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Stability of Singular Equilibria in Quasilinear Implicit Differential Equations

✍ Scribed by Ricardo Riaza; Pedro J. Zufiria

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
204 KB
Volume
171
Category
Article
ISSN
0022-0396

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✦ Synopsis

This paper addresses stability properties of singular equilibria arising in quasilinear implicit ODEs. Under certain assumptions, local dynamics near a singular point may be described through a continuous or directionally continuous vector field. This fact motivates a classification of geometric singularities into weak and strong ones. Stability in the weak case is analyzed through certain linear matrix equations, a singular version of the Lyapunov equation being especially relevant in the study. Weak stable singularities include singular zeros having a spherical domain of attraction which contains other singular points. Regarding strong equilibria, stability is proved via a Lyapunov Schmidt approach under additional hypotheses. The results are shown to be relevant in singular root-finding problems.

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On the approximation of singular source
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✍ Johan WaldΓ©n πŸ“‚ Article πŸ“… 1999 πŸ› John Wiley and Sons 🌐 English βš– 263 KB πŸ‘ 2 views

We study differential equations with singular source terms. For such equations classical convergence results do not apply, as these rely on the regularity of the solution and the source terms. We study some elliptic and parabolic problems numerically and theoretically, and show that, with the right