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Structural stability and generic properties of planar polynomial vector fields. (English) Zbl 0677.58029

Denote by \({\mathcal P}_ n\) the set of polynomial vector fields of degree \(\leq n\) on \({\mathbb{R}}^ 2\). The present paper is concerned with structural stability of elements of \({\mathcal P}_ n\) both with respect to polynomial perturbations (coefficient topology) as well as with respect to general \(C^ k\) perturbations, \(k\geq 1\) (Whitney topology). For \(X\in {\mathcal P}_ n\) denote by \(\pi\) (X) the Poincaré vector field induced by X on \(S^ 2\). Firstly, a general genericity result is proved, namely openness and denseness of the set of polynomial vector fields X satisfying the following conditions: The Poincaré vector field \(\pi\) (X) has only finitely many critical points and closed orbits, all hyperbolic, \(\pi\) (X) has no saddle connections except at infinity, and \(\Omega (X)=Per(X)\), where \(\Omega\) (X) and Per(X) are the nonwandering and the periodic points, resp., of X. Next, a criterion for structural stability of \(X\in {\mathcal P}_ n\) with respect to perturbations in \({\mathcal P}_ n\) is derived in terms of conditions on the orbit structures of X and \(\pi\) (X). These conditions are then proved to be necessary also, provided X has no non-hyperbolic limit cycles of odd multiplicity. The existence of an open and dense subset of \({\mathcal P}_ n\) all of whose elements are structurally stable (with respect to either \(C^ k\) or \({\mathcal P}_ n\) perturbations) is then deduced from the general genericity result. Finally, the author generalizes a result proved by Peixoto for compact spaces: In the definition of structural stability of \(X\in {\mathcal P}_ n\) with respect to \(C^ k\) perturbations, \(k\geq 1\), the requirement that the equivalence inducing homeomorphism lies in a neighbourhood of \(id_{{\mathbb{R}}^ 2}\) is redundant.
Reviewer: H.Crauel

MSC:

37C75 Stability theory for smooth dynamical systems
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
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