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Global aspects of homoclinic bifurcations of vector fields. (English) Zbl 0862.34042
Let $𝒳$ be a class of smooth vector fields on a smooth $n$-dimensional manifold $M$ such that every $X\in 𝒳$ admits a hyperbolic singular point $p$ with the weakest stable and unstable eigenvalues ${\lambda }^{s}$, ${\lambda }^{u}\in ℝ$ of multiciplity one and a homoclinic orbit ${\Gamma }:=\left\{{\Gamma }\left(t\right):t\in ℝ\right\}$ converging to this singularity in positive and negative time. Let ${X}_{\mu }$ be a one parameter family of vector fields on $M$ with ${X}_{0}\in 𝒳$. Assume that ${\lambda }^{s}+{\lambda }^{u}\ne 0$ and ${lim}_{t\to \infty }{{\Gamma }}^{\text{'}}\left(t\right)/|{{\Gamma }}^{\text{'}}\left(t\right)|=±{e}_{s}$, ${lim}_{t\to -\infty }{{\Gamma }}^{\text{'}}\left(t\right)/|{{\Gamma }}^{\text{'}}\left(t\right)|=±{e}_{u}$, where ${e}_{s}$ and ${e}_{u}$ denote the unit vectors in the eigenspaces of ${\lambda }_{s}$ and ${\lambda }_{u}$. Define the stable set of $\overline{{\Gamma }}$ as the set of points in $M$ whose $\omega$-limit set is contained in $\overline{{\Gamma }}$. It is first shown for generic families ${X}_{\mu }$ the existence of a two-dimensional invariant manifold (called centre manifold) which is tangent to the direct sum of the weak stable and unstable eigenspaces. Next it is assumed that the centre manifold is orientable and it is shown that if the strong unstable eigenspace of $p$ vanishes then the existence of a transversal intersection of the stable set of $\overline{{\Gamma }}$ and the unstable manifold of $p$ (a generalized homoclinc orbit) implies the occurrence of intermittency bifurcations. In particular in a small neighborhood $𝒱$ of $\overline{{\Gamma }}$ the vector field ${X}_{\mu }$ has a periodic attractor on one side of $\mu =0$ and on the other side an orbit of ${X}_{\mu }$ spends most of its time in $𝒱$ but traverses parts of the state space outside $𝒱$ for bounded intervals of time. The rest of the paper is devoted to the study of the case when the strong unstable manifold does not vanish. In particular it is shown that under certain conditions the set of bifurcation values may contain a Cantor set.

##### MSC:
 34C37 Homoclinic and heteroclinic solutions of ODE 34-02 Research monographs (ordinary differential equations) 34C23 Bifurcation (ODE) 37C10 Vector fields, flows, ordinary differential equations 37-99 Dynamic systems and ergodic theory (MSC2000) 37D45 Strange attractors, chaotic dynamics