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] 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 $\mu\to 0$, 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

$\mathbf{g}(\mathbf{x},\mathbf{z},t) = 0. \,$

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

1. http://www.math.bme.hu/~jtoth/frk/Roussel.pdf

External links

• Singular Perturbation Theory, by Marc Roussel