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.

Consider this system of differential equations:

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

where the second equation is the solution of the algebraic equation

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"

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