Chaotic behaviour of the map x ↦ ω(x, f)

2014 ◽  
Vol 12 (4) ◽  
Author(s):  
Emma D’Aniello ◽  
Timothy Steele
Keyword(s):  
Hausdorff Metric ◽  
Chaotic Behaviour ◽  
Limit Set ◽  
Cantor Space ◽  
Compact Subsets

AbstractLet K(2ℕ) be the class of compact subsets of the Cantor space 2ℕ, furnished with the Hausdorff metric. Let f ∈ C(2ℕ). We study the map ω f: 2ℕ → K(2ℕ) defined as ω f (x) = ω(x, f), the ω-limit set of x under f. Unlike the case of n-dimensional manifolds, n ≥ 1, we show that ω f is continuous for the generic self-map f of the Cantor space, even though the set of functions for which ω f is everywhere discontinuous on a subsystem is dense in C(2ℕ). The relationships between the continuity of ω f and some forms of chaos are investigated.

scholarly journals Li-Yorke Sensitivity of Set-Valued Discrete Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Heng Liu ◽  
Fengchun Lei ◽  
Lidong Wang
Keyword(s):  
Metric Space ◽  
Short Proof ◽  
Hausdorff Metric ◽  
Discrete Systems ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Nonempty Compact ◽  
Compact Subsets

Consider the surjective, continuous mapf:X→Xand the continuous mapf¯of𝒦(X)induced byf, whereXis a compact metric space and𝒦(X)is the space of all nonempty compact subsets ofXendowed with the Hausdorff metric. In this paper, we give a short proof that iff¯is Li-Yoke sensitive, thenfis Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity offdoes not imply Li-Yorke sensitivity off¯.

Fermat-Steiner problem in the space of compact subsets of [IMG align=ABSMIDDLE alt=$ \mathbb R^m$]tex_sm_4905_img1[/IMG], endowed with the Hausdorff metric

2021 ◽  
Vol 212 (1) ◽  
Author(s):  
Arsen Khachaturovich Galstyan ◽  
Alexandr Olegovich Ivanov ◽  
Alexey Avgustinovich Tuzhilin
Keyword(s):  
Hausdorff Metric ◽  
Steiner Problem ◽  
Compact Subsets

scholarly journals Some Chaotic Properties of Discrete Fuzzy Dynamical Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Yaoyao Lan ◽  
Qingguo Li ◽  
Chunlai Mu ◽  
Hua Huang
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Continuous Map ◽  
Weakly Mixing ◽  
Dynamical Properties ◽  
Fuzzy Dynamical Systems ◽  
Compact Subsets

Letting(X,d)be a metric space,f:X→Xa continuous map, and(ℱ(X),D)the space of nonempty fuzzy compact subsets ofXwith the Hausdorff metric, one may study the dynamical properties of the Zadeh's extensionf̂:ℱ(X)→ℱ(X):u↦f̂u. In this paper, we present, as a response to the question proposed by Román-Flores and Chalco-Cano 2008, some chaotic relations betweenfandf̂. More specifically, we study the transitivity, weakly mixing, periodic density in system(X,f), and its connections with the same ones in its fuzzified system.

scholarly journals A note on compact sets in spaces of subsets

1988 ◽  
Vol 38 (3) ◽  
pp. 393-395 ◽  
Author(s):  
Phil Diamond ◽  
Peter Kloeden
Keyword(s):  
Metric Space ◽  
Complete Metric Space ◽  
Hausdorff Metric ◽  
Compact Sets ◽  
Selection Theorem ◽  
Starshaped Sets ◽  
Nonempty Compact ◽  
Compact Subsets

A simple characterisation is given of compact sets of the space K(X), of nonempty compact subsets of a complete metric space X, with the Hausdorff metric dH. It is used to give a new proof of the Blaschke selection theorem for compact starshaped sets.

THE CRAB: A CONNECTED FRACTILE OF INFINITE CONNECTIVITY

Fractals ◽  
2011 ◽  
Vol 19 (03) ◽  
pp. 367-377 ◽  
Author(s):  
GILBERT HELMBERG
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Metric ◽  
Unit Interval ◽  
Limit Set ◽  
Space Filling ◽  
Polygonal Path ◽  
Space Filling Curve ◽  
Dimension Formula ◽  
Filling Curve ◽  
Similar Copy

In the plane IR2, let A0 be the unit interval on the x-axis, and let A(1) be the polygonal path with nodes (0, 0), [Formula: see text], (½, 0), [Formula: see text], (1, 0). Let S be the operator which, applied to a segment B(0) in IR2, replaces it by a polygonal path B(1) = SB(0), a similar copy of A(1), but with the same endpoints as B(0). Denote by S(n) the n-th iterate of S. The limit set (with respect to the Hausdorff metric) A(∞) = lim n → ∞ S(n)A(0) is a space-filling curve which is the closure of its interior and the union of four half-size copies of itself, intersecting only in their boundaries. Although A(∞) is of infinite connectivity, it is a tile tessellating the plane. It is related to the set of Eisenstein fractions and has a boundary of Hausdorff dimension [Formula: see text]

On multi-sensitivity with respect to a vector

2018 ◽  
Vol 32 (15) ◽  
pp. 1850166 ◽  
Author(s):  
Lixin Jiao ◽  
Lidong Wang ◽  
Fengquan Li ◽  
Heng Liu
Keyword(s):  
Dynamical System ◽  
Metric Space ◽  
Hausdorff Metric ◽  
Sufficient Conditions ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Continuous Map ◽  
Periodic Sets ◽  
Vector Formula ◽  
Compact Subsets

Consider the surjective continuous map [Formula: see text]: [Formula: see text] defined on a compact metric space X. Let [Formula: see text] be the space of all non-empty compact subsets of X equipped with the Hausdorff metric and define [Formula: see text]: [Formula: see text] by [Formula: see text] for any [Formula: see text]. In this paper, we introduce several stronger versions of sensitivities, such as multi-sensitivity with respect to a vector, [Formula: see text]-sensitivity, strong multi-sensitivity. We obtain some basic properties of the concepts of these sensitivities and discuss the relationships with other sensitivities for continuous self-map on [0,[Formula: see text]1]. Some sufficient conditions for a dynamical system to be [Formula: see text]-sensitive are presented. Also, it is shown that the strong multi-sensitivity of f implies that [Formula: see text] is [Formula: see text]-sensitive. In turn, the [Formula: see text]-sensitivity of [Formula: see text] implies that [Formula: see text] is [Formula: see text]-sensitive. In particular, it is proved that if [Formula: see text] is a multi-transitive map with dense periodic sets, then f is [Formula: see text]-sensitive. Finally, we give a multi-sensitive example which is not [Formula: see text]-sensitive.

scholarly journals Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n-Valued Impulses

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1701
Author(s):  
Jan Andres
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Vector Fields ◽  
Hausdorff Metric ◽  
Invariant Sets ◽  
Devaney’S Chaos ◽  
Nielsen Numbers ◽  
Multiple Periodic Solutions ◽  
Translation Operators ◽  
Compact Subsets

Ordinary differential equations with n-valued impulses are examined via the associated Poincaré translation operators from three perspectives: (i) the lower estimate of the number of periodic solutions on the compact subsets of Euclidean spaces and, in particular, on tori; (ii) weakly locally stable (i.e., non-ejective in the sense of Browder) invariant sets; (iii) fractal attractors determined implicitly by the generating vector fields, jointly with Devaney’s chaos on these attractors of the related shift dynamical systems. For (i), the multiplicity criteria can be effectively expressed in terms of the Nielsen numbers of the impulsive maps. For (ii) and (iii), the invariant sets and attractors can be obtained as the fixed points of topologically conjugated operators to induced impulsive maps in the hyperspaces of the compact subsets of the original basic spaces, endowed with the Hausdorff metric. Five illustrative examples of the main theorems are supplied about multiple periodic solutions (Examples 1–3) and fractal attractors (Examples 4 and 5).

Compositional Analysis of III-V Interface Lattice Images

1980 ◽  
Vol 38 ◽  
pp. 318-319
Author(s):  
A. Olsen ◽  
J.C.H. Spence ◽  
P. Petroff
Keyword(s):  
Crystal Structure ◽  
Image Matching ◽  
Compositional Analysis ◽  
Depth Of Field ◽  
Limit Set ◽  
High Contrast ◽  
Resolution Limit ◽  
Fourier Image ◽  
Lattice Images ◽  
True Structure

Since the point resolution of the JEOL 200CX electron microscope is up = 2.6Å it is not possible to obtain a true structure image of any of the III-V or elemental semiconductors with this machine. Since the information resolution limit set by electronic instability (1) u0 = (2/πλΔ)½ = 1.4Å for Δ = 50Å, it is however possible to obtain, by choice of focus and thickness, clear lattice images both resembling (see figure 2(b)), and not resembling, the true crystal structure (see (2) for an example of a Fourier image which is structurally incorrect). The crucial difficulty in using the information between Up and u0 is the fractional accuracy with which Af and Cs must be determined, and these accuracies Δff/4Δf = (2λu2Δf)-1 and ΔCS/CS = (λ3u4Cs)-1 (for a π/4 phase change, Δff the Fourier image period) are strongly dependent on spatial frequency u. Note that ΔCs(up)/Cs ≈ 10%, independent of CS and λ. Note also that the number n of identical high contrast spurious Fourier images within the depth of field Δz = (αu)-1 (α beam divergence) decreases with increasing high voltage, since n = 2Δz/Δff = θ/α = λu/α (θ the scattering angle). Thus image matching becomes easier in semiconductors at higher voltage because there are fewer high contrast identical images in any focal series.

Nonlinear dynamics of a shell encapsulated microbubble near solid boundary in an ultrasonic field

2020 ◽  
Vol 14 (3) ◽  
pp. 7235-7243
Author(s):  
N.M. Ali ◽  
F. Dzaharudin ◽  
E.A. Alias
Keyword(s):  
Oscillation Amplitude ◽  
Ultrasonic Field ◽  
Dynamical Behaviour ◽  
Ultrasound Contrast Agents ◽  
Chaotic Behaviour ◽  
Solid Boundary ◽  
Driving Frequency ◽  
Pressure Amplitude ◽  
Radial Expansion ◽  
Therapeutic Delivery

Microbubbles have the potential to be used for diagnostic imaging and therapeutic delivery. However, the transition from microbubbles currently being used as ultrasound contrast agents to achieve its’ potentials in the biomedical field requires more in depth understanding. Of particular importance is the influence of microbubble encapsulation of a microbubble near a vessel wall on the dynamical behaviour as it stabilizes the bubble. However, many bubble studies do not consider shell encapsulation in their studies. In this work, the dynamics of an encapsulated microbubble near a boundary was studied by numerically solving the governing equations for microbubble oscillation. In order to elucidate the effects of a boundary to the non-linear microbubble oscillation the separation distances between microbubble will be varied along with the acoustic driving. The complex nonlinear vibration response was studied in terms of bifurcation diagrams and the maximum radial expansion. It was found that the increase in distance between the boundary and the encapsulated bubble will increase the oscillation amplitude. When the value of pressure amplitude increased the single bubble is more likely to exhibit the chaotic behaviour and maximum radius also increase as the inter wall-bubble distance is gradually increased. While, with higher driving frequency the maximum radial expansion decreases and suppress the chaotic behaviour.

Influence of Applied Phytosanitary Treatments and Climatic Crop Year on Zinc Content in Wines from Tohani Vineyard

2017 ◽  
Vol 68 (9) ◽  
pp. 2092-2097
Author(s):  
Catalina Calin ◽  
Gina Vasile Scaeteanu ◽  
Roxana Maria Madjar ◽  
Otilia Cangea
Keyword(s):  
Zinc Content ◽  
Soil Samples ◽  
Limit Set ◽  
Red Wines ◽  
Metallic Ions ◽  
Sauvignon Blanc ◽  
Before And After ◽  
Haze Formation ◽  
Zinc Levels ◽  
Concentration Levels

Metallic ions present a great importance in oenological practice and usually are present in wines in levels that are not hazardous. Among all metallic ions, zinc presents a great interest because may cause the persistence of the wine sour taste and by the side of Al, Cu, Fe and Ni, contribute to the haze formation and the change of color. The present study was focused on measuring the concentration levels of mobile zinc from vineyard soil before and after phytosanitary treatments and zinc content from white (Feteasca Alba - FA, Riesling Italian - RI, Sauvignon Blanc - SB, Tamaioasa Rom�neasca - TR), rose (Busuioaca de Bohotin - BB) and red (Feteasca Neagra - FN) wines within the wine-growing Tohani area, Romania. Other objective was to investigate of the influence of crop year and variety on zinc levels found in wine samples. Mobile zinc content for all analyzed soil samples is low ([1.5 mg/kg). Analyses indicated that zinc content found in wines was below 5 mg/L, limit set by Organisation Internationale of Vine and Wine (OIV). Also, it was found that red wines contain zinc in higher concentrations than white ones.

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