Existence of blenders in a Hénon-like family: geometric insights from invariant manifold computations

Hittmeyer, Stefanie; Krauskopf, Bernd; Osinga, Hinke M.; Shinohara, Katsutoshi

doi:10.1088/1361-6544/aacd66

Cited by 14 publications

(34 citation statements)

References 31 publications

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“…There are a number of related definitions of the concept of blender [3,4,5,6,7,8]; see also the discussion in [17,Sec. 2.1].…”

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confidence: 99%

“…Throughout this work, we follow [8] and use [6,Definition 6.11]. For the case of a diffeomorphism with a three-dimensional phase space, it can be stated as follows [17]: a hyperbolic set Λ of unstable index 2 is called a blender if there exists a C 1 -open set of curve segments in the three-dimensional phase space that each intersect the one-dimensional stable manifold W s (Λ) locally near Λ. Moreover, this property must be robust, that is, hold for the corresponding hyperbolic set of every sufficiently C 1 -close diffeomorphism.…”

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confidence: 99%

“…We present and extend here a complementary approach that was first suggested in [17] and applied to the study of a specific example of a family of diffeomorphisms with a blender. Its novel aspect is that we check and illustrate the carpet property directly by computing and visualizing (an approximation of) Λ and its one-dimensional stable manifold W s (Λ).…”

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confidence: 99%

“…Moreover, a numerical test for denseness in projection is presented, which is based on computing increasingly longer pieces of the respective one-dimensional global manifold. These techniques were applied in [17] to a specific example of a family of diffeomorphisms with a blender.…”

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confidence: 99%

“…Here, we use the same example that was introduced in [17], namely, the Hénonlike family H(x, y, z) = (y, µ + y 2 + βx, ξz + y),…”

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confidence: 99%

See 4 more Smart Citations

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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“…There are a number of related definitions of the concept of blender [3,4,5,6,7,8]; see also the discussion in [17,Sec. 2.1].…”

mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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confidence: 99%

“…Here, we use the same example that was introduced in [17], namely, the Hénonlike family H(x, y, z) = (y, µ + y 2 + βx, ξz + y),…”

mentioning

confidence: 99%

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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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Blender-horseshoes in Center-unstable Hénon-like Families

Díaz

Pérez

2019

New Trends in One-Dimensional Dynamics

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A blender-horseshoe is a locally maximal transitive hyperbolic set that appears in dimension at least three carrying a distinctive geometrical property: its local stable manifold "behaves" as a manifold of topological dimension greater than the expected one (the dimension of the stable bundle). This property persists under perturbations turning this kind of dynamics an important piece in the global description of robust non-hyperbolic systems. In this paper, we consider a parameterized family of center-unstable Hénonlike of endomorphisms in dimension three and show how blender-horseshoes naturally occur in a specific parameter range.

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Computing parametrised large intersection sets of 1D invariant manifolds: a tool for blender detection

C’Julio,

Krauskopf,

Osinga

2024

Numer Algor

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A dynamical system given by a diffeomorphism with a three-dimensional phase space may have a blender, which is a hyperbolic set $$\Lambda $$ Λ with, say, a one-dimensional stable invariant manifold that behaves like a surface. This means that the stable manifold of any fixed or periodic point in $$\Lambda $$ Λ weaves back and forth as a curve in phase space such that it is dense in some projection; we refer to this as the carpet property. We present a method for computing very long pieces of such a one-dimensional manifold so efficiently and accurately that a very large number of intersection points with a specified section can reliably be identified. We demonstrate this with the example of a family of Hénon-like maps $$\mathcal {H}$$ H on $$\mathbb {R}^3$$ R 3 , which is the only known, explicit example of a diffeomorphism with proven existence of a blender. The code for this example is available as a Matlab script as supplemental material. In contrast to earlier work, our method allows us to determine a very large number of intersection points of the respective one-dimensional stable manifold with a chosen planar section and render each as individual curves when a parameter is changed. With suitable accuracy settings, we not only compute these parametrised curves for the fixed points of $$\mathcal {H}$$ H over the relevant parameter interval, but we also compute the corresponding parametrised curves of the stable manifolds of a period-two orbit (with negative eigenvalues) and of a period-three orbit (with positive eigenvalues). In this way, we demonstrate that our algorithm can handle large expansion rates generated by (up to) the fourth iterate of $$\mathcal {H}$$ H .

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Existence of blenders in a Hénon-like family: geometric insights from invariant manifold computations

Cited by 14 publications

References 31 publications

How to identify a hyperbolic set as a blender

How to identify a hyperbolic set as a blender

Blender-horseshoes in Center-unstable Hénon-like Families

Computing parametrised large intersection sets of 1D invariant manifolds: a tool for blender detection

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