2025/10/31 by Bernold Fiedler, Fiedler, Bernold
Mathematics · #32M25 #34M35 #35B44 #37F50 #37F75 #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2510.27453
openalex publication_date 2025/10/31 · openalex created_date 2025/11/05 · openalex updated_date 2026/07/28
For real b, consider quadratic heat equations like wt=w\boldsymbolξ\boldsymbolξ + b(\boldsymbolξ) w2 on \boldsymbolξ∈(0,π) with Neumann boundary conditions. For 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 w=x+y cos\boldsymbolξ leads us to an ODE caricature. As in the PDE, spatially homogeneous solutions y=0≠ x∈ℝ starting at x0 blow up at finite real time t=T=1/x0. We aim for ODE "linearization at infinity". Since iterated complex time loops are not feasible, for parabolic PDEs, our PDE-motivated approach is currently limited to ODEs. On the other hand, all ODE results of the present paper exactly embed into certain PDEs of parabolic type, which possess a PDE-invariant Galerkin subspace. In the spirit of Masuda, we extend real analytic ODE solutions to complex time, and to real 4-dimensional (x,y)∈ℂ2, to circumvent the real blow-up singularity at t=T. We therefore study complex foliations of general polynomial ODEs for (x,y)∈ℂ2, 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.