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:

{\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 φ.

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).

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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