# Is there an entire solution for the Van der pol equation?

Is there a non constant entire function $\gamma(t)=(x(t),y(t)): \mathbb{C} \to \mathbb{C}^{2}$ which satisfy the following Vander pol dififferential equation?

$$\begin{cases}\dot{x}=y-x^{3}\\\dot y=-x\end{cases}$$

For a related question see the last part of the following post:

The error in Petrovski and Landis' proof of the 16th Hilbert problem

No, there is not. Eliminate $y$ to get $x''=-x-3x^2x'$. There is no entire function $x(t)$ which satisfies this. Idea of the proof. Let $$M(r,f)=\max_{|t|\leq r}|f(t)|.$$ At the point $w$, $|w|=r$ where this maximum is achieved we have $\log|x''(w)|\sim \log|x'(w)|\sim\log|x(w)|$ for most values of $r$. This gives a contradiction: the last term is much larger than the rest.
• Am I mistaken to think that the same argument should work for $x"=-x-x'$ rather than $x"=-x-x^2x'$? Sep 11 '17 at 7:28