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Is there a polynomial vector field on $\mathbb{C}^2$ which possesses a bounded regular leaf? By bounded, I mean a bounded subset of $\mathbb{C}^2$.

We consider the Van der Pol vector field $$(1) \;\;\;\;\;\; \begin{cases} x'=y-(x^3-x)\\ y'=-x\end{cases}$$ on $\mathbb{R}^2.$ It is well known that this equation has a unique limit ...

Let $V=\sum_{i=1}^{n}a_i(z_1,\ldots z_n)\frac{\partial}{\partial z_i}$ be a holomorphic vector field defined on a neighborhood $U\subset \mathbb{C}^n$ of the origin, such that the common zero point of ...

Edit 1: For a related discussion see this MSE post I apologize in advance, if this question is obvious: 1)What is an example of a polynomial vector field on $\mathbb{R}^{2}$ with at least two ...

Suppose we have a complex vector field on $\mathbb{C}^n$ which is analytic and has $|DV| < L$ on ball $B_r$ with radius r. I would like to understand: 1) if there exists an analytic flow $\phi_t(x)...