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[00988] Treatment of infinity and finite-time singularities in differential equations

  • Session Time & Room :
    • 00988 (1/2) : 1C (Aug.21, 13:20-15:00) @G402
    • 00988 (2/2) : 1D (Aug.21, 15:30-17:10) @G402
  • Type : Proposal of Minisymposium
  • Abstract : Finite-time singularities arise in various problems in differential equations and have been ones of the most important issues towards the comprehensive understanding of the global nature of systems for decades.
 In recent years, various universal machineries from geometry, dynamical systems and numerical analysis have been proposed and applied to unraveling wide variety of finite-time singularities, as well as appropriate treatments of infinity, and the common nature among them. This symposium aims at sharing state-of-the-art topics of singularity, instability and unboundedness manifesting in finite times in differential equations towards new foundations of these complex and rich characteristics.
  • Organizer(s) : Kaname Matsue
  • Classification : 34D05, 34M35, 35B44, 37C60, 70F16, Blow-up in ODEs and asymptotic expansions of solutions, 
Blow-up in parabolic and hyperbolic PDEs, 
Collision in celestial mechanics, 
Monodromy matrices for ODEs in complex domain, 
nonautonomous instability in ODEs and PDEs
  • Minisymposium Program :
    • 00988 (1/2) : 1C @G402 [Chair: Kaname Matsue]
      • [01480] Finite-time singularity and dynamics at infinity: characterization and asymptotic expansions
        • Format : Talk at Waseda University
        • Author(s) :
          • Kaname Matsue (Kyushu University)
        • Abstract : Finite-time singularities in differential equations, in particular finite-time blow-up, from the viewpoint of dynamical systems are discussed in this talk. Using compactifications of phase spaces and time-scale desingularization naturally introduced by the quasi-homogeneity of vector fields in an asymptotic sense, blow-up characterization is reduced to dynamics at infinity. We also discuss a systematic calculations of multi-order asymptotic expansion of blow-up solutions with a natural correspondence to dynamical properties of invariant sets at infinity.
      • [01461] Compactification for Asymptotically Autonomous Dynamical Systems with Applications to Tipping Points.
        • Format : Talk at Waseda University
        • Author(s) :
          • Sebastian Wieczorek (University College Cork)
          • Christopher K.R.T. Jones (University of North Carolina)
        • Abstract : We develop a general compactification framework for non-autonomous ODEs, where non-autonomous terms decay asymptotically. The aim is to use compact invariant sets of the autonomous limit systems from infinity to analyse non-autonomous instabilities in the original problem, in the spirit of dynamical systems theory. We illustrate our framework using rate-induced tipping instability that occurs in natural systems when external inputs, such as climatic conditions, vary faster than some critical rate.
      • [01512] Rate-induced tipping in heterogeneous reaction-diffusion systems
        • Format : Talk at Waseda University
        • Author(s) :
          • Cris Hasan (University of Glasgow)
          • Sebastian Wieczorek (University College Cork)
          • Ruaidhrí Mac Cárthaigh (University College Cork)
        • Abstract : We propose a framework to study nonlinear waves in reaction-diffusion equations (RDEs) based on a compactification technique and Lin’s method for constructing heteroclinic orbits. We identify generic instabilities of travelling pulses in an RDE with a fold of heteroclinic orbits in the compactified system. In an illustrative model of a habitat patch that is geographically shrinking or shifting due to climate change, we combine our framework with numerical continuation to study tipping points to extinction.
      • [01527] Using Geometric Singular Perturbation Theory to Understand Singular Shocks
        • Format : Talk at Waseda University
        • Author(s) :
          • Barbara Lee Keyfitz (The Ohio State University)
        • Abstract : Solutions to hyperbolic conservation laws (quasilinear hyperbolic partial differential equations) typically satisfy the equations in the sense of distributions. But there are examples of systems whose solutions have even lower regularity, solutions known as singular shocks or delta shocks. In some of these examples, candidates for solutions that exhibit singular shocks have been found as limits of approximations. An unusual model in two-component chromatography, discovered by Marco Mazzotti, provides the first physically significant example of a system where singular shocks appear. This model does not fit into the existing theory. Here, I present new approach. Singular perturbation theory (SPT), long a mainstay of classical applied mathematics, has been put on a new footing by an approach developed by Fenichel in the 1970's and since then extended by many other researchers. This approach uses manifold and dynamical systems theory to replace the formal constructions of SPT. It was first applied to singular shocks by Stephen Schecter. Using geometric singular perturbation theory, Ting-Hao Hsu, Martin Krupa, Charis Tsikkou and I can give a singular shock structure to Mazzotti's unusual chromatography equations.
    • 00988 (2/2) : 1D @G402 [Chair: Kaname Matsue]
      • [01810] Traveling wave solutions for certain 1D degenerate parabolic equation
        • Format : Talk at Waseda University
        • Author(s) :
          • Yu Ichida (Meiji University)
        • Abstract : In this talk, the speaker will report results on the classification of the traveling wave solutions of the 1D degenerate parabolic equation and porous medium equation, and give observations and suggestions on phenomena corresponding to bifurcations of equilibria at infinity. These are obtained through dynamical systems theory and Poincar\'e compactification. This talk includes a collaborative work with Professor Takashi Sakamoto at Meiji University.
      • [02005] Blow-up Rates for Solutions of a Quasi-Linear Parabolic Equation
        • Format : Talk at Waseda University
        • Author(s) :
          • Koichi Anada (Waseda University)
          • Tetsuya Ishiwata (Shibaura Institute of Technology)
          • Takeo Ushijima (Tokyo University of Science)
        • Abstract : The motion of curves by the power of their curvatures with positive exponent has been studied. The motion is described by a parabolic equation and some solutions blow up with Type II singularity. In this talk, we discuss the blow-up rates of solutions with Type II singularity. Precisely, we derive an asymptotic expansion of the traveling wave which plays a significant role and then we provide an upper estimate for the blow-up rates.
      • [04119] Rigorous numerics for finding the monodromy of Picard-Fuchs differential equations for a family of K3 toric hypersurfaces
        • Format : Talk at Waseda University
        • Author(s) :
          • Akitoshi Takayasu (University of Tsukuba)
          • Toshimasa Ishige (Chiba University)
        • Abstract : In this talk, we present a method for finding monodromy matrices of linear differential equations with finite-dimensional solution spaces via rigorous numerics. We also provide a computational result that gives monodromy matrices, which represent the fundamental group of the differential equation, of Picard-Fuchs differential equations for a certain family of K3 toric hypersurfaces.
      • [02492] Computation of collision and near-collision orbits in Celestial Mechanics problems.
        • Format : Talk at Waseda University
        • Author(s) :
          • Shane Kepley (Vrije Universiteit)
          • Jason Desmond Mireles James (Florida Atlantic University)
          • Maciej Capinski (AGH University of Science and Technology)
        • Abstract : Understanding connecting and collision/ejection orbits is central to the study of transport in Celestial Mechanics. Finding and validating connecting orbits can be difficult in general, and is complicated even more by the presence of ejection/collisions which are singularities of the flow. We present a rigorous Levi-Cevita regularization algorithm which, combined with existing analytic continuation techniques, allows us to overcome this obstruction. This regularization is performed dynamically allowing invariant manifolds to be parameterized dynamically and globally, even near singularities.