On the structure of the set of bifurcation points for ordinary differential equations
Suppose \(\Phi: \mathbb{R}^ 2\to \mathbb{R}\) and \(G: \mathbb{R}^ 2\to \mathbb{R}\) are analytic, \(\Phi(0)= G(0) =0\), and 0 is an isolated singular point of \(\Phi\). Let \(b_ + (G,\Phi)\) (resp. \(b_ -(G,\Phi)\)) be the number of branches of \(\Phi^{-1}(0)\) through 0 on which \(G\) is positive (resp. negative). Define \(H: \mathbb{R}^ 2\to \mathbb{R}^ 2\) by \(H(\lambda)= (\text{det } D(G,\Phi) (\lambda), \Phi(\lambda))\). Then \textit{Z. Szafraniec} [On the number of branches of a 1-dimensional semianalytic set, Kodai Math. J. 11, 78-85 (1988; Zbl 0652.58011)] showed: Theorem. If \(0\in \mathbb{R}^ 2\) is isolated in \(G^{-1}(0)\cap \Phi^{- 1}(0)\), then (1) 0 is isolated in \(H^{-1} (0)\), (2) \(b_ + (G,\Phi)- b_ -(G, \Phi)=2\cdot \deg H\), where \(\deg H\) denotes the topological degree of \(H\) with respect to a small disc around 0 in \(\mathbb{R}^ 2\). By taking various choices of \(G\), one can use this result to count, e.g., the number of branches of \(\Phi^{-1}(0)\) in each quadrant of \(\mathbb{R}^ 2\). The authors use it to study the bifurcation of solutions of the boundary value problem (BVP) \(\dot x= A(\lambda) x(t)+ \varphi(t,x(t), \lambda)\), \(x(0)= x(1)\) with \(x\in \mathbb{R}^ n\), \(\lambda\in \mathbb{R}^ 2\), \(\varphi(t,0,\lambda) =0\), \(D_ x \varphi(t, 0,\lambda) =0\), under the assumption that 0 is an isolated singular point of \(\Phi(\lambda)= \text{det } A(\lambda)\). (BVP) has the trivial solution \(x=0\) for each \(\lambda\), and its bifurcation set is \(\Phi^{-1}(0)\).
scientific article; zbMATH DE number 553871