Second Minimal Orbits, Sharkovski Ordering and Universality in Chaos

Abdulla, Ugur G.; Abdulla, Rashad U.; Abdulla, Muhammad U.; Iqbal, Naveed

doi:10.1142/s021812741730018x

Cited by 3 publications

(2 citation statements)

References 21 publications

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“…Theorems 1.12 & 1.13 in the particular case k = 3 was proved in [2]. Proof of Theorems 1.12 & 1.13 is constructive, and provides explicit description of all types of second minimal odd orbits in terms of cyclic permutations and digraphs.…”

Section: Introduction and Main Resultsmentioning

confidence: 97%

Classification of the Second Minimal Odd Periodic Orbits in the Sharkovskii Ordering

Abdulla,

Abdulla

et al. 2017

Preprint

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This paper presents full classification of second minimal odd periodic orbits of a continuous endomorphisms on the real line. A (2k + 1)-periodic orbitin the Sharkovski ordering. We prove that second minimal odd orbits either have a Stefan structure like minimal odd orbits, or have one of the 4k − 3 types, each characterized with unique cyclic permutation and directed graph of transitions with accuracy up to inverses.This result played a fundamental role in the development of the theory of discrete dynamical systems. Following the standard approach ([7, 5, 3]), we characterize each periodic orbit with cyclic permutations and directed graphs of transitions or digraphs. Consider the m-orbit:

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Section: Introduction and Main Resultsmentioning

confidence: 97%

Classification of the Second Minimal Odd Periodic Orbits in the Sharkovskii Ordering

Abdulla,

Abdulla

et al. 2017

Preprint

Self Cite

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“…The crystallographic data with selected bond distances and angles for 1 are listed in Tables S1 and S2. Crystal data for 1: C 28 H 36 F 6 MnN 4 O 6 S 2 , monoclinic, P2/n, Z = 4, a = 11.119( 4), b = 17.531 (10), c = 17.329(7) Å, α = 90, β = 103.752 (12), γ = 90°, V = 3281(3) Å 3 , μ = 0.611 mm −1 , ρ calcd = 1.534 g/cm 3 , R 1 = 0.0468, wR 2 = 0.1440 for 8171 unique reflections, 424 variables. CCDC-2191811 for 1 contains the supplementary crystallographic data for this paper.…”

Section: ■ Introductionmentioning

confidence: 99%

Aliphatic and Aromatic C–H Bond Oxidation by High-Valent Manganese(IV)-Hydroxo Species

Lee

Tripodi²,

Jeong

et al. 2022

J. Am. Chem. Soc.

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The strong C−H bond activation of hydrocarbons is a difficult reaction in environmental and biological chemistry. Herein, a high-valent manganese(IV)-hydroxo complex, [Mn IV (CHDAP-O)(OH)] 2+ (2), was synthesized and characterized by various physicochemical measurements, such as ultraviolet−visible (UV−vis), electrospray ionization-mass spectrometry (ESI-MS), electron paramagnetic resonance (EPR), and helium-tagging infrared photodissociation (IRPD) methods. The one-electron reduction potential (E red ) of 2 was determined to be 0.93 V vs SCE by redox titration. 2 is formed via a transient green species assigned to a manganese(IV)-bis(hydroxo) complex, [Mn IV (CHDAP)(OH) 2 ] 2+ (2′), which performs intramolecular aliphatic C−H bond activation. The kinetic isotope effect (KIE) value of 4.8 in the intramolecular oxidation was observed, which indicates that the C−H bond activation occurs via rate-determining hydrogen atom abstraction. Further, complex 2 can activate the C−H bonds of aromatic compounds, anthracene and its derivatives, under mild conditions. The KIE value of 1.0 was obtained in the oxidation of anthracene. The rate constant (k et ) of electron transfer (ET) from N,N′-dimethylaniline derivatives to 2 is fitted by Marcus theory of electron transfer to afford the reorganization energy of ET (λ = 1.59 eV). The driving force dependence of log k et for oxidation of anthracene derivatives by 2 is well evaluated by Marcus theory of electron transfer. Detailed kinetic studies, including the KIE value and Marcus theory of outer-sphere electron transfer, imply that the mechanism of aromatic C−H bond hydroxylation by 2 proceeds via the rate-determining electron-transfer pathway.

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Minimal set of periods for continuous self-maps of the eight space

Llibre

Sá

2021

Fixed Point Theory Algorithms Sci Eng

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Let $G_{k}$ G k be a bouquet of circles, i.e., the quotient space of the interval $[0,k]$ [ 0 , k ] obtained by identifying all points of integer coordinates to a single point, called the branching point of $G_{k}$ G k . Thus, $G_{1}$ G 1 is the circle, $G_{2}$ G 2 is the eight space, and $G_{3}$ G 3 is the trefoil. Let $f: G_{k} \to G_{k}$ f : G k → G k be a continuous map such that, for $k>1$ k > 1 , the branching point is fixed.If $\operatorname{Per}(f)$ Per ( f ) denotes the set of periods of f, the minimal set of periods of f, denoted by $\operatorname{MPer}(f)$ MPer ( f ) , is defined as $\bigcap_{g\simeq f} \operatorname{Per}(g)$ ⋂ g ≃ f Per ( g ) where $g:G_{k}\to G_{k}$ g : G k → G k is homological to f.The sets $\operatorname{MPer}(f)$ MPer ( f ) are well known for circle maps. Here, we classify all the sets $\operatorname{MPer}(f)$ MPer ( f ) for self-maps of the eight space.

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Second Minimal Orbits, Sharkovski Ordering and Universality in Chaos

Cited by 3 publications

References 21 publications

Classification of the Second Minimal Odd Periodic Orbits in the Sharkovskii Ordering

Classification of the Second Minimal Odd Periodic Orbits in the Sharkovskii Ordering

Aliphatic and Aromatic C–H Bond Oxidation by High-Valent Manganese(IV)-Hydroxo Species

Minimal set of periods for continuous self-maps of the eight space

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