Tikhonov's theorem on slow-fast systems

Remarks

In the fast time $\tau = t / \varepsilon$, the variable $x$ hardly moves, and $y$ follows the boundary layer equation, in which $\varphi(x)$ is an asymptotically stable equilibrium point by hypothesis 2. So $y$ first jumps quickly to near the slow manifold $y = \varphi(x)$, and then the system moves slowly along it according to the reduced problem. The proof turns this into estimates with a Lyapunov function for the fast subsystem, uniform in $x$ on a compact neighbourhood of the reduced trajectory.

Example: $\dot x = -x + y$, $\varepsilon \dot y = x - 2y$, with $x(0) = 1$, $y(0) = 5$. Here $\varphi(x) = x / 2$ and $\partial_y g = -2 < 0$, and the reduced problem $\dot{\bar x} = -\bar x / 2$ gives $\bar x(t) = e^{-t/2}$. The theorem gives $x_\varepsilon(t) \to e^{-t/2}$ and $y_\varepsilon(t) \to e^{-t/2} / 2$ for $t > 0$, while $y_\varepsilon(0) = 5$, so $y$ has a boundary layer at $t = 0$.

Stability of the slow manifold is needed. For $\dot x = 0$, $\varepsilon \dot y = y$ with $y(0) = y^0 \neq 0$, the root is $\varphi = 0$ but $\partial_y g = 1 > 0$, and $y_\varepsilon(t) = y^0 e^{t/\varepsilon}$ does not converge.

Going deeper

For a student

Tikhonov's theorem justifies the quasi-steady-state approximation of chemical kinetics, such as the Michaelis–Menten law in enzyme kinetics. It is a result on finite time intervals; Fenichel's theorem, the basis of geometric singular perturbation theory, gives the invariant slow manifold for $\varepsilon > 0$. The approximation can be refined to all orders by the method of matched asymptotic expansions, under extra smoothness.