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30 pages, 456 KB

Open AccessArticle

Classification of the Second Minimal Orbits in the Sharkovski Ordering

by Ugur G. Abdulla, Naveed H. Iqbal, Muhammad U. Abdulla and Rashad U. Abdulla

Axioms 2025, 14(3), 222; https://doi.org/10.3390/axioms14030222 - 17 Mar 2025

Viewed by 704

We prove a conjecture on the second minimal odd periodic orbits with respect to Sharkovski ordering for the continuous endomorphisms on the real line. A

(2 k + 1)

-periodic orbit [...] Read more.

We prove a conjecture on the second minimal odd periodic orbits with respect to Sharkovski ordering for the continuous endomorphisms on the real line. A

(2 k + 1)

-periodic orbit

{β_{1} < β_{2} < \dots < β_{2 k + 1}}

, (

k \geq 3

) is called second minimal for the map f, if

2 k - 1

is a minimal period of

{f |}_{[β_{1}, β_{2 k + 1}]}

in the Sharkovski ordering. Full classification of second minimal orbits is presented in terms of cyclic permutations and directed graphs of transitions. It is proved that second minimal odd orbits either have a Stefan-type structure like minimal odd orbits or one of the

4 k - 3

types, each characterized with unique cyclic permutations and directed graphs of transitions with an accuracy up to the inverses. The new concept of second minimal orbits and its classification have an important application towards an understanding of the universal structure of the distribution of the periodic windows in the bifurcation diagram generated by the chaotic dynamics of nonlinear maps on the interval. Full article

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14 pages, 723 KB

Open AccessArticle

Dynamic Injection and Permutation Coding for Enhanced Data Transmission

by Kehinde Ogunyanda, Opeyemi O. Ogunyanda and Thokozani Shongwe

Entropy 2024, 26(8), 685; https://doi.org/10.3390/e26080685 - 13 Aug 2024

Cited by 1 | Viewed by 1035

Abstract

In this paper, we propose a novel approach to enhance spectral efficiency in communication systems by dynamically adjusting the mapping between cyclic permutation coding (CPC) and its injected form. By monitoring channel conditions such as interference levels and impulsive noise strength, the system optimises the coding scheme to maximise data transmission reliability and efficiency. The CPC method employed in this work maps information bits onto non-binary symbols in a cyclic manner, aiming to improve the Hamming distance between mapped symbols. To address challenges such as low data rates inherent in permutation coding, injection techniques are introduced by removing

δ

column(s) from the CPC codebook. Comparative analyses demonstrate that the proposed dynamic adaptation scheme outperforms conventional permutation coding and injection schemes. Additionally, we present a generalised mathematical expression to describe the relationship between the spectral efficiencies of both coding schemes. This dynamic approach ensures efficient and reliable communication in environments with varying levels of interference and impulsive noise, highlighting its potential applicability to systems like power line communications. Full article

(This article belongs to the Special Issue New Advances in Error-Correcting Codes)

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14 pages, 289 KB

Open AccessArticle

Artificial Neural Networks Using Quiver Representations of Finite Cyclic Groups

by Lucky Cahya Wanditra, Intan Muchtadi-Alamsyah and Dellavitha Nasution

Symmetry 2023, 15(12), 2110; https://doi.org/10.3390/sym15122110 - 24 Nov 2023

Cited by 1 | Viewed by 1747

Abstract

In this paper, we propose using quiver representations as a tool for understanding artificial neural network algorithms. Specifically, we construct these algorithms by utilizing the group algebra of a finite cyclic group as vertices and convolution transformations as maps. We will demonstrate the neural network using convolution operation in the group algebra. The convolution operation in the group algebra that is formed by a finite cyclic group can be seen as a circulant matrix. We will represent a circulant matrix as a map from a cycle permutation matrix to a polynomial function. Using the permutation matrix, we will see some properties of the circulant matrix. Furthermore, we will examine some properties of circulant matrices using representations of finite symmetric groups as permutation matrices. Using the properties, we also examine the properties of moduli spaces formed by the actions of the change of basis group on the set of quiver representations. Through this analysis, we can compute the dimension of the moduli spaces. Full article

(This article belongs to the Special Issue Selected Papers of International Conference on Mathematical and Statistical Sciences: A Selçuk Meeting (Dedicated to Victims of Türkiye Earthquakes))

18 pages, 14201 KB

Open AccessArticle

Image Encryption Using a New Hybrid Chaotic Map and Spiral Transformation

by Mingfang Jiang and Hengfu Yang

Entropy 2023, 25(11), 1516; https://doi.org/10.3390/e25111516 - 5 Nov 2023

Cited by 16 | Viewed by 4156

Abstract

Image encryption based on chaotic maps is an important method for ensuring the secure communication of digital multimedia on the Internet. To improve the encryption performance and security of image encryption systems, a new image encryption algorithm is proposed that employs a compound chaotic map and random cyclic shift. First, a new hybrid chaotic system is designed by coupling logistic, ICMIC, Tent, and Chebyshev (HLITC) maps. Comparison tests with previous chaotic maps in terms of chaotic trajectory, Lyapunov exponent, and approximate entropy illustrate that the new hybrid chaotic map has better chaotic performance. Then, the proposed HLITC chaotic system and spiral transformation are used to develop a new chaotic image encryption scheme using the double permutation strategy. The new HLITC chaotic system is used to generate key sequences used in the image scrambling and diffusion stages. The spiral transformation controlled by the chaotic sequence is used to scramble the pixels of the plaintext image, while the XOR operation based on a chaotic map is used for pixel diffusion. Extensive experiments on statistical analysis, key sensitivity, and key space analysis were conducted. Experimental results show that the proposed encryption scheme has good robustness against brute-force attacks, statistical attacks, and differential attacks and is more effective than many existing chaotic image encryption algorithms. Full article

(This article belongs to the Special Issue Cryptography and Data Security Based on Information Theory)

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9 pages, 259 KB

Open AccessArticle

Distributional Chaos and Sensitivity for a Class of Cyclic Permutation Maps

by Yu Zhao, Waseem Anwar, Risong Li, Tianxiu Lu and Zhiwen Mo

Mathematics 2023, 11(15), 3310; https://doi.org/10.3390/math11153310 - 27 Jul 2023

Cited by 1 | Viewed by 1009

Abstract

Several chaotic properties of cyclic permutation maps are considered. Cyclic permutation maps refer to p-dimensional dynamical systems of the form [...] Read more.

Several chaotic properties of cyclic permutation maps are considered. Cyclic permutation maps refer to p-dimensional dynamical systems of the form

φ (b_{1}, b_{2}, \dots, b_{p}) = (u_{p} (b_{p}), u_{1} (b_{1}), \dots, u_{p - 1} (b_{p - 1})),

where

b_{j} \in H_{j}

(

j \in {1, 2, \dots, p}

p \geq 2

is an integer, and

H_{j}

(

j \in {1, 2, \dots, p}

) are compact subintervals of the real line

R = (- \infty, + \infty)

u_{j} : H_{j} \to H_{j + 1} (j = 1, 2, \dots, p - 1)

and

u_{p} : H_{p} \to H_{1}

are continuous maps. Necessary and sufficient conditions for a class of cyclic permutation maps to have Li–Yorke chaos, distributional chaos in a sequence, distributional chaos, or Li–Yorke sensitivity are given. These results extend the existing ones. Full article

(This article belongs to the Special Issue Applied Mathematical Modelling and Dynamical Systems)

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