On loops in the hyperbolic locus of the complex Hénon map and their monodromies

Arai, Zin

doi:10.1016/j.physd.2016.02.006

Cited by 15 publications

(21 citation statements)

References 27 publications

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“…We also study several of the area-preserving Hénon maps in Section 5.1, which have been well-studied in the Physics community [12,23,7]. Recently, some precise rigorous results emerged as well [2]. We find that our rigorous lower bounds match or are very close to estimates given in previous work, and match the rigorous results exactly when applicable.…”

Section: Introductionsupporting

confidence: 74%

“…While we have computed a large array of lower bounds, covering a vast portion of the parameter space of the Hénon map, a recent result of Arai gives a method of rigorously computing exact entropy values for (uniformly) hyperbolic Hénon [2]. In fact, in that work he computes values for plateaus numbered 5, 7, 11, and 12 in Figure 4, which match our lower bounds precisely.…”

Section: Hyperbolic Plateaus Of Hénonmentioning

confidence: 66%

“…Given some compact set N ⊆ X, its maximal invariant set is the set of points x ∈ N such that there exists a trajectory of f through x which stays entirely within N for all forward and backward time. 2 A central concept of Conley index theory is that of isolation:…”

Section: Combinatorial Structures and The Discrete Conley Indexmentioning

confidence: 99%

“…The red horizontal line in each represents the maximum lower bound computed (also printed below each plot). Plots are omitted for 0-entropy plateaus (2,3,4,19,25,27,32,35,38). Proof.…”

Section: Hyperbolic Plateaus Of Hénonmentioning

confidence: 99%

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Topological Entropy Bounds for Hyperbolic Plateaus of the Henon Map

Frongillo¹

2014

SIURO

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Combining two existing rigorous computational methods, for verifying hyperbolicity (due to Arai) and for computing topological entropy bounds (due to Day et al.), we prove lower bounds on topological entropy for 43 hyperbolic plateaus of the Hénon map. We also examine the 16 area-preserving plateaus studied by Arai and compare our results with related work. Along the way, we augment the algorithms of Day et al. with routines to optimize the algorithmic parameters and simplify the resulting semi-conjugate subshift.

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Section: Introductionsupporting

confidence: 74%

Section: Hyperbolic Plateaus Of Hénonmentioning

confidence: 66%

Section: Combinatorial Structures and The Discrete Conley Indexmentioning

confidence: 99%

“…The red horizontal line in each represents the maximum lower bound computed (also printed below each plot). Plots are omitted for 0-entropy plateaus (2,3,4,19,25,27,32,35,38). Proof.…”

Section: Hyperbolic Plateaus Of Hénonmentioning

confidence: 99%

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Topological Entropy Bounds for Hyperbolic Plateaus of the Henon Map

Frongillo¹

2014

SIURO

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“…The Henon map takes a point ( n x , n y )n the plane and maps it to a new point. It can defined as followed [15]: (8) Figure 1 are respectively the chaotic phenomena of Sinusodial map, Logistic map, Tinkerbell map, Henon map and so on. The Sinusodial chaotic map with 251 iterations is showed in Figure 1a, and here the initial point of variable is .…”

Section: Sinusodial Mapmentioning

confidence: 99%

The Chaos and Stability of Firefly Algorithm Adjacent Individual

Wang

et al. 2017

TELKOMNIKA

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On Parameter Loci of the Hénon Family

Arai

Ishii

2018

Commun. Math. Phys.

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The purpose of the current article is to investigate the dynamics of the Hénon family f a,b : (x, y) → (x 2 − a − by, x), where (a, b) ∈ R × R × is the parameter [H]. We are interested in certain geometric and topological structures of two loci of parameters (a, b) ∈ R × R × for which f a,b share common dynamical properties; one is the hyperbolic horseshoe locus where the restriction of f a,b to its non-wandering set is hyperbolic and topologically conjugate to the full shift with two symbols, and the other is the maximal entropy locus where the topological entropy of f a,b attains the maximal value log 2 among all Hénon maps.The main result of this paper states that these two loci are characterized by the graph of a real analytic function from the b-axis to the a-axis of the parameter space R × R × , which extends in full generality the previous result of Bedford and Smillie [BS2] for |b| < 0.06. As consequences of this result, we show that (i) the two loci are both connected and simply connected in {b > 0} and in {b < 0}, (ii) the closure of the hyperbolic horseshoe locus coincides with the maximal entropy locus, (iii) the boundaries of both loci are identical and piecewise analytic with two analytic pieces. Among others, the consequence (i) indicates a weak form of monotonicity of the topological entropy as a function of the parameter (a, b) → htop(f a,b ) at its maximal value.The proof consists of both theoretical and computational parts. In the theoretical part we extend both the dynamical and the parameter spaces over C, investigate their complex dynamical and complex analytic properties, and reduce them to obtain the conclusion over R as in [BS2]. One of our new ingredients is to employ a flexible family of "boxes" in C 2 that are intrinsically two-dimensional and works for all values of b. In the computational part we use interval arithmetic together with some numerical algorithms such as set-oriented computations and the interval Krawczyk method to verify certain numerical criteria which imply analytic, combinatorial and dynamical consequences.

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On loops in the hyperbolic locus of the complex Hénon map and their monodromies

Cited by 15 publications

References 27 publications

Topological Entropy Bounds for Hyperbolic Plateaus of the Henon Map

Topological Entropy Bounds for Hyperbolic Plateaus of the Henon Map

The Chaos and Stability of Firefly Algorithm Adjacent Individual

On Parameter Loci of the Hénon Family

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