzimierz Klonowski |title=Simplifying Principles for Chemical and Enzyme Reaction Kinetics |journal=Biophysical Chemistry |volume=18 |issue=2 |year=1983 |pages=73–87 |doi=10.1016/0301-4622(83)85001-7 |pmid=6626688 }}{{cite journal |first=Marc R. |last=Roussel |title=Singular perturbation theory |date=October 19, 2005 |journal=Lecture Notes |url=http://people.uleth.ca/~roussel/nld/singpert.pdf }} The theorem is named after Andrey Nikolayevich Tikhonov.
Statement
Consider this system of differential equations:
:
\begin{align}
\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{align}
Taking the limit as , this becomes the "degenerate system":
:
\begin{align}
\frac{d\mathbf{x}}{dt} & = \mathbf{f}(\mathbf{x},\mathbf{z},t), \\
\mathbf{z} & = \varphi(\mathbf{x},t),
\end{align}
where the second equation is the solution of the algebraic equation
:
Note that there may be more than one such function .
Tikhonov's theorem states that as the solution of the system of two differential equations above approaches the solution of the degenerate system if is a stable root of the "adjoined system"
: