Hénon-Like Families and Blender-Horseshoes at Nontransverse Heterodimensional Cycles

Díaz, Lorenzo J.; Pérez, Sebastián A.

doi:10.1142/s0218127419300064

Cited by 10 publications

(19 citation statements)

References 38 publications

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“…We remark that our way of representing a blender-as the intersection set of certain stable and unstable manifolds that also convey the carpet property-is complementary to illustrating the threedimensional blender-horseshoe construction itself. Illustrations in [3] and, especially, the three-dimensional sketches in [9], show how a box in phase space maps back to itself under the map in forward and backward time; under suitable conditions, the limit of repeating this process is a blender. In contrast, we present here images of global one-dimensional manifolds that illustrate the carpet property of the hyperbolic set.…”

Section: The Figures and Their Different Sub-panels That We Present Amentioning

confidence: 99%

How to identify a hyperbolic set as a blender

Hittmeyer¹,

Krauskopf²,

Osinga³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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A blender is a hyperbolic set with a stable or unstable invariant manifold that behaves as a geometric object of a dimension larger than that of the respective manifold itself. Blenders have been constructed in diffeomorphisms with a phase space of dimension at least three. We consider here the question of how one can identify, characterize and also visualize the underlying hyperbolic set of a given diffeomorphism to verify whether it actually is a blender or not. More specifically, we employ advanced numerical techniques for the computation of global manifolds to identify the hyperbolic set and its stable and unstable manifolds in an explicit Hénon-like family of three-dimensional diffeomorphisms. This allows to determine and illustrate whether the hyperbolic set is a blender; in particular, we consider as a distinguishing feature the self-similar structure of the intersection set of the respective global invariant manifold with a plane. By checking and illustrating a denseness property, we are able to identify a parameter range over which the hyperbolic set is a blender, and we discuss and illustrate how the blender disappears.

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Section: The Figures and Their Different Sub-panels That We Present Amentioning

confidence: 99%

How to identify a hyperbolic set as a blender

Hittmeyer¹,

Krauskopf²,

Osinga³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…Remark 2. A partial case of Theorem 1 can be derived from the result in [17] about renormalization near heterodimensional cycles of three-dimensional diffeomorphisms with two saddle-foci. For two-dimensional endomorphisms under the partial hyperbolicity condition, the result of Theorem 1 is Theorem B in [7].…”

Section: Introductionmentioning

confidence: 99%

Persistence of Heterodimensional Cycles

Li¹,

Turaev²

2021

Preprint

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A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least C 2 , we show that bifurcations of a coindex-1 heterodimensional cycle within a generic 2-parameter family always create robust heterodimensional dynamics, i.e., chain-transitive sets which contain coexisting orbits with different numbers of positive Lyapunov exponents and persist for an open set of parameter values. In particular, we solve the so-called C r -stabilization problem for the coindex-1 heterodimensional cycles in any regularity class r = 2, . . . , ∞, ω. The results are based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-Díaz blenders.

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“…Here we rely on preliminary results in [16,17] towards the development of the strategy (a')-(d'). In our context, the "limit" family is the center-unstable Hénonlike family given by G (x, y, z) def = (y, µ + y 2 + η 1 y z + η 2 z 2 , ξ z + y), = (ξ, µ, η 1 , η 2 ).…”

Section: Introductionmentioning

confidence: 99%

“…For diffeomorphisms in H r BH (M ), the renormalisation scheme and its convergence to the family G (steps (a') and (b')). were obtained in [16]. A crucial property (step (c')) is that there is an open set O BH of parameters for which the family G exhibits blender-horseshoes, see [17].…”

Section: Introductionmentioning

confidence: 99%

Nontransverse heterodimensional cycles: stabilisation and robust tangencies

Díaz¹,

Pérez²

2020

Preprint

Self Cite

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We consider three-dimensional diffeomorphisms having simultaneously heterodimensional cycles and heterodimensional tangencies associated to saddle-foci. These cycles lead to a completely nondominated bifurcation setting. For every r 2, we exhibit a class of such diffeomorphisms whose heterodimensional cycles can be C r stabilised and (simultaneously) approximated by diffeomorphisms with C r robust homoclinic tangencies. The complexity of our nondominated setting with plenty of homoclinic and heteroclinic intersections is used to overcome the difficulty of performing C r perturbations, r 2, which are remarkably more difficult than C 1 ones. Our proof is reminiscent of the Palis-Takens' approach to get surface diffeomorphisms with infinitely many sinks (Newhouse phenomenon) in the unfolding of homoclinic tangencies of surface diffeomorphisms. This proof involves a scheme of renormalisation along nontransverse heteroclinic orbits converging to a center-unstable Hénonlike family displaying blender-horseshoes. A crucial step is the analysis of the embeddings of these blender-horseshoes in a nondominated context.

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Hénon-Like Families and Blender-Horseshoes at Nontransverse Heterodimensional Cycles

Cited by 10 publications

References 38 publications

How to identify a hyperbolic set as a blender

How to identify a hyperbolic set as a blender

Persistence of Heterodimensional Cycles

Nontransverse heterodimensional cycles: stabilisation and robust tangencies

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