Beyond real blow-up: Masuda detours and complex holonomy

TL;DR

The work develops a complex-time framework for bypassing real blow-up in quadratic heat-type dynamics by embedding a two-mode Galerkin caricature in complex space and analysing its complex foliations on \mathbb{C}P^2. Through Euler multipliers and projective compactification, it constructs blow-up loops in complex time and relates their winding to real-time blow-up branches (blow-up stars). The analysis hinges on nonresonant Poincaré and Siegel domains, yielding precise linearization results and holonomy multipliers that determine when Masuda detours close discrepancy-free in real time, or produce quasiperiodic or multibranched continuations. These findings bridge finite-dimensional complex dynamics with PDE blow-up phenomena, offering insights into global behavior, discretization effects, and potential applications where complex-time continuations govern real-time outcomes. Overall, the paper provides a rigorous geometric and analytic treatment of Masuda detours, holonomy, and branching in a tractable ODE caricature that informs both theory and numerical practice in nonlinear blow-up problems.

Abstract

For real $\mathbf{b}$, consider quadratic heat equations like \begin{equation*} \mathbf{w}_t=\mathbf{w}_{\boldsymbolξ\boldsymbolξ} + \mathbf{b}(\boldsymbolξ)\,\mathbf{w}^2 \end{equation*} on $\boldsymbolξ\in(0,π)$ with Neumann boundary conditions. For $\mathbf{b}$=1, pioneering work by Kyûya Masuda in the 1980s aimed to circumvent PDE blow-up, which occurs in finite real time, by a detour which ventures through complex time. Naive projection onto the first two Galerkin modes $\mathbf{w}=x+y \cos\boldsymbolξ$ leads us to an ODE caricature. As in the PDE, spatially homogeneous solutions $y=0\neq x\in\mathbb{R}$ starting at $x_0$ blow up at finite real time $t=T=1/x_0$. In the spirit of Masuda, we extend real analytic ODE solutions to complex time, and to real 4-dimensional $(x,y)\in\mathbb{C}^2$, to circumvent the real blow-up singularity at $t=T$. We therefore study complex foliations of general polynomial ODEs for $(x,y)\in\mathbb{C}^2$, in projective compactifications like $u=1/x,\ z=y/x$, including their holonomy at blow-up $u=0$. We obtain linearizations, at blow-up equilibria of Poincaré and Siegel type, based on spectral nonresonance. We discuss the consequences of rational periodic nonresonance, and of irrational quasiperiodic nonresonance of Diophantine type, for iterated Masuda detours in the ODE caricature. We conclude with some comments on global aspects, PDEs, discretizations, and other applications.

Figures (5)

• Figure 1.1: Euler multipliers $\varrho$ associated to coordinate transformations $[\xi\!:\!\eta\!:\!\zeta]=[x\!:\!y\!:\!1]=[1\!:\!z\!:\!u]=[w\!:\!1\!:\!v]$ of the projective compactification $\mathbb{C}\mathrm{P}^2$, for $(x,y)\in\mathbb{C}^2$. Time variables in \ref{['ODEfF']}, \ref{['ODExy']} -- \ref{['ODEvw']} are $\tau$ and $t=t_0, t_1, t_2$ , respectively. Multipliers are indicated in the direction of arrows. For example, $dt=\zeta^{m-1}d\tau=u^{m-1}dt_1$ , and so on. See \ref{['ttl']} and \ref{['tt1']}, \ref{['tt2']}. Note multiplicativity \ref{['rho12']}, and hence commutativity, in the diagram.
• Figure 2.1: Phase portrait of the quadratic complex ODE $\,\dot x=x^2-1$, i.e. \ref{['ODERic']} with $a=1,\ e_1=+1,\ e_2=-1$. In real time $t$, the equilibrium $e_1=+1$ is a source (red) and $e_2=-1$ (blue) is a sink. All other real-time orbits (blue) are heteroclinic from $+1$ to $-1$. They foliate the cylinder leaf $\mathfrak{L}=\widehat{\mathbb{C}}\setminus\{\pm 1\}$ into circular segments with midpoints on the imaginary axis. The blow-up and blow-down real-time orbits $x_\infty\subset\mathbb{C}^2$ (cyan) are parts of a single heteroclinic orbit through $x=\infty$, on the Riemann sphere $x\in\widehat{\mathbb{C}}$. Hence blow-up can be circumnavigated by a blow-up loop. Note that the definition and selection of the cyan orbits is not invariant under Möbius transformations. Nonstationary orbits in imaginary time (orange), in contrast, provide an iso-periodic foliation of the complex leaf $\mathfrak{L}$ with minimal period $\pi$. The periodic orbits are nested around $\pm 1$. On the Riemann sphere, this includes the iso-periodic orbit on the imaginary axis, which experiences $\mathbb{C}^2$ blow-up and blow-down in finite imaginary time. Blow-up and blow-down in imaginary time are related to real heteroclinicity of $x_0$ in real time, just as in the PDE context of FiedlerFila; see section \ref{['PDE']}. Since the flow $t\mapsto\Phi^t(x_0)$ is conformal, for any fixed $x_0\in\widehat{\mathbb{C}}\setminus\{\pm1\}$, the blue and orange circle families are mutually orthogonal.
• Figure 3.1: Schematic phase portraits, in real time, of complex-valued ODEs \ref{['ODEP']}, for cyclotomic polynomial vector fields $\dot{x}=P(x):=x^m-1$. The equilibria $x=e_j$ are $m$-th roots of unity. Blue indicates blow-down orbits and their sink targets. Blow-up orbits, and the sources from which they emanate in reverse time, for $t\searrow -\infty$, are red. For $m=3$ see $x$ in (a), and $u=1/x$ in (b). Similarly, $x$ in (c) and $u=1/x$ in (d) refer to $m=4$. The invariant Poincaré circle $r=|u|=0,\ \varphi\in\mathbb{S}^1$ of the Poincaré compactification of $x$, by a circle $u=r\exp(\mathrm{i}\varphi)$ at infinity, is marked black in (a), (c). In (b), (d) a black dot marks the one-point compactification $x=\infty$ by $u=1/x=0$ in the Riemann sphere $\widehat{\mathbb{S}}\cong\mathbb{S}^2$. Note the real-time blow-up stars of $m-1$ red blow-up orbits towards $u=0$, which alternate with $m-1$ blue blow-down orbits around $u=0$, in (b), (d). In (a), (c) these become separatrices associated to saddles on the circle $u=0$ at infinity. Saddles are marked by integers $\mathbf{0},\ldots, \mathbf{2}m\mathbf{-3}$. Two Lyapunov centers (purple) of purely imaginary $P'(e_j)=\pm4\,\mathrm{i}$ occur in (c), (d), by nongeneric degeneracy. They are surrounded by foliated counter-rotating nests of periodic orbits with minimal period $\pi/2$. Each nest, in turn, is bounded by a homoclinic orbit (also purple) of conflated blow-down-up orbits; see (d). In (c), these become heteroclinic orbits between boundary saddles on the circle $u=0$ at infinity.
• Figure 3.2: The $14$ planar trees $\mathcal{T}$, alias reduced connection graphs $\mathcal{C}^*$, with $m=7$ vertices, up to orientation preserving equivalence. Note the two pairs which are mirror-symmetric to each other, but not mirror-symmetric, individually. Terminal vertices, of edge degree one, are distinguished by circles. Solid dots mark all other vertices. The trees are enumerated by increasing number $2,\ldots,6$ of terminal vertices. Each tree possesses two bi-colorations which mark the vertices as red sources and blue sinks, alternatingly. These also defines edge orientations, from sources to sinks. Color swaps correspond to time reversal.
• Figure 4.1: Phase portraits of blow-up loops and real-time blow-up stars near blow-up equilibrium $u=0$, in the complex plane $u\in\mathbb{C}$ of the stable leaf $\mathfrak{L}\subset\{z=0\}$. Time rescaling to original time $t\sim -u^{m-1}$ is by the Euler multiplier $\varrho=u^{m-1}$ of \ref{['tt1']}. See \ref{['tu']} for $m=2,3,4$. The real-time blow-up stars consist of red blow-up and blue blow-down separatrix orbits. Other orbits (black) follow parallel real time directions of original time $t\in\mathbb{C}$, with fixed nonzero imaginary parts $\mathrm{Im}\, t$. Blow-up of real $0<x=1/u\nearrow\infty$ occurs in finite real time $0>t\nearrow T=0$. The blow-up loop $\gamma^{uz}=\gamma^u\times\{0\}$ (black dashed) arises from $m-1$ iterates $\gamma^t$ of a simple loop in $t$, and then without any resulting discrepancy. The alternating $m-1$ red and $m-1$ blue separatrices of the hyperbolic sectors mark real-time blow-up stars. Parallel time rays $t$ in the real direction, i.e. at fixed $\mathrm{Im}\, t$, are indicated in black. Note the parallel flow, for the quadratic case $m=2$ (left). Real-time blow-down $-\infty\swarrow x=1/u<0$ occurs from continuing $0=T\swarrow t>0$. The continuation can be reached, discrepancy-free, by single complex Masuda detours in time along the upper or the lower half of the blow-up loop $\gamma^t$. Also compare the cyan orbits in the Riccati case of figure \ref{['figRiccati']}, for a more global view. The cubic case $m=3$ (center) requires two cycles through the complex loop $\gamma^t$. Accordingly, we traverse the negative axis of original time $t<0$ twice, resulting in two red blow-up orbits along real $u\neq0$. Indeed, cubic $\dot x=x^3$ blows up for positive and negative $x=1/u$. Any blue blow-down continuation to $0=T\swarrow t>0$, along the blow-up loop $\gamma^u$, requires purely imaginary $x,u$. Moreover, there are two conflicting options for such continuations beyond blow-up. For $m=4$ (right), as for all even $m$, the real time axis again allows for a continuation beyond blow-up of real $x=1/u>0$, by blow-down along real $u<0$. However, triple passage through $t<0$ and $t>0$, each, along the triply iterated blow-up loop cycle $\gamma^t$ now leads to three blow-up and three blow-down separatrices, each. For further examples see figure \ref{['figm34']} (b), (d).

Theorems & Definitions (33)

• Definition 1.1
• Remark 1.2
• Proposition 1.3
• proof
• Corollary 1.4
• proof
• Definition 1.5
• Proposition 1.6
• proof
• proof