Global stability of saddle-node bifurcation of a periodic orbit for vector fields
In a bifurcation value, the global stability of families of \(C^\infty\) vector fields, which have a generically unfolding saddle-node periodic orbit is studied. Let \(M\) be a \(C^\infty\) closed manifold, \({\mathcal X}^\infty (M)\) be the space of \(C^\infty\) vector fields on \(M\) and \({\mathcal X}_1^\infty (M)\) be the space of \(C^\infty\) arcs \(\xi: I=[-1, 1]\to {\mathcal X}^\infty (M)\) of vector fields, both endowed with the \(C^\infty\) Whitney topology. Assuming that the arc \(\xi= \{X_\mu \}\), \(\mu\in I\), satisfies the properties: (1) for each \(\mu\in I\), the non-wandering set of \(X_\mu\) consists of a finite number of critical elements (i.e. singularities and periodic orbits) of \(X_\mu\), (2) there are no cycles among the critical elements of \(X_\mu\), (3) there is \(\overline {\mu}\in I\) such that \(X_{\overline {\mu}}\) has a saddle-node periodic orbit, under certain additional conditions, the author obtains results on the global stability, non-stability of \(\{X_\mu\}\) at a bifurcation value \(\overline {\mu}\) and a modulus of stability for \(X_{\overline {\mu}}\). The results of the paper generalize some results of \textit{I. Malta} and \textit{J. Palis} [Lect. Notes Math. 898, 212-229 (1981; Zbl 0482.58023)]\ obtained for the case \(\dim M=2\).
scientific article; zbMATH DE number 778924
scientific article