Structurally stable differential systems of the form \(A(x)x'=F(x)\)
Consider the implicit system \[ A(x)x'=f(x)\tag{1} \] on a compact plane region \(M\subset\mathbb R^2\) with smooth boundary. The entries of the \(2\times 2\) matrix \(A\) and \(f\) are assumed to be of class \(C^r\). The set \({\mathcal S}_A\) consists of points in the plane, where \(\det A(x)\) vanishes (impass points of (1)). The phase portrait of (1) is an ordered pair whose first element is \({\mathcal S}_A\) and the second element is the phase portrait of \[ x'=A^{-1}(x)f(x),\quad x\in M\setminus{\mathcal S}_A. \] The author introduces the notion of topological equivalence of two systems \((A,f)\) and \((\widetilde A,\widetilde f)\) defined by (1). He endows this class of systems with the \(C^r\) topology on \({\mathcal M}\) yielding a Banach space \(Z^r(M)\). By this way, the author extends the fundamental structural stability results due to Andronov-Pontrjagin and Peixoto-Peixoto to the class of considered systems (also called descriptor systems).
scientific article; zbMATH DE number 1158170