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Saddle-Type Blow-Up Solutions with Computer-Assisted Proofs: Validation and Extraction of Global Nature.
- Source :
-
Journal of Nonlinear Science . Jun2023, Vol. 33 Issue 3, p1-76. 76p. - Publication Year :
- 2023
-
Abstract
- In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery (e.g., compactifications, timescale desingularizations of vector fields) with tools from computer-assisted proofs (e.g., rigorous integrators, the parameterization method for invariant manifolds), these blow-up solutions are obtained as trajectories on local stable manifolds of hyperbolic saddle equilibria at infinity. With the help of computer-assisted proofs, global trajectories on stable manifolds, inducing blow-up solutions, provide a global picture organized by global-in-time solutions and blow-up solutions simultaneously. Using the proposed methodology, intrinsic features of saddle-type blow-ups are observed: locally smooth dependence of blow-up times on initial points, level set distribution of blow-up times and decomposition of the phase space playing a role as separatrixes among solutions, where the magnitude of initial points near those blow-ups does not matter for asymptotic behavior. Finally, singular behavior of blow-up times on initial points belonging to different family of blow-up solutions is addressed. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 09388974
- Volume :
- 33
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Journal of Nonlinear Science
- Publication Type :
- Academic Journal
- Accession number :
- 162521710
- Full Text :
- https://doi.org/10.1007/s00332-023-09900-6