Research

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12 pages, 5177 KiB

Open AccessArticle

Memristor-Based Lozi Map with Hidden Hyperchaos

by Jiang Wang, Yang Gu, Kang Rong, Quan Xu and Xi Zhang

Mathematics 2022, 10(19), 3426; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193426 - 21 Sep 2022

Cited by 14 | Viewed by 1336

Recently, the application of memristors to improve chaos complexity in discrete chaotic systems has been paid more and more attention to. To enrich the application examples of discrete memristor-based chaotic systems, this article proposes a new three-dimensional (3-D) memristor-based Lozi map by introducing [...] Read more.

Recently, the application of memristors to improve chaos complexity in discrete chaotic systems has been paid more and more attention to. To enrich the application examples of discrete memristor-based chaotic systems, this article proposes a new three-dimensional (3-D) memristor-based Lozi map by introducing a discrete memristor into the original two-dimensional (2-D) Lozi map. The proposed map has no fixed points but can generate hidden hyperchaos, so it is a hidden hyperchaotic map. The dynamical effects of the discrete memristor on the memristor-based Lozi map and two types of coexisting hidden attractors boosted by the initial conditions are demonstrated using some numerical methods. The numerical results clearly show that the introduced discrete memristor allows the proposed map to have complicated hidden dynamics evolutions and also exhibit heterogeneous and homogeneous hidden multistability. Finally, a digital platform is used to realize the memristor-based Lozi map, and its experimental phase portraits are obtained to confirm the numerical ones. Full article

(This article belongs to the Special Issue Chaotic Systems: From Mathematics to Real-World Applications)

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13 pages, 1085 KiB

Open AccessArticle

The Discrete Fractional Variable-Order Tinkerbell Map: Chaos, 0–1 Test, and Entropy

by Souad Bensid Ahmed, Adel Ouannas, Mohammed Al Horani and Giuseppe Grassi

Mathematics 2022, 10(17), 3173; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173173 - 03 Sep 2022

Cited by 5 | Viewed by 1630

Abstract

The dynamics of the Caputo-fractional variable-order difference form of the Tinkerbell map are studied. The phase portraits, bifurcation, and largest Lyapunov exponent (LLE) were employed to demonstrate the presence of chaos over a different fractional variable-order and establish the nature of the dynamics. [...] Read more.

The dynamics of the Caputo-fractional variable-order difference form of the Tinkerbell map are studied. The phase portraits, bifurcation, and largest Lyapunov exponent (LLE) were employed to demonstrate the presence of chaos over a different fractional variable-order and establish the nature of the dynamics. In addition, the 0–1 test tool was used to detect chaos. Finally, the numerical results were confirmed using the approximate entropy. Full article

(This article belongs to the Special Issue Chaotic Systems: From Mathematics to Real-World Applications)

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20 pages, 3217 KiB

Open AccessArticle

Control of Multistability in an Erbium-Doped Fiber Laser by an Artificial Neural Network: A Numerical Approach

by Daniel A. Magallón, Rider Jaimes-Reátegui, Juan H. García-López, Guillermo Huerta-Cuellar, Didier López-Mancilla and Alexander N. Pisarchik

Mathematics 2022, 10(17), 3140; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173140 - 01 Sep 2022

Cited by 9 | Viewed by 1288

Abstract

A recurrent wavelet first-order neural network (RWFONN) is proposed to select a desired attractor in a multistable erbium-doped fiber laser (EDFL). A filtered error algorithm is used to classify coexisting EDFL states and train RWFONN. The design of the intracavity laser power controller [...] Read more.

A recurrent wavelet first-order neural network (RWFONN) is proposed to select a desired attractor in a multistable erbium-doped fiber laser (EDFL). A filtered error algorithm is used to classify coexisting EDFL states and train RWFONN. The design of the intracavity laser power controller is developed according to the RWFONN states with the block control linearization technique and the super-twisting control algorithm. Closed-loop stability analysis is performed using the boundedness of synaptic weights. The efficiency of the control method is demonstrated through numerical simulations. Full article

(This article belongs to the Special Issue Chaotic Systems: From Mathematics to Real-World Applications)

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9 pages, 3783 KiB

Open AccessArticle

Fractional-Order Memristive Wilson Neuron Model: Dynamical Analysis and Synchronization Patterns

by Gayathri Vivekanandan, Mahtab Mehrabbeik, Hayder Natiq, Karthikeyan Rajagopal and Esteban Tlelo-Cuautle

Mathematics 2022, 10(16), 2827; https://0-doi-org.brum.beds.ac.uk/10.3390/math10162827 - 09 Aug 2022

Cited by 6 | Viewed by 1370

Abstract

Fractional nonlinear systems have been considered in many fields due to their ability to bring memory-dependent properties into various systems. Therefore, using fractional derivatives to model real-world phenomena, such as neuronal dynamics, is of significant importance. This paper presents the fractional memristive Wilson [...] Read more.

Fractional nonlinear systems have been considered in many fields due to their ability to bring memory-dependent properties into various systems. Therefore, using fractional derivatives to model real-world phenomena, such as neuronal dynamics, is of significant importance. This paper presents the fractional memristive Wilson neuron model and studies its dynamics as a single neuron. Furthermore, the collective behavior of neurons is researched when they are locally and diffusively coupled in a ring topology. It is found that the fractional-order neurons are bistable in some values of the fractional order. Additionally, complete synchronization, lag synchronization, phase synchronization, and sine-like synchronization patterns can be observed in the constructed network with different fractional orders. Full article

(This article belongs to the Special Issue Chaotic Systems: From Mathematics to Real-World Applications)

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22 pages, 11994 KiB

Open AccessArticle

A Novel Chaos-Based Cryptography Algorithm and Its Performance Analysis

by Ahmed A. Abd El-Latif, Janarthanan Ramadoss, Bassem Abd-El-Atty, Hany S. Khalifa and Fahimeh Nazarimehr

Mathematics 2022, 10(14), 2434; https://0-doi-org.brum.beds.ac.uk/10.3390/math10142434 - 12 Jul 2022

Cited by 15 | Viewed by 1860

Abstract

Data security represents an essential task in the present day, in which chaotic models have an excellent role in designing modern cryptosystems. Here, a novel oscillator with chaotic dynamics is presented and its dynamical properties are investigated. Various properties of the oscillator, like [...] Read more.

Data security represents an essential task in the present day, in which chaotic models have an excellent role in designing modern cryptosystems. Here, a novel oscillator with chaotic dynamics is presented and its dynamical properties are investigated. Various properties of the oscillator, like equilibria, bifurcations, and Lyapunov exponents (LEs), are discussed. The designed system has a center point equilibrium and an interesting chaotic attractor. The existence of chaotic dynamics is proved by calculating Lyapunov exponents. The region of attraction for the chaotic attractor is investigated by plotting the basin of attraction. The oscillator has a chaotic attractor in which its basin is entangled with the center point. The complexity of the chaotic dynamic and its entangled basin of attraction make it a proper choice for image encryption. Using the effective properties of the chaotic oscillator, a method to construct pseudo-random numbers (PRNGs) is proposed, then utilizing the generated PRNG sequence for designing secure substitution boxes (S-boxes). Finally, a new image cryptosystem is presented using the proposed PRNG mechanism and the suggested S-box approach. The effectiveness of the suggested mechanisms is evaluated using several assessments, in which the outcomes show the characteristics of the presented mechanisms for reliable cryptographic applications. Full article

(This article belongs to the Special Issue Chaotic Systems: From Mathematics to Real-World Applications)

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13 pages, 4373 KiB

Open AccessArticle

Extreme Multistability and Its Incremental Integral Reconstruction in a Non-Autonomous Memcapacitive Oscillator

by Bei Chen, Xinxin Cheng, Han Bao, Mo Chen and Quan Xu

Mathematics 2022, 10(5), 754; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050754 - 26 Feb 2022

Cited by 7 | Viewed by 1492

Abstract

Extreme multistability has frequently been reported in autonomous circuits involving memory-circuit elements, since these circuits possess line/plane equilibrium sets. However, this special phenomenon has rarely been discovered in non-autonomous circuits. Luckily, extreme multistability is found in a simple non-autonomous memcapacitive oscillator in this [...] Read more.

Extreme multistability has frequently been reported in autonomous circuits involving memory-circuit elements, since these circuits possess line/plane equilibrium sets. However, this special phenomenon has rarely been discovered in non-autonomous circuits. Luckily, extreme multistability is found in a simple non-autonomous memcapacitive oscillator in this paper. The oscillator only contains a memcapacitor, a linear resistor, a linear inductor, and a sinusoidal voltage source, which are connected in series. The memcapacitive system model is firstly built for further study. The equilibrium points of the memcapacitive system evolve between a no equilibrium point and a line equilibrium set with the change in time. This gives rise to the emergence of extreme multistability, but the forming mechanism is not clear. Thus, the incremental integral method is employed to reconstruct the memcapacitive system. In the newly reconstructed system, the number and stability of the equilibrium points have complex time-varying characteristics due to the presence of fold bifurcation. Furthermore, the forming mechanism of the extreme multistability is further explained. Note that the initial conditions of the original memcapacitive system are mapped onto the controlling parameters of the newly reconstructed system. This makes it possible to achieve precise control of the extreme multistability. Furthermore, an analog circuit is designed for the reconstructed system, and then PSIM circuit simulations are performed to verify the numerical results. Full article

(This article belongs to the Special Issue Chaotic Systems: From Mathematics to Real-World Applications)

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25 pages, 14429 KiB

Open AccessArticle

Chaotic Oscillations in Cascoded and Darlington-Type Amplifier Having Generalized Transistors

by Jiri Petrzela and Miroslav Rujzl

Mathematics 2022, 10(3), 532; https://0-doi-org.brum.beds.ac.uk/10.3390/math10030532 - 08 Feb 2022

Cited by 3 | Viewed by 1704

Abstract

This paper describes, based on both numerical and experimental bases, the evolution of chaotic and, in some cases, hyperchaotic attractors within mathematical models of two two-port analog functional blocks commonly used inside radio-frequency systems. The first investigated electronic circuit is known as the [...] Read more.

This paper describes, based on both numerical and experimental bases, the evolution of chaotic and, in some cases, hyperchaotic attractors within mathematical models of two two-port analog functional blocks commonly used inside radio-frequency systems. The first investigated electronic circuit is known as the cascoded class C amplifier and the second network represents a resonant amplifier with Darlington’s active part. For the analysis of each mentioned block, fundamental configurations that contain coupled generalized bipolar transistors are considered; without driving force or interactions with other lumped circuits. The existence of the structurally stable strange attractors is proved via the high-resolution composition plots of the Lyapunov exponents, numerical sensitivity analysis and captured oscilloscope screenshots. Full article

(This article belongs to the Special Issue Chaotic Systems: From Mathematics to Real-World Applications)

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16 pages, 5508 KiB

Open AccessArticle

D₃ Dihedral Logistic Map of Fractional Order

by Marius-F. Danca and Nikolay Kuznetsov

Mathematics 2022, 10(2), 213; https://0-doi-org.brum.beds.ac.uk/10.3390/math10020213 - 11 Jan 2022

Cited by 4 | Viewed by 1882

Abstract

In this paper, the D₃ dihedral logistic map of fractional order is introduced. The map presents a dihedral symmetry D₃. It is numerically shown that the construction and interpretation of the bifurcation diagram versus the fractional order requires special attention. [...] Read more.

In this paper, the D₃ dihedral logistic map of fractional order is introduced. The map presents a dihedral symmetry D₃. It is numerically shown that the construction and interpretation of the bifurcation diagram versus the fractional order requires special attention. The system stability is determined and the problem of hidden attractors is analyzed. Furthermore, analytical and numerical results show that the chaotic attractor of integer order, with D₃ symmetries, looses its symmetry in the fractional-order variant. Full article

(This article belongs to the Special Issue Chaotic Systems: From Mathematics to Real-World Applications)

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15 pages, 1769 KiB

Open AccessArticle

An Oscillator without Linear Terms: Infinite Equilibria, Chaos, Realization, and Application

by Othman Abdullah Almatroud, Victor Kamdoum Tamba, Giuseppe Grassi and Viet-Thanh Pham

Mathematics 2021, 9(24), 3315; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243315 - 20 Dec 2021

Cited by 8 | Viewed by 2168

Abstract

Oscillations and oscillators appear in various fields and find applications in numerous areas. We present an oscillator with infinite equilibria in this work. The oscillator includes only nonlinear elements (quadratic, absolute, and cubic ones). It is different from common oscillators, in which there [...] Read more.

Oscillations and oscillators appear in various fields and find applications in numerous areas. We present an oscillator with infinite equilibria in this work. The oscillator includes only nonlinear elements (quadratic, absolute, and cubic ones). It is different from common oscillators, in which there are linear elements. Special features of the oscillator are suitable for secure applications. The oscillator’s dynamics have been discovered via simulations and an electronic circuit. Chaotic attractors, bifurcation diagrams, Lyapunov exponents, and the boosting feature are presented while measurements of the implemented oscillator are reported by using an oscilloscope. We introduce a random number generator using such an oscillator, which is applied in biomedical image encryption. Moreover, the security and performance analysis are considered to confirm the correctness of encryption and decryption processes. Full article

(This article belongs to the Special Issue Chaotic Systems: From Mathematics to Real-World Applications)

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9 pages, 3858 KiB

Open AccessArticle

Dynamical Behavior of a New Chaotic System with One Stable Equilibrium

by Vijayakumar M.D., Anitha Karthikeyan, Jozef Zivcak, Ondrej Krejcar and Hamidreza Namazi

Mathematics 2021, 9(24), 3217; https://0-doi-org.brum.beds.ac.uk/10.3390/math9243217 - 13 Dec 2021

Cited by 6 | Viewed by 2213

Abstract

This paper reports a simple three-dimensional autonomous system with a single stable node equilibrium. The system has a constant controller which adjusts the dynamic of the system. It is revealed that the system exhibits both chaotic and non-chaotic dynamics. Moreover, chaotic or periodic [...] Read more.

This paper reports a simple three-dimensional autonomous system with a single stable node equilibrium. The system has a constant controller which adjusts the dynamic of the system. It is revealed that the system exhibits both chaotic and non-chaotic dynamics. Moreover, chaotic or periodic attractors coexist with a single stable equilibrium for some control parameter based on initial conditions. The system dynamics are studied by analyzing bifurcation diagrams, Lyapunov exponents, and basins of attractions. Beyond a fixed-point analysis, a new analysis known as connecting curves is provided. These curves are one-dimensional sets of the points that are more informative than fixed points. These curves are the skeleton of the system, which shows the direction of flow evolution. Full article

(This article belongs to the Special Issue Chaotic Systems: From Mathematics to Real-World Applications)

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Review

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25 pages, 8147 KiB

Open AccessReview

Quantum–Classical Mechanics: Nano-Resonance in Polymethine Dyes

by Vladimir V. Egorov

Mathematics 2022, 10(9), 1443; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091443 - 25 Apr 2022

Cited by 4 | Viewed by 2489

Abstract

It is well known in quantum mechanics that the theory of quantum transitions is based on the convergence of the series of time-dependent perturbation theory. This series converges in atomic and nuclear physics. However, in molecular and chemical physics, this series converges only [...] Read more.

It is well known in quantum mechanics that the theory of quantum transitions is based on the convergence of the series of time-dependent perturbation theory. This series converges in atomic and nuclear physics. However, in molecular and chemical physics, this series converges only in the Born–Oppenheimer adiabatic approximation and due to the application of the Franck–Condon principle, and it diverges as a result of going beyond the adiabatic approximation and the Franck–Condon principle. This divergence (singularity) is associated with the incommensurability of the masses of light electrons and heavy nuclei which jointly participate in the full-fledged movement in the transient state of molecular “quantum” transitions. In a new physical theory—quantum–classical mechanics (Egorov, V.V. Heliyon Physics 2019, 5, e02579)—this singularity is damped by introducing chaos into the transient state. This transient chaos is introduced by replacing the infinitely small imaginary additive in the energy denominator of the spectral representation of the total Green’s function of the system with a finite value and is called dozy chaos. In this article, resonance at the nanoscale (nano-resonance) between electron and nuclear reorganization motions in the quantum–classical (dozy-chaos) mechanics of elementary electron transfers in condensed media and their applications to polymethine dyes and J-aggregates in solutions are reviewed. Nano-resonance explains the resonant character of the transformation of the shape of the optical absorption band in a series of polymethine dyes in which the length of the polymethine chain changes, as well as the nature of the red-shifted absorption band of the J-aggregates of polymethine dyes (J-band), which is narrow and intense. The process of dye aggregation in an aqueous solution with an increase in its concentration by the formation of J-aggregates is considered a structural tuning of the “polymethine dye + environment” system into resonance with light absorption. For J-aggregates in Langmuir films, the asymmetry of the luminescence and absorption bands, as well as the small value of their Stokes shifts, are explained. The parasitic transformation of the resonant shape of the optical absorption band of a polymethine dye in solution during the transition from one-photon to two-photon absorption is also explained, and the conditions for the restoration of this nano-resonance shape are predicted. Full article

(This article belongs to the Special Issue Chaotic Systems: From Mathematics to Real-World Applications)

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