Stability of singular equilibria in quasilinear implicit differential equations (Q5933460)

Stability properties of singular equilibria arising in quasilinear implicit ordinary differential equations of the form NEWLINE\[NEWLINEA(x)\dot x= f(x),\tag{1}NEWLINE\]NEWLINE with \(\Lambda\in C^k(\mathbb{R}^n, \mathbb{R}^{n\times n})\), \(\mathbb{R}^{n\times n}\) being the set of all real \(n\times n\)-matrices, \(f\in C^1(\mathbb{R}^n, \mathbb{R}^n)\), with \(k,l\geq 2\) are investigated.NEWLINENEWLINENEWLINEFrom a mathematical point of view, it is of interest to investigate the dynamic behavior of (1) near singular points \(x^*\), where \(A(x^*)\) is noninvertible. If \(A(x)\) has constant rank \(r< n\) on a neighborhood of \(x^*\), the equation can be reduced to a regular system in the theory of differential-algebraic equations.NEWLINENEWLINENEWLINEHere, the interest is focused on the case in which \(A(x)\) is singular on a hypersurface including \(x^*\). Under this hypothesis, existence of solutions, normal forms and phase portraits are analyzed and under certain assumptions local dynamics near a singular point may be described by a continuous or directionally continuous vector field. This fact motivates a classification of geometric singularities into weak and strong ones. The stability in the weak case is analyzed through certain linear matrix equations, a singular version of the Lyapunov equation. Weak stable singularities include zeros having a spherical domain of attraction that contains other singular points. Regarding strong equilibria, the stability is proved via a Lyapunov-Schmidt approach under additional hypotheses. The results are shown to be relevant in singular root-finding problems.