Liénard's theorem

Remarks

The proof uses the energy $E(x, y) = \frac{1}{2} y^2 + G(x)$, where $G(x) = \int_0^x g(s)\, ds$. Along solutions, $E' = -g(x) F(x)$. By the symmetry $(x, y) \mapsto (-x, -y)$, it is enough to follow a solution from a point $(0, y_0)$ with $y_0 > 0$ to its next crossing of the $y$-axis. Let $\Delta(y_0)$ be the change of $E$ on this half turn. It is positive when the half turn stays in $0 \le x \le a$. For the half turns that go beyond $x = a$, it decreases strictly in $y_0$ and tends to $-\infty$. So $\Delta$ vanishes for exactly one $y_0$, and this gives the unique closed orbit.

For the van der Pol equation $x'' - \mu (1 - x^2) x' + x = 0$ with $\mu > 0$, one has $F(x) = \mu (x^3/3 - x)$ and $g(x) = x$. Here $a = \sqrt{3}$, $F < 0$ on $(0, \sqrt{3})$, and $F' = \mu (x^2 - 1) > 0$ for $x > \sqrt{3}$. So the van der Pol oscillator has exactly one limit cycle, and it is stable.

Going deeper

For a student

Proving that a limit cycle exists is often done with the Poincaré–Bendixson theorem, but that theorem says nothing about how many there are. Liénard's theorem gives existence and uniqueness at once for a whole family of equations. Its proof is a careful estimate of how the energy changes on a half turn around the origin. The van der Pol oscillator is the main application.