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](https://mathoverflow.net/questions/171988/the-error-in-petrovski-and-landis-proof-of-the-16th-hilbert-problem)



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


This is called Wiman-Valiron theory. See
MR1438606  

Hayman, Walter K.
The growth of solutions of algebraic differential equations. 
Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 7 (1996), no. 2, 67–73. 


or Valiron's book Fonctions analytiques.


