Scalar polynomial vector fields in real and complex time

TL;DR

This work analyzes scalar polynomial ODEs of the form $\\dot w=f(w)$ with $f$ degree $d$ and simple zeros, deriving explicit separation-of-variables solutions and a global complex-time flow on a Riemann surface $\\mathcal{R}$. A central device is the period map built from residues $\\eta_j=1/f'(e_j)$, which governs multivalued time data and enables prescribing residues under a zero-sum constraint; this yields a rich global geometry including a Riemann-surface description, branch structure, and a classification of planar real-time portraits via Poincaré compactification. The real-time dynamics are encoded combinatorially by planar trees or, equivalently, noncrossing chord diagrams, with a realization theorem ensuring that every such tree is realized by some polynomial $f$; the counts relate to OEIS sequence $A_{d-1}$ and provide explicit formulas. The paper also develops both the Morse-Smale-type orbit-structure theory for the compactified flows and a 1,000€-offer framework to address the existence of complex-entire homoclinic orbits and ultra-exponential splitting phenomena under forcing. Altogether, the results give a detailed bridge between complex-analytic structure of ODE flows and real-time planar portraits, with broad implications for interpolation, dynamics, and potential higher-dimensional extensions.

Abstract

In the present paper, the simplest scalar ODE case is studied for polynomials $$ \dot{w}=f(w)=(w-e_0)\cdot\ldots\cdot(w-e_{d-1}) $$ of degree $d$ with $d$ simple complex zeros. The explicit solution by separation of variables and explicit integration is an almost trivial matter. In a classical spirit, indeed, we describe the complex Riemann surface $\mathcal{R}$ of the global nontrivial solution $(t,w(t))$ in complex time, as an unbranched cover of the punctured Riemann sphere. The flow property, however, fails at $w=\infty$. The global consequences depend on the period map of the residues $2π\mathrm{i}/f'(e_j)$ of $1/f$ at the punctures, in detail. We therefore show that polynomials $f$ exist for arbitrarily prescribed residues with zero sum. This result is not covered by standard interpolation theory. Motivated by the PDE case, we also classify the planar real-time phase portraits of the above ODE. Poincaré compactification of $w$ regularizes $w=\infty$ by $2(d-1)$ equilibria, alternatingly stable and unstable within the invariant circle boundary at infinity. In structurally stable cases, we classify all compactified phase portraits, up to orientation preserving orbit equivalence and time reversal. Combinatorially, their connection graphs are equivalent to certain unrooted, unlabeled, undirected planar trees of $d$ vertices or, dually, to certain chord diagrams with $d-1$ nonintersecting chords. We sketch a proof that all planar trees are actually realized by Poincaré compactifications. Not least, we offer a 1,000 Euro reward for the discovery, or refutation, of complex entire homoclinic orbits.

Figures (3)

• Figure 1.1: Phase portrait of quadratic complex ODEs \ref{['ODEw2']} -- \ref{['ODEuv2']} with equilibria $w=W_\mp=\pm1$ and the heteroclinic orbit $\Gamma:W_-\leadsto W_+$ . Orbits $r\mapsto\Gamma(r+\mathrm{i} s)=u+\mathrm{i} v$ of \ref{['ODEuv2']}, \ref{['ODEw2']}, in real time $r$, are circular arcs (blue). They are heteroclinic from source $W_-=+1$ (red dot) to sink $W_+=-1$ (blue dot). The $\pi$-periodic orbits $s\mapsto\psi(s)=\Gamma(r-\mathrm{i} s)=u+\mathrm{i} v$ of \ref{['ODEpsi2']}, in contrast, are full circles (orange) in $\mathbb{C}$, each surrounding one of the two equilibria $W_\mp=\pm 1$. By \ref{['flow']}, the flows in real and imaginary time commute. Therefore the orange circles also serve as isochrones which globally synchronize the blue heteroclinics. Conversely, the blue circle segments are isochrones which globally synchronize the orange periodic solutions. For any fixed time $t$, the holomorphic flow map $w_0\mapsto \Phi^t(w_0)$ is conformal, i.e. angle-preserving. In particular, the blue and orange circle families are mutually orthogonal. The imaginary $v$-axis (orange) blows up in finite time $s^*=\pm\pi/2$; see \ref{['s*']}. On the Riemann sphere $\widehat{\mathbb{C}}$ , this is just a longitude circle of period $\pi$ through the South Pole at $w=0$ and the North Pole at $w=\infty$. The real $u$-axis features the real heteroclinic orbit $u_0: +1 \leadsto -1$, in blue, as well as the two cyan parts of the blow-up heteroclinic $u_\infty:+1\leadsto\infty$ and the blow-down heteroclinic $u_\infty:\infty\leadsto -1$. On the Riemann sphere $\widehat{\mathbb{C}}$ , these three line segments combine to another longitude circle, perpendicular to the first at the polar intersections 0 and $\infty$. In conclusion, the blue bounded heteroclinic orbit $u_0$ on the real $u$-axis, in real time, gives rise to finite time blow-up and blow-down on the orange imaginary $v$-axis, in imaginary time, when started at $u=v=0$. The complex viewpoint also reveals how the blue real heteroclinicity $u_0$ and the two cyan real blow-up/blow-down segments $u_\infty$ , in real time, become three aspects of one and the same underlying trajectory $\Gamma$, in complex time $t=r+\mathrm{i} s$.
• Figure 1.2: Schematic phase portraits, in real time, of complex-valued ODEs \ref{['ODEw']}, \ref{['gpolyzEuler']} -- \ref{['alpha']} for cyclotomic vector fields $\dot{w}=w^d-1$; see \ref{['cyclotomic']}. For $d=3$ see $w$ in (a), and $z=1/w$ in (b). Similarly, $w$ in (c) and $z=1/w$ in (d) refer to $d=4$. The invariant circle $\varrho=|z|=0,\ \alpha\in\mathbb{S}^1$ of the Poincaré compactification at $w=\infty$ is marked black in (a), (c). Interior equilibria $e_j\in\mathbb{D}$ are stable sinks (blue), Lyapunov centers (purple) surrounded by periodic orbits, or unstable sources (red). Unstable blow-down separatrices (blue) emanate from $w=\infty$ at odd-labeled vertices $\mathbf{k}=\mathbf{1},\mathbf{3},\mathbf{5}$. Stable blow-up separatrices (red) run towards the even-labeled saddles $\mathbf{k}=\mathbf{0},\mathbf{2},\mathbf{4}$. In (c), two pairs of interior saddle separatrices coincide (purple) in each of the interior saddle-connections $\mathbf{1}\leadsto \mathbf{2}$ and $\mathbf{5}\leadsto \mathbf{4}$. In (d), the two purple separatrices become homoclinic to $w=\infty$. The black trajectories marked $\pm\mathrm{i}\varepsilon$ are complex perturbations of the red real blow-up separatrices $e_0\leadsto \mathbf{0}$, say with initial conditions $w(0)=2\pm\mathrm{i}\varepsilon$. For $d=3$ in (a), they closely follow the heteroclinic chains $e_0\leadsto \mathbf{0}\leadsto \mathbf{1}\leadsto e_1$ and $e_0\leadsto \mathbf{0}\leadsto \mathbf{3}\leadsto e_2$, respectively. The intermediate boundary connections $\mathbf{0}\leadsto \mathbf{1}$ and $\mathbf{0}\leadsto \mathbf{3}$ within the invariant boundary circle $\mathbb{S}^1$ concatenate initial red blow-up to terminal blue blow-down. In the polar view (b), centered at $w=\infty,\ z=0$, the boundary parts are conflated into $z=0$. Note the markedly distinct limits of the two perturbations, for $\varepsilon\searrow 0$, given by the two distinct remaining blow-up-down concatenations $e_0\leadsto \infty\leadsto e_1$ and $e_0\leadsto \infty\leadsto e_2$ . In case $d=4$ (d), the initial red blow-up and terminal blue blow-down limits of both perturbations $\pm\varepsilon$ coincide. Each limit $e_0\leadsto \infty\leadsto\infty\leadsto e_3$ contains an additional interior blow-down-up separatrix $\infty\leadsto\infty$ (purple) which is homoclinic to $z=0$. The two counter-rotating purple homoclinic separatrices, however, are markedly distinct: their lobes surround the distinct centers $e_1$ and $e_3$, respectively, in opposite direction. In (c), this is manifest by the two purple interior saddle-connections $\mathbf{1}\leadsto \mathbf{2}$ and $\mathbf{5}\leadsto \mathbf{4}$. Together with the heteroclinic boundary connections within the black invariant boundary circle $\mathbb{S}^1$ which run between the same saddles, in opposite direction, we obtain two heteroclinic cycles. Their interiors are properly foliated by families of nested, synchronously iso-periodic orbits of minimal periods $\mp\pi/2$ around the Lyapunov centers $e_1$ and $e_3$ ; see lemma \ref{['perlem']} (iv),(v).
• Figure 5.1: Schematic splitting region for a homoclinic orbit $\Gamma$ of the ODE-flow \ref{['odef']} (hashed), under discretization with step size $\varepsilon$ or, equivalently, under $\varepsilon$-periodic non-autonomous forcing \ref{['odefg']}; see FiedlerScheurle. At fixed levels of $\varepsilon$, the horizontal splitting intervals $\lambda\in I(\varepsilon)$ mark parameters $\lambda$ for which single-round homoclinic orbits occur near $\Gamma$. For analyticity of $\Gamma(r+\mathrm{i} s)$ in a strip $|s|\leq a$, the width $\ell(\varepsilon)$ of the splitting interval $I(\varepsilon)$ is exponentially small in $\varepsilon$ with exponent $a$; see \ref{['split']}. For better visibility, the horizontal width $\ell(\varepsilon)$ of the exponentially flat splitting region has therefore been much exaggerated, in our schematic illustration. We call the dynamics in the resulting chaotic region "invisible chaos". For complex entire homoclinic orbits $\Gamma(t)$, the exponent $a$ could be chosen arbitrarily large. Such ultra-exponentially small splittings would therefore lead to ultra-invisible chaos.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Lemma 2.2
• Lemma 2.3
• Corollary 2.4
• Lemma 4.1