Has the system of ODEs $$\frac{dx}{dt}=P(x,y)\\ \frac{dy}{dt}=Q(x,y) $$ been studied for the special case of the polynomials $P$ and $Q$ being a harmonic pair, i.e. the real and imaginary part of a holomorphic polynomial $F=F(z)$, $z=x+iy$?

I am looking to learn a bit about (complex) ODEs and their interplay with algebraic geometry by some examples, but I couldn't find anything on this special case in Ilyashenko's survey on Hilbert 16 (I guess this case is too special and/or not very interesting as far as Hilbert 16 is concerned).

Nontheless, it seems very natural. If we set $\gamma(t)=x(t)+iy(t)$, this amounts to the equation $$ \int_{\gamma_t}\frac{dz}{F(z)}=t $$ where $\gamma_t$ is the curve $\gamma$ "truncated" at $t$ and the RHS is in particular real. This can be taken further, for example by assuming $\gamma$ is closed and using the residue theorem to obtain constraints on (the coefficients of) $F$.

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    $\begingroup$ In this case you can consider the heat equation $\partial_t u(z,t)=\Delta u(z,t)$. If the number of the limit cycle change, it must be the time to pass a singularity. and take $t=\infty$, the dynamic system coverage to a very simlpe one and in particular it do not have limit cycle. So we need only look at the moments passing singularity. $\endgroup$ – Hu xiyu Nov 18 '17 at 17:05

First, this case is totally uninteresting regarding Hilbert XVI. Indeed, there are no limit cycles in such systems. The $\alpha / \omega$-limit of a trajectory is either a point or a non-isolated cycle (center case).

A singularity at $a\in \mathbb C$ (i.e. a root of $F$) can only be of three types, according to the value of $F'(a)$:

  1. Source/focus: $F'(a)\notin i\mathbb R$.
  2. Center: $F'(a)\in i\mathbb R_{\neq 0}$.
  3. Flower with $2k$ petals: $F'(a)=0$ with multiplicity $k$.

In addition there is a pole at infinity (if $\deg(F)>0$) with exactly $2\deg(F)$ separatrices, reaching the singularity in finite time. The bassins of attraction / center regions attached to the above singularities are delimited by the separatrices.

enter image description here

S. Smale began to get interested in the question in the early 80's while laying the foundations for BSS computational model (The fundamental theorem of algebra and complexity theory, 1981). He proposed a numerical root solver for polynomials by following the flow of $\frac{F}{F'}$. This started some works on the topic, for instance by Schub, Tischler, William (The Newtonian graph of a complex polynomial, 1988) or Benzinger (Plane autonomous systems with rational vector fields, 1991)…

In the case of these vector fields, the topological class is entirely encoded by their Newtonian graph (or the «dual» spinal graph) given by the incidence graph of the $\alpha / \omega$-limits of trajectories (in red on the picture). The main result for polynomials is that it is a tree. See e.g. Sverdlove (Inverse problems for dynamical systems,1981) and Schecter, Singer (A class of vectorfields on $\mathbb S^2$ that are topologically equivalent to polynomial vectorfields,1985) and Jongen, Jonker, Twilt (On the classification of plane graphs representing structurally stable rational Newton flows,1991).

The conformal classification has been initiated by Douady, Estrada and Sentenac (unpublished monograph, 2005) for the generic case (only focus/source singularities) and completed by Branner and Dias (Classification of complex polynomial vector fields in one complex variable, 2010). In addition to the combinatorial (topological) invariant, a complex «time-shift» (related to the integrals $\int_\gamma\frac{1}{F(z)} dz$) is associated to the separatrices, providing a complete conformal invariant.

In that latter context, the function $\int\frac{1}{F(z)} dz$ is called a Fatou coordinates. It is a rectifying chart for the vector field, and has many interesting dynamical properties.

Notice also the deep and beautiful relationship between spinal graph and Dessins d'enfants, as established by Pilgrim (Polynomial vector fields, dessins d'enfants, and circle packings,2006), related to this question.

  • $\begingroup$ Thanks for the detailed and clarifying response! Branner and Dias' monograph seems like exactly the thing I was looking for. To be honest, I did not expect that this specific equation type can be the source of such broad mathematics. It is certainly a lot of information to digest. $\endgroup$ – M.G. Nov 15 '17 at 17:13
  • $\begingroup$ Sorry for slightly off-topic question: do you happen to know of a good beginner introduction to limit cycles and related topics? $\endgroup$ – M.G. Nov 17 '17 at 10:19
  • $\begingroup$ Not as such, sorry (I mainly know research articles on the subject). Try beginning with Bendixon-Poincaré theorem, which is classical stuff and belongs to a lot of textbooks on vector fields. There may be mention of introductory texts on the specific topic. $\endgroup$ – Loïc Teyssier Nov 17 '17 at 14:39
  • $\begingroup$ Ok, thanks for the hint, I will check it out. $\endgroup$ – M.G. Nov 17 '17 at 15:23

Yes, they have been studied. Look into the recent paper Langley, J. K. Trajectories escaping to infinity in finite time. Proc. Amer. Math. Soc. 145 (2017), no. 5, 2107–2117, and the reference list in this paper.

They were also studied by physicists:

Bender, Carl M.; Hook, Daniel W. Complex classical motion in potentials with poles and turning points. Stud. Appl. Math. 133 (2014), no. 3, 318–336.

EDIT. I forgot to mention this:

B. Branner, K. Dias, Classification of complex polynomial vector fields in one complex variable, Journal Journal of Difference Equations and Applications Volume 16, 2010 - Issue 5-6:

  • $\begingroup$ The first reference you indicate (Langley) deals explicitely with transcendal functions (as most of its references do), so I believe this does not quite answer the OP. Nice paper, though. $\endgroup$ – Loïc Teyssier Nov 15 '17 at 22:09
  • $\begingroup$ @Loic Teysser: Yes, I mainly meant the reference list of this paper, to show that this is indeed an active research subject. $\endgroup$ – Alexandre Eremenko Nov 15 '17 at 23:16
  • $\begingroup$ @Loic Teysser: I added a reference which is more relevant. $\endgroup$ – Alexandre Eremenko Nov 16 '17 at 13:00

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