Relaxation oscillation

Remarks

Along the arcs of $S$ the motion takes a time of order $1$, and along the horizontal segments, the jumps, it takes a time of order $\varepsilon$.

Example: the van der Pol equation $u'' - \mu(1 - u^2) u' + u = 0$ with large $\mu$. With $F(x) = x^3/3 - x$, $\varepsilon = 1/\mu^2$ and a rescaled time, it becomes $\varepsilon x' = y - F(x)$, $y' = -x$. The critical manifold is the cubic $y = F(x)$; its attracting part is $\abs{x} > 1$ and its fold points are $(\pm 1, \mp 2/3)$. The limit curve $\Gamma_0$ runs down the right branch from $(2, 2/3)$ to $(1, -2/3)$, jumps to $(-2, -2/3)$, runs up the left branch to $(-1, 2/3)$, and jumps back to $(2, 2/3)$. The period of the van der Pol oscillation is about $(3 - 2 \ln 2)\, \mu$ for large $\mu$.

Non-example: the harmonic oscillator $x' = y$, $y' = -x$ has periodic orbits, but no small parameter and no fast variable, so it is not of the slow–fast form.

Some authors use the term more loosely, for any oscillation with slow phases and fast jumps.