Displaying similar documents to “Persistent homoclinic tangencies and the unfolding of cycles”

Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles

Jacob Palis, Jean-Christophe Yoccoz (2009)

Publications Mathématiques de l'IHÉS

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In the present paper, we advance considerably the current knowledge on the topic of bifurcations of heteroclinic cycles for smooth, meaning C ∞, parametrized families {g t ∣t∈ℝ} of surface diffeomorphisms. We assume that a quadratic tangency q is formed at t=0 between the stable and unstable lines of two periodic points, not belonging to the same orbit, of a (uniformly hyperbolic) horseshoe K (see an example at the Introduction) and that such lines cross each other with positive relative...

Heterodimensional cycles, partial hyperbolicity and limit dynamics

L. J. Diaz, J. Rocha (2002)

Fundamenta Mathematicae

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We study one-parameter families of diffeomorphisms unfolding heterodimensional cycles (i.e. cycles containing periodic points of different indices). We construct an open set of such arcs such that, for a subset of the parameter space with positive relative density at the bifurcation value, the resulting nonwandering set is the disjoint union of two hyperbolic basic sets of different indices and a strong partially hyperbolic set which is robustly transitive. The dynamics of the diffeomorphisms...

The explosion of singular cycles

Rodrigo Bamon, Rafael Labarca, Ricardo Mañé, Maria-José Pacífico (1993)

Publications Mathématiques de l'IHÉS

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Simple examples of one-parameter planar bifurcations.

Armengol Gasull, Rafel Prohens (2000)

Extracta Mathematicae

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In this paper we give simple and low degree examples of one-parameter polynomial families of planar differential equations which present generic, codimension one, isolated, compact bifurcations. In contrast with some examples which appear in the usual text books each bifurcation occurs when the bifurcation parameter is zero. We study the total number of limit cycles that the examples present and we also make their phase portraits on the Poincaré sphere.