分野別セミナー

The main concern in the present talk is blow-up solutions for autonomous ordinary differential equations (ODEs). The main ideas to characterize blow-ups are compactifications of phase spaces associated with asymptotic quasi-homogeneity of vector fields and time-scale desingularization at infinity to obtaining the desingularized vector field, in which case blow-up solutions of the original ODE correspond to trajectories on stable manifolds of invariant sets on the geometric object expressing the infinity in the compactified phase space, which is referred to as the horizon. As a consequence, dynamical properties of invariant sets on the horizon characterize blow-up behavior. Here theoretical results in a series of studies, such as a blow-up criterion, hyperbolicity and blow-up rates, and multiple-order asymptotic expansions of blow-ups, are mainly focused. If the time permits, numerical results (rigorous numerics) are also mentioned.


References:
[1] K. Matsue, SIAM Journal on Applied Dynamical Systems, 17(3):2249--2288, 2018.
[2] K. Matsue, Journal of Differential Equations, 267(12):7313--7368, 2019.
[3] T. Asai, H. Kodani, K. Matsue, H. Ochiai, T. Sasaki and A. Takayasu, in preparation.(Rigorous Numerics)
[4] K. Matsue and A. Takayasu, Numerische Mathematik, 145:605--654, 2020.
[5] K. Matsue and A. Takayasu, Journal of Computational and Applied Mathematics, 374:112607, 2020.
[6] A. Takayasu, K. Matsue, T. Sasaki, K. Tanaka, M. Mizuguchi, and S. Oishi, Journal of Computational and Applied Mathematics, 314:10--29, 2017.
[7] J.-P. Lessard, K. Matsue, and A. Takayasu, arXiv:2103.12390, 2021.