Blow-up behavior for ODEs with normally hyperbolic nature in dynamics at infinity
Abstract
We describe blow-up behavior for ODEs by means of dynamics at infinity with complex asymptotic behavior in autonomous systems, as well as in nonautonomous systems. Based on preceding studies, a variant of closed embeddings of phase spaces and the time-scale transformation determined by the structure of vector fields at infinity reduce our description of blow-ups to unravel the shadowing property of (pre)compact trajectories on the horizon, the geometric object expressing the infinity, with the specific convergence rates. Geometrically, this description is organized by asymptotic phase of invariant sets on the horizon. Blow-up solutions in nonautonomous systems can be described in a similar way. As a corollary, normally, or partially hyperbolic invariant manifolds on the horizon possessing asymptotic phase are shown to induce blow-ups.
Cite
@article{arxiv.2307.09201,
title = {Blow-up behavior for ODEs with normally hyperbolic nature in dynamics at infinity},
author = {Kaname Matsue},
journal= {arXiv preprint arXiv:2307.09201},
year = {2024}
}
Comments
Accepted version, which will be appeared in SIAM Journal on Applied Dynamical Systems. 42 pages, no figures, Organization as well as title are changed from v2