# 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:

Taking the limit as , 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 .

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"

## References

- 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. - Roussel, Marc R. (October 19, 2005). "Singular perturbation theory" (PDF).
*Lecture notes*.

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