Can a holomorphic vector field have an attractor homoclinic loop?

Question

It is well known that a holomorphic vector field $z'=f(z), z\in \mathbb{C}$ does not have any limit cycle.See the last paragraph of this post

Orbits space of real-analytic planar foliations

One can imagine several reason for this fact. For example: The flow map is a holomorphic map. So if the limit cycle is of period $T$ then $\phi_T$ admits an infinite number of fixed point hence it is identically equal to the identity map.

Now we ask the following question

Is there a holomorphic vector field $z'=f(z)$ on an open region of $\mathbb{C}$ which admit an attractor homoclinic loop?

Answer 1

The answer is 'no' for much the same reason that the OP indicates: the existence of a homoclinic or heteroclinic connection implies that neighboring trajectories are periodic.

First, one needs to have poles in $f$ if one wishes to have heteroclinic connections between (real) saddle singularities. Indeed if $f$ is holomorphic, then the stationnary points of the underlying real vector field are either centers, foci or multiple singularities locally conformally conjugate to $z'=\frac{z^{k+1}}{1+\mu z^k}$. None of these singularities can be part of a candidate limit (poly)cycle.

Next, consider the time-coordinate $$T(z)=\int_{z_*}^z \frac{dz}{f(z)}.$$ Real-time trajectories of the vector field correspond to level sets $\mathrm{Im}(T)=\textrm{cst}$. The presence of a (poly)cycle $\Gamma$ connecting poles of $f$ implies that $\int_\Gamma \frac{dz}{f(z)}=:\tau\in \mathbb{R}$ is the period of $\Gamma$.

Let $\gamma$ be a neighboring trajectory connecting a transverse section $\Sigma$ to itself "after making one turn". Without loss of generality one may choose $\Sigma$ as a small piece of $\mathrm{Re}(T)=\textrm{cst}$. Then there exists a small curve $\sigma\subset\Sigma$ such that the concatenation $\sigma\gamma$ is a closed loop. Cauchy's formula implies that $\int_{\sigma\gamma} \frac{dz}{f(z)}=\tau$. But only $\sigma$ contributes to the imaginary part of the integral, therefore $\sigma$ is just one point and $\gamma$ is closed.

Answer 2

There is no any kind of, generally speaking, "Charactristic curve" for the vector field $z'=f(z)$ when $f$ is a holomorphic function. By charactristic curve I mean any kind of particular curve which is really rare!. Mathematically speaking: it is isolated in its neighborhood: there is no any thing similar to it, in its small neighborhood. For example a limit cycle, an attracor loop, a UINIQE hetroclinic or homoclinic connection, etc:

>Proof: observe that $[f,if]=0$ and $f$ and $if$ are independent vector fields \QED

Remark: The above 1-line proof can be accepted via the same argument as page 2 item 1 of my paper https://arxiv.org/abs/math/0507516 which says: an attracting limit cycle for a vector field $X$ is an invariant curve for any vector field $Y$ with $[X,Y]=0$. a common observation in investigation of limit cycles, isochronois centers, etc.

When I was asking this MO question I forgot the simple fact that $[f,if]=0$ when $f$ is a holomorphic function. I observed it imeddiate after my first meet with the concept "Lie bracket of vector fields" when I was a master student of the course "Differentiable manifold"(about 27 years ago). In that period I was not familliar with the concept limit cycle but it seems that "limit cycle" was in my inconscient.

Any way I lookforward to hear the answer to my commented questions to other existing answer to this question. a possible answer to my commented questions would help me to understand:

In the holomorphic setting, is it possible we have a familly of homoclinic loop $\gamma_t$ which homotopically vanish at the vertex of homoclinic loop? So this would be a contradiction to possible common belife that in the interior of a homoclinic loop we necessarilly have a singularity(of course different from the vertex).