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Symmetry
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Symmetry in Chaotic Systems and Circuits
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Symmetry in Chaotic Systems and Circuits

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Computer".

Deadline for manuscript submissions: closed (30 September 2020) | Viewed by 38515

Printed Edition Available!
A printed edition of this Special Issue is available here.

Special Issue Editor

Dr. Christos Volos

E-Mail Website
Guest Editor
Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece
Interests: electrical and electronics engineering; mathematical modeling; control theory; engineering, applied and computational mathematics; numerical analysis; mathematical analysis; numerical modeling; modeling and simulation; robotics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Chaos theory is one of the most fascinating fields in modern science, revolutionizing our understanding of order and pattern in nature. Symmetry, a traditional and highly developed area of mathematics, would seem to lie at the opposite end of the spectrum. However, in the last few years, scientists have found connections between these two areas, connections which can have profound consequences for our understanding of the complex behavior in many physical, chemical, biological, and mechanical chaotic systems.

Therefore, symmetry can play an important role in the field of nonlinear systems and especially in the field of designing nonlinear circuits that produce chaos. In more detail, from designing chaotic systems and circuits with symmetric nonlinear terms to the study of system’s equilibria with symmetry in the case of self-excited attractors or symmetric line of equilibria in the case of hidden attractors, the feature of symmetry can play significant role in this kind of systems.

The aim of this Special Issue is to collect contributions in nonlinear chaotic systems and circuits that present as special a kind of symmetries as the aforementioned. Applications of chaotic systems and circuits where symmetry, or the deliberate lack of symmetry, is present, are also welcome.

Prof. Dr. Christos Volos
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • chaos
  • nonlinear systems
  • nonlinear circuits
  • control
  • synchronization
  • symmetric nonlinearities
  • self-excited attractors
  • hidden attractors
  • applications of chaotic systems and circuits with symmetries

Published Papers (15 papers)

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Editorial

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5 pages, 187 KiB  
Editorial
Symmetry in Chaotic Systems and Circuits
by Christos Volos
Symmetry 2022, 14(8), 1612; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14081612 - 05 Aug 2022
Cited by 1 | Viewed by 902
Abstract
Chaos theory is currently one of the most fascinating fields in modern science, revolutionizing our understanding of organization and patterns in nature [...] Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)

Research

Jump to: Editorial, Other

18 pages, 5203 KiB  
Article
A Simple Chaotic Flow with Hyperbolic Sinusoidal Function and Its Application to Voice Encryption
by Saleh Mobayen, Christos Volos, Ünal Çavuşoğlu and Sezgin S. Kaçar
Symmetry 2020, 12(12), 2047; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12122047 - 10 Dec 2020
Cited by 12 | Viewed by 2317
Abstract
In this article, a new chaotic system with hyperbolic sinusoidal function is introduced. This chaotic system provides a new category of chaotic flows which gives better perception of chaotic attractors. In the proposed chaotic flow with hyperbolic sinusoidal function, according to the changes [...] Read more.
In this article, a new chaotic system with hyperbolic sinusoidal function is introduced. This chaotic system provides a new category of chaotic flows which gives better perception of chaotic attractors. In the proposed chaotic flow with hyperbolic sinusoidal function, according to the changes of parameters of the system, the self-excited attractor and two forms of hidden attractors are occurred. Dynamic behavior of the offered chaotic flow is studied through eigenvalues, bifurcation diagrams, phase portraits, and spectrum of Lyapunov exponents. Moreover, the existence of double-scroll attractors in real word is considered via the Orcard-PSpice software through an electronic execution of the new chaotic flow and illustrative results between the numerical simulation and Orcard-PSpice outcomes are obtained. Lastly, random number generator (RNG) design is completed with the new chaos. Using the new RNG design, a novel voice encryption algorithm is suggested and voice encryption use and encryption analysis are performed. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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17 pages, 14832 KiB  
Article
A New Hyperchaotic Map for a Secure Communication Scheme with an Experimental Realization
by Nadia M. G. Al-Saidi, Dhurgham Younus, Hayder Natiq, M. R. K. Ariffin, M. A. Asbullah and Z. Mahad
Symmetry 2020, 12(11), 1881; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12111881 - 15 Nov 2020
Cited by 20 | Viewed by 2171
Abstract
Using different chaotic systems in secure communication, nonlinear control, and many other applications has revealed that these systems have several drawbacks in different aspects. This can cause unfavorable effects to chaos-based applications. Therefore, presenting a chaotic map with complex behaviors is considered important. [...] Read more.
Using different chaotic systems in secure communication, nonlinear control, and many other applications has revealed that these systems have several drawbacks in different aspects. This can cause unfavorable effects to chaos-based applications. Therefore, presenting a chaotic map with complex behaviors is considered important. In this paper, we introduce a new 2D chaotic map, namely, the 2D infinite-collapse-Sine model (2D-ICSM). Various metrics including Lyapunov exponents and bifurcation diagrams are used to demonstrate the complex dynamics and robust hyperchaotic behavior of the 2D-ICSM. Furthermore, the cross-correlation coefficient, phase space diagram, and Sample Entropy algorithm prove that the 2D-ICSM has a high sensitivity to initial values and parameters, extreme complexity performance, and a much larger hyperchaotic range than existing maps. To empirically verify the efficiency and simplicity of the 2D-ICSM in practical applications, we propose a symmetric secure communication system using the 2D-ICSM. Experimental results are presented to demonstrate the validity of the proposed system. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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16 pages, 1518 KiB  
Article
Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode
by Pattrawut Chansangiam
Symmetry 2020, 12(11), 1803; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12111803 - 30 Oct 2020
Cited by 3 | Viewed by 1879
Abstract
This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to [...] Read more.
This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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22 pages, 16185 KiB  
Article
The Effect of a Non-Local Fractional Operator in an Asymmetrical Glucose-Insulin Regulatory System: Analysis, Synchronization and Electronic Implementation
by Jesus M. Munoz-Pacheco, Cornelio Posadas-Castillo and Ernesto Zambrano-Serrano
Symmetry 2020, 12(9), 1395; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12091395 - 21 Aug 2020
Cited by 21 | Viewed by 2365
Abstract
For studying biological conditions with higher precision, the memory characteristics defined by the fractional-order versions of living dynamical systems have been pointed out as a meaningful approach. Therefore, we analyze the dynamics of a glucose-insulin regulatory system by applying a non-local fractional operator [...] Read more.
For studying biological conditions with higher precision, the memory characteristics defined by the fractional-order versions of living dynamical systems have been pointed out as a meaningful approach. Therefore, we analyze the dynamics of a glucose-insulin regulatory system by applying a non-local fractional operator in order to represent the memory of the underlying system, and whose state-variables define the population densities of insulin, glucose, and β-cells, respectively. We focus mainly on four parameters that are associated with different disorders (type 1 and type 2 diabetes mellitus, hypoglycemia, and hyperinsulinemia) to determine their observation ranges as a relation to the fractional-order. Like many preceding works in biosystems, the resulting analysis showed chaotic behaviors related to the fractional-order and system parameters. Subsequently, we propose an active control scheme for forcing the chaotic regime (an illness) to follow a periodic oscillatory state, i.e., a disorder-free equilibrium. Finally, we also present the electronic realization of the fractional glucose-insulin regulatory model to prove the conceptual findings. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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11 pages, 823 KiB  
Article
A Nonlinear Five-Term System: Symmetry, Chaos, and Prediction
by Vo Phu Thoai, Maryam Shahriari Kahkeshi, Van Van Huynh, Adel Ouannas and Viet-Thanh Pham
Symmetry 2020, 12(5), 865; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12050865 - 25 May 2020
Cited by 14 | Viewed by 2549
Abstract
Chaotic systems have attracted considerable attention and been applied in various applications. Investigating simple systems and counterexamples with chaotic behaviors is still an important topic. The purpose of this work was to study a simple symmetrical system including only five nonlinear terms. We [...] Read more.
Chaotic systems have attracted considerable attention and been applied in various applications. Investigating simple systems and counterexamples with chaotic behaviors is still an important topic. The purpose of this work was to study a simple symmetrical system including only five nonlinear terms. We discovered the system’s rich behavior such as chaos through phase portraits, bifurcation diagrams, Lyapunov exponents, and entropy. Interestingly, multi-stability was observed when changing system’s initial conditions. Chaos of such a system was predicted by applying a machine learning approach based on a neural network. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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11 pages, 7222 KiB  
Article
A Two-Parameter Modified Logistic Map and Its Application to Random Bit Generation
by Lazaros Moysis, Aleksandra Tutueva, Christos Volos, Denis Butusov, Jesus M. Munoz-Pacheco and Hector Nistazakis
Symmetry 2020, 12(5), 829; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12050829 - 18 May 2020
Cited by 31 | Viewed by 3253
Abstract
This work proposes a modified logistic map based on the system previously proposed by Han in 2019. The constructed map exhibits interesting chaos related phenomena like antimonotonicity, crisis, and coexisting attractors. In addition, the Lyapunov exponent of the map can achieve higher values, [...] Read more.
This work proposes a modified logistic map based on the system previously proposed by Han in 2019. The constructed map exhibits interesting chaos related phenomena like antimonotonicity, crisis, and coexisting attractors. In addition, the Lyapunov exponent of the map can achieve higher values, so the behavior of the proposed map is overall more complex compared to the original. The map is then successfully applied to the problem of random bit generation using techniques like the comparison between maps, X O R , and bit reversal. The proposed algorithm passes all the NIST tests, shows good correlation characteristics, and has a high key space. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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10 pages, 6900 KiB  
Article
Symmetry Evolution in Chaotic System
by Chunbiao Li, Jiayu Sun, Tianai Lu and Tengfei Lei
Symmetry 2020, 12(4), 574; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12040574 - 05 Apr 2020
Cited by 17 | Viewed by 1964
Abstract
A comprehensive exploration of symmetry and conditional symmetry is made from the evolution of symmetry. Unlike other chaotic systems of conditional symmetry, in this work it is derived from the symmetric diffusionless Lorenz system. Transformation from symmetry and asymmetry to conditional symmetry is [...] Read more.
A comprehensive exploration of symmetry and conditional symmetry is made from the evolution of symmetry. Unlike other chaotic systems of conditional symmetry, in this work it is derived from the symmetric diffusionless Lorenz system. Transformation from symmetry and asymmetry to conditional symmetry is examined by constant planting and dimension growth, which proves that the offset boosting of some necessary variables is the key factor for reestablishing polarity balance in a dynamical system. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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13 pages, 2028 KiB  
Article
A Novel Method for Performance Improvement of Chaos-Based Substitution Boxes
by Fırat Artuğer and Fatih Özkaynak
Symmetry 2020, 12(4), 571; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12040571 - 05 Apr 2020
Cited by 46 | Viewed by 3339
Abstract
Symmetry plays an important role in nonlinear system theory. In particular, it offers several methods by which to understand and model the chaotic behavior of mathematical, physical and biological systems. This study examines chaotic behavior in the field of information security. A novel [...] Read more.
Symmetry plays an important role in nonlinear system theory. In particular, it offers several methods by which to understand and model the chaotic behavior of mathematical, physical and biological systems. This study examines chaotic behavior in the field of information security. A novel method is proposed to improve the performance of chaos-based substitution box structures. Substitution box structures have a special role in block cipher algorithms, since they are the only nonlinear components in substitution permutation network architectures. However, the substitution box structures used in modern block encryption algorithms contain various vulnerabilities to side-channel attacks. Recent studies have shown that chaos-based designs can offer a variety of opportunities to prevent side-channel attacks. However, the problem of chaos-based designs is that substitution box performance criteria are worse than designs based on mathematical transformation. In this study, a postprocessing algorithm is proposed to improve the performance of chaos-based designs. The analysis results show that the proposed method can improve the performance criteria. The importance of these results is that chaos-based designs may offer opportunities for other practical applications in addition to the prevention of side-channel attacks. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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17 pages, 5422 KiB  
Article
A Symmetric Controllable Hyperchaotic Hidden Attractor
by Xin Zhang, Chunbiao Li, Tengfei Lei, Zuohua Liu and Changyuan Tao
Symmetry 2020, 12(4), 550; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12040550 - 04 Apr 2020
Cited by 16 | Viewed by 2257
Abstract
By introducing a simple feedback, a hyperchaotic hidden attractor is found in the newly proposed Lorenz-like chaotic system. Some variables of the equilibria-free system can be controlled in amplitude and offset by an independent knob. A circuit experiment based on Multisim is consistent [...] Read more.
By introducing a simple feedback, a hyperchaotic hidden attractor is found in the newly proposed Lorenz-like chaotic system. Some variables of the equilibria-free system can be controlled in amplitude and offset by an independent knob. A circuit experiment based on Multisim is consistent with the theoretic analysis and numerical simulation. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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16 pages, 23598 KiB  
Article
Two New Asymmetric Boolean Chaos Oscillators with No Dependence on Incommensurate Time-Delays and Their Circuit Implementation
by Jesus M. Munoz-Pacheco, Tonatiuh García-Chávez, Victor R. Gonzalez-Diaz, Gisela de La Fuente-Cortes and Luz del Carmen Gómez-Pavón
Symmetry 2020, 12(4), 506; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12040506 - 01 Apr 2020
Cited by 11 | Viewed by 2872
Abstract
This manuscript introduces two new chaotic oscillators based on autonomous Boolean networks (ABN), preserving asymmetrical logic functions. That means that the ABNs require a combination of XOR-XNOR logic functions. We demonstrate analytically that the two ABNs do not have fixed points, and therefore, [...] Read more.
This manuscript introduces two new chaotic oscillators based on autonomous Boolean networks (ABN), preserving asymmetrical logic functions. That means that the ABNs require a combination of XOR-XNOR logic functions. We demonstrate analytically that the two ABNs do not have fixed points, and therefore, can evolve to Boolean chaos. Using the Lyapunov exponent’s method, we also prove the chaotic behavior, generated by the proposed chaotic oscillators, is insensitive to incommensurate time-delays paths. As a result, they can be implemented using distinct electronic circuits. More specifically, logic-gates–, GAL–, and FPGA–based implementations verify the theoretical findings. An integrated circuit using a CMOS 180nm fabrication technology is also presented to get a compact chaos oscillator with relatively high-frequency. Dynamical behaviors of those implementations are analyzed using time-series, time-lag embedded attractors, frequency spectra, Poincaré maps, and Lyapunov exponents. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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14 pages, 2678 KiB  
Article
Image Encryption Algorithm Based on Tent Delay-Sine Cascade with Logistic Map
by Guidong Zhang, Weikang Ding and Lian Li
Symmetry 2020, 12(3), 355; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12030355 - 01 Mar 2020
Cited by 34 | Viewed by 3447
Abstract
We propose a new chaotic map combined with delay and cascade, called tent delay-sine cascade with logistic map (TDSCL). Compared with the original one-dimensional simple map, the proposed map has increased initial value sensitivity and internal randomness and a larger chaotic parameter interval. [...] Read more.
We propose a new chaotic map combined with delay and cascade, called tent delay-sine cascade with logistic map (TDSCL). Compared with the original one-dimensional simple map, the proposed map has increased initial value sensitivity and internal randomness and a larger chaotic parameter interval. The chaotic sequence generated by TDSCL has pseudo-randomness and is suitable for image encryption. Based on this chaotic map, we propose an image encryption algorithm with a symmetric structure, which can achieve confusion and diffusion at the same time. Simulation results show that after encryption using the proposed algorithm, the entropy of the cipher is extremely close to the ideal value of eight, and the correlation coefficients between the pixels are lower than 0.01, thus the algorithm can resist statistical attacks. Moreover, the number of pixel change rate (NPCR) and the unified average changing intensity (UACI) of the proposed algorithm are very close to the ideal value, which indicates that it can efficiently resist chosen-plain text attack. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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12 pages, 831 KiB  
Article
Symmetric Key Encryption Based on Rotation-Translation Equation
by Borislav Stoyanov and Gyurhan Nedzhibov
Symmetry 2020, 12(1), 73; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12010073 - 02 Jan 2020
Cited by 18 | Viewed by 3398
Abstract
In this paper, an improved encryption algorithm based on numerical methods and rotation–translation equation is proposed. We develop the new encryption-decryption algorithm by using the concept of symmetric key instead of public key. Symmetric key algorithms use the same key for both encryption [...] Read more.
In this paper, an improved encryption algorithm based on numerical methods and rotation–translation equation is proposed. We develop the new encryption-decryption algorithm by using the concept of symmetric key instead of public key. Symmetric key algorithms use the same key for both encryption and decryption. Most symmetric key encryption algorithms use either block ciphers or stream ciphers. Our goal in this work is to improve an existing encryption algorithm by using a faster convergent iterative method, providing secure convergence of the corresponding numerical scheme, and improved security by a using rotation–translation formula. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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10 pages, 664 KiB  
Article
New Chaotic Systems with Two Closed Curve Equilibrium Passing the Same Point: Chaotic Behavior, Bifurcations, and Synchronization
by Xinhe Zhu and Wei-Shih Du
Symmetry 2019, 11(8), 951; https://0-doi-org.brum.beds.ac.uk/10.3390/sym11080951 - 26 Jul 2019
Cited by 5 | Viewed by 2315
Abstract
In this work, we introduce a chaotic system with infinitely many equilibrium points laying on two closed curves passing the same point. The proposed system belongs to a class of systems with hidden attractors. The dynamical properties of the new system were investigated [...] Read more.
In this work, we introduce a chaotic system with infinitely many equilibrium points laying on two closed curves passing the same point. The proposed system belongs to a class of systems with hidden attractors. The dynamical properties of the new system were investigated by means of phase portraits, equilibrium points, Poincaré section, bifurcation diagram, Kaplan–Yorke dimension, and Maximal Lyapunov exponents. The anti-synchronization of systems was obtained using the active control. This study broadens the current knowledge of systems with infinite equilibria. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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19 pages, 1841 KiB  
Concept Paper
Dynamic Symmetry in Dozy-Chaos Mechanics
by Vladimir V. Egorov
Symmetry 2020, 12(11), 1856; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12111856 - 11 Nov 2020
Cited by 4 | Viewed by 1954
Abstract
All kinds of dynamic symmetries in dozy-chaos (quantum-classical) mechanics (Egorov, V.V. Challenges 2020, 11, 16; Egorov, V.V. Heliyon Physics 2019, 5, e02579), which takes into account the chaotic dynamics of the joint electron-nuclear motion in the transient state of molecular “quantum” transitions, are [...] Read more.
All kinds of dynamic symmetries in dozy-chaos (quantum-classical) mechanics (Egorov, V.V. Challenges 2020, 11, 16; Egorov, V.V. Heliyon Physics 2019, 5, e02579), which takes into account the chaotic dynamics of the joint electron-nuclear motion in the transient state of molecular “quantum” transitions, are discussed. The reason for the emergence of chaotic dynamics is associated with a certain new property of electrons, consisting in the provocation of chaos (dozy chaos) in a transient state, which appears in them as a result of the binding of atoms by electrons into molecules and condensed matter and which provides the possibility of reorganizing a very heavy nuclear subsystem as a result of transitions of light electrons. Formally, dozy chaos is introduced into the theory of molecular “quantum” transitions to eliminate the significant singularity in the transition rates, which is present in the theory when it goes beyond the Born–Oppenheimer adiabatic approximation and the Franck–Condon principle. Dozy chaos is introduced by replacing the infinitesimal imaginary addition in the energy denominator of the full Green’s function of the electron-nuclear system with a finite value, which is called the dozy-chaos energy γ. The result for the transition-rate constant does not change when the sign of γ is changed. Other dynamic symmetries appearing in theory are associated with the emergence of dynamic organization in electronic-vibrational transitions, in particular with the emergence of an electron-nuclear-reorganization resonance (the so-called Egorov resonance) and its antisymmetric (chaotic) “twin”, with direct and reverse transitions, as well as with different values of the electron–phonon interaction in the initial and final states of the system. All these dynamic symmetries are investigated using the simplest example of quantum-classical mechanics, namely, the example of quantum-classical mechanics of elementary electron-charge transfers in condensed media. Full article
(This article belongs to the Special Issue Symmetry in Chaotic Systems and Circuits)
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