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Global stability of saddle-node bifurcation of a periodic orbit for vector fields. (English) Zbl 0831.58036
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 I. Malta and J. Palis [Lect. Notes Math. 898, 212-229 (1981; Zbl 0482.58023)] obtained for the case $$\dim M=2$$.

##### MSC:
 37G99 Local and nonlocal bifurcation theory for dynamical systems
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##### References:
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