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22 pages, 315 KiB

Open AccessArticle

Fixed Point Dynamics in a New Type of Contraction in b-Metric Spaces

by María A. Navascués and Ram N. Mohapatra

Symmetry 2024, 16(4), 506; https://doi.org/10.3390/sym16040506 - 22 Apr 2024

Viewed by 210

Since the topological properties of a b-metric space (which generalizes the concept of a metric space) seem sometimes counterintuitive due to the fact that the “open” balls may not be open sets, we review some aspects of these spaces concerning compactness, metrizability, continuity [...] Read more.

Since the topological properties of a b-metric space (which generalizes the concept of a metric space) seem sometimes counterintuitive due to the fact that the “open” balls may not be open sets, we review some aspects of these spaces concerning compactness, metrizability, continuity and fixed points. After doing so, we introduce new types of contractivities that extend the concept of Banach contraction. We study some of their properties, giving sufficient conditions for the existence of fixed points and common fixed points. Afterwards, we consider some iterative schemes in quasi-normed spaces for the approximation of these critical points, analyzing their convergence and stability. We apply these concepts to the resolution of a model of integral equation of Urysohn type. In the last part of the paper, we refine some results about partial contractivities in the case where the underlying set is a strong b-metric space, and we establish some relations between mutual weak contractions and quasi-contractions and the new type of contractivity. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

22 pages, 9762 KiB

Open AccessArticle

Influence of the Plan Structural Symmetry on the Non-Linear Seismic Response of Framed Reinforced Concrete Buildings

by Juan Carlos Vielma-Quintero, Edgar Giovanny Diaz-Segura and Juan Carlos Vielma

Symmetry 2024, 16(3), 370; https://doi.org/10.3390/sym16030370 - 19 Mar 2024

Viewed by 669

Abstract

Seismic-resistant design incorporates measures to ensure that structures perform adequately under specific limit states, focusing on seismic forces derived from both the equivalent static and spectral modal methods. This study examined buildings on slopes in densely built urban areas, a common scenario in [...] Read more.

Seismic-resistant design incorporates measures to ensure that structures perform adequately under specific limit states, focusing on seismic forces derived from both the equivalent static and spectral modal methods. This study examined buildings on slopes in densely built urban areas, a common scenario in Latin American cities with high seismic risks. The adjustment of high-rise buildings to sloping terrains induces structural asymmetry, leading to plan and elevation irregularities that significantly impact their seismic response. This paper explores the asymmetry in medium-height reinforced concrete frame buildings on variable inclines (0°, 15°, 30°, and 45°) and its effect on their nonlinear response, assessed via displacements, rotations, and damage. Synthetic accelerograms matched with Chile’s high seismic hazard design spectrum, scaled for different performance states and seismic records from the Chilean subduction zone, were applied. The findings highlight structural asymmetry’s role in influencing nonlinear response parameters such as ductility, transient interstory drifts, and roof rotations, and uncover element demand distributions surpassing conventional analysis and in earthquake-resistant design expectations. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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21 pages, 1007 KiB

Open AccessArticle

Symmetry Properties of Models for Reversible and Irreversible Thermodynamic Processes

by S. A. Lurie, P. A. Belov and H. A. Matevossian

Symmetry 2023, 15(12), 2173; https://doi.org/10.3390/sym15122173 - 07 Dec 2023

Viewed by 741

Abstract

The problem of formulating variational models for irreversible processes of media deformation is considered in this paper. For reversible processes, the introduction of variational models actually comes down to defining functionals with a given list of arguments of various tensor dimensions. For irreversible [...] Read more.

The problem of formulating variational models for irreversible processes of media deformation is considered in this paper. For reversible processes, the introduction of variational models actually comes down to defining functionals with a given list of arguments of various tensor dimensions. For irreversible processes, an algorithm based on the principle of stationarity of the functional is incorrect. In this paper, to formulate a variational model of irreversible deformation processes with an expanded range of coupled effects, an approach is developed based on the idea of the introduction of the non-integrable variational forms that clearly separate dissipative processes from reversible deformation processes. The fundamental nature of the properties of symmetry and anti-symmetry of tensors of physical properties in relation to multi-indices characterizing independent arguments of bilinear forms in the variational formulation of models of thermomechanical processes has been established. For reversible processes, physical property tensors must necessarily be symmetric with respect to multi-indices. On the contrary, for irreversible thermomechanical processes, the tensors of physical properties that determine non-integrable variational forms must be antisymmetric with respect to the permutation of multi-indices. As a result, an algorithm for obtaining variational models of dissipative irreversible processes is proposed. This algorithm is based on determining the required number of dissipative channels and adding them to the known model of a reversible process. Dissipation channels are introduced as non-integrable variational forms that are linear in the variations of the arguments. The hydrodynamic models of Darcy, Navier–Stokes, and Brinkman are considered, each of which is determined by a different set of dissipation channels. As another example, a variational model of heat transfer processes is presented. The equations of heat conduction laws are obtained as compatibility equations by excluding the introduced thermal potential from the constitutive equations for temperature and heat flux. The Fourier and Maxwell–Cattaneo equations and the generalized heat conduction laws of Gaer–Krumhansl and Jeffrey are formulated. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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20 pages, 24078 KiB

Open AccessArticle

Chaotic Maps with Tunable Mean Value—Application to a UAV Surveillance Mission

by Lazaros Moysis, Marcin Lawnik, Christos Volos, Murilo S. Baptista and Sotirios K. Goudos

Symmetry 2023, 15(12), 2138; https://doi.org/10.3390/sym15122138 - 01 Dec 2023

Cited by 1 | Viewed by 811

Abstract

Chaos-related applications are abundant in the literature, and span the fields of secure communications, encryption, optimization, and surveillance. Such applications take advantage of the unpredictability of chaotic systems as an alternative to using true random processes. The chaotic systems used, though, must showcase [...] Read more.

Chaos-related applications are abundant in the literature, and span the fields of secure communications, encryption, optimization, and surveillance. Such applications take advantage of the unpredictability of chaotic systems as an alternative to using true random processes. The chaotic systems used, though, must showcase the statistical characteristics suitable for each application. This may often be hard to achieve, as the design of maps with tunable statistical properties is not a trivial task. Motivated by this, the present study explores the task of constructing maps, where the statistical measures like the mean value can be appropriately controlled by tuning the map’s parameters. For this, a family of piecewise maps is considered, with three control parameters that affect the endpoint interpolations. Numerous examples are given, and the maps are studied through a collection of numerical simulations. The maps can indeed achieve a range of values for their statistical mean. Such maps may find extensive use in relevant chaos-based applications. To showcase this, the problem of chaotic path surveillance is considered as a potential application of the designed maps. Here, an autonomous agent follows a predefined trajectory but maneuvers around it in order to imbue unpredictability to potential hostile observers. The trajectory inherits the randomness of the chaotic map used as a seed, which results in chaotic motion patterns. Simulations are performed for the designed strategy. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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13 pages, 6261 KiB

Open AccessArticle

Assessment of Stochastic Numerical Schemes for Stochastic Differential Equations with “White Noise” Using Itô’s Integral

by Alina Bogoi, Cătălina-Ilinca Dan, Sergiu Strătilă, Grigore Cican and Daniel-Eugeniu Crunteanu

Symmetry 2023, 15(11), 2038; https://doi.org/10.3390/sym15112038 - 09 Nov 2023

Viewed by 616

Abstract

Stochastic Differential Equations (SDEs) model physical phenomena dominated by stochastic processes. They represent a method for studying the dynamic evolution of a physical phenomenon, like ordinary or partial differential equations, but with an additional term called “noise” that represents a perturbing factor that [...] Read more.

Stochastic Differential Equations (SDEs) model physical phenomena dominated by stochastic processes. They represent a method for studying the dynamic evolution of a physical phenomenon, like ordinary or partial differential equations, but with an additional term called “noise” that represents a perturbing factor that cannot be attached to a classical mathematical model. In this paper, we study weak and strong convergence for six numerical schemes applied to a multiplicative noise, an additive, and a system of SDEs. The Efficient Runge–Kutta (ERK) technique, however, comes out as the top performer, displaying the best convergence features in all circumstances, including in the difficult setting of multiplicative noise. This result highlights the importance of researching cutting-edge numerical techniques built especially for stochastic systems and we consider to be of good help to the MATLAB function code for the ERK method. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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19 pages, 1541 KiB

Open AccessArticle

The Numerical Solution of Nonlinear Fractional Lienard and Duffing Equations Using Orthogonal Perceptron

by Akanksha Verma, Wojciech Sumelka and Pramod Kumar Yadav

Symmetry 2023, 15(9), 1753; https://doi.org/10.3390/sym15091753 - 13 Sep 2023

Viewed by 631

Abstract

This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. [...] Read more.

This paper proposes an approximation algorithm based on the Legendre and Chebyshev artificial neural network to explore the approximate solution of fractional Lienard and Duffing equations with a Caputo fractional derivative. These equations show the oscillating circuit and generalize the spring–mass device equation. The proposed approach transforms the given nonlinear fractional differential equation (FDE) into an unconstrained minimization problem. The simulated annealing (SA) algorithm minimizes the mean square error. The proposed techniques examine various non-integer order problems to verify the theoretical results. The numerical results show that the proposed approach yields better results than existing methods. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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14 pages, 7132 KiB

Open AccessArticle

Symmetry in a Fractional-Order Multi-Scroll Chaotic System Using the Extended Caputo Operator

by A. E. Matouk, D. K. Almutairi, M. A. E. Herzallah, M. A. Abdelkawy and T. N. Abdelhameed

Symmetry 2023, 15(8), 1582; https://doi.org/10.3390/sym15081582 - 13 Aug 2023

Cited by 1 | Viewed by 779

Abstract

In this work, complex dynamics are found in a fractional-order multi-scroll chaotic system based on the extended Gamma function. Firstly, the extended left and right Caputo fractional differential operators are introduced. Then, the basic features of the extended left Caputo fractional differential operator [...] Read more.

In this work, complex dynamics are found in a fractional-order multi-scroll chaotic system based on the extended Gamma function. Firstly, the extended left and right Caputo fractional differential operators are introduced. Then, the basic features of the extended left Caputo fractional differential operator are outlined. The proposed operator is shown to have a new fractional parameter (higher degree of freedom) that increases the system’s ability to display more varieties of complex dynamics than the corresponding case of the Caputo fractional differential operator. Numerical results are performed to show the effectiveness of the proposed fractional operators. Then, rich complex dynamics are obtained such as coexisting one-scroll chaotic attractors, coexisting two-scroll chaotic attractors, or approximate periodic cycles, which are shown to persist in a shorter range as compared with the corresponding states of the integer-order counterpart of the multi-scroll system. The bifurcation diagrams, basin sets of attractions, and Lyapunov spectra are used to confirm the existence of the various scenarios of complex dynamics in the proposed systems. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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17 pages, 3560 KiB

Open AccessFeature PaperArticle

Spontaneous Symmetry Breaking in Systems Obeying the Dynamics of On–Off Intermittency and Presenting Bimodal Amplitude Distributions

by Stelios M. Potirakis, Pericles Papadopoulos, Niki-Lina Matiadou, Michael P. Hanias, Stavros G. Stavrinides, Georgios Balasis and Yiannis Contoyiannis

Symmetry 2023, 15(7), 1448; https://doi.org/10.3390/sym15071448 - 20 Jul 2023

Cited by 2 | Viewed by 812

Abstract

In this work, first, it is confirmed that a recently introduced symbolic time-series-analysis method based on the prime-numbers-based algorithm (PNA), referred to as the “PNA-based symbolic time-series analysis method” (PNA-STSM), can accurately determine the exponent of the distribution of waiting times in the [...] Read more.

In this work, first, it is confirmed that a recently introduced symbolic time-series-analysis method based on the prime-numbers-based algorithm (PNA), referred to as the “PNA-based symbolic time-series analysis method” (PNA-STSM), can accurately determine the exponent of the distribution of waiting times in the symbolic dynamics of two symbols produced by the 3D Ising model in its critical state. After this numerical verification of the reliability of PNA-STSM, three examples of how PNA-STSM can be applied to the category of systems that obey the dynamics of the on–off intermittency are presented. Usually, such time series, with on–off intermittency, present bimodal amplitude distributions (i.e., with two lobes). As has recently been found, the phenomenon of on–off intermittency is associated with the spontaneous symmetry breaking (SSB) of the second-order phase transition. Thus, the revelation that a system is close to SSB supports a deeper understanding of its dynamics in terms of criticality, which is quite useful in applications such as the analysis of pre-earthquake fracture-induced electromagnetic emission (also known as fracture-induced electromagnetic radiation) (FEME/FEMR) signals. Beyond the case of on–off intermittency, PNA-STSM can provide credible results for the dynamics of any two-symbol symbolic dynamics, even in cases in which there is an imbalance in the probability of the appearance of the two respective symbols since the two symbols are not considered separately but, instead, simultaneously, considering the information from both branches of the symbolic dynamics. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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19 pages, 421 KiB

Open AccessArticle

Nonlinear Dynamics of a Piecewise Modified ABC Fractional-Order Leukemia Model with Symmetric Numerical Simulations

by Hasib Khan, Jehad Alzabut, Wafa F. Alfwzan and Haseena Gulzar

Symmetry 2023, 15(7), 1338; https://doi.org/10.3390/sym15071338 - 30 Jun 2023

Cited by 10 | Viewed by 1009

Abstract

In this study, we introduce a nonlinear leukemia dynamical system for a piecewise modified ABC fractional-order derivative and analyze it for the theoretical as well computational works and examine the crossover effect of the model. For the crossover behavior of the operators, we [...] Read more.

In this study, we introduce a nonlinear leukemia dynamical system for a piecewise modified ABC fractional-order derivative and analyze it for the theoretical as well computational works and examine the crossover effect of the model. For the crossover behavior of the operators, we presume a division of the period of study

[0, t_{2}]

in two subclasses as

I_{1} = [0, t_{1}]

,

I_{2} = [t_{1}, t_{2}]

, for

t_{1}, t_{2} \in R

with

t_{1} < t_{2}

. In

I_{1}

, the classical derivative is considered for the study of the leukemia growth while in

I_{2}

we presume modified ABC fractional differential operator. As a result, the study is initiated in the piecewise modified ABC sense of derivative for the dynamical systems. The novel constructed model is then studied for the solution existence and stability as well computational results. The symmetry in dynamics for all the three classes can be graphically observed in the presented six plots. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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14 pages, 9517 KiB

Open AccessEditor’s ChoiceArticle

A Family of 1D Chaotic Maps without Equilibria

by Marcin Lawnik, Lazaros Moysis and Christos Volos

Symmetry 2023, 15(7), 1311; https://doi.org/10.3390/sym15071311 - 27 Jun 2023

Cited by 4 | Viewed by 1157

Abstract

In this work, a family of piecewise chaotic maps is proposed. This family of maps is parameterized by the nonlinear functions used for each piece of the mapping, which can be either symmetric or non-symmetric. Applying a constraint on the shape of each [...] Read more.

In this work, a family of piecewise chaotic maps is proposed. This family of maps is parameterized by the nonlinear functions used for each piece of the mapping, which can be either symmetric or non-symmetric. Applying a constraint on the shape of each piece, the generated maps have no equilibria and can showcase chaotic behavior. This family thus belongs to the category of systems with hidden attractors. Numerous examples of chaotic maps are provided, showcasing fractal-like, symmetrical patterns at the interchange between chaotic and non-chaotic behavior. Moreover, the application of the proposed maps to a pseudorandom bit generator is successfully performed. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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27 pages, 13869 KiB

Open AccessEditor’s ChoiceArticle

Chaotification of 1D Maps by Multiple Remainder Operator Additions—Application to B-Spline Curve Encryption

by Lazaros Moysis, Marcin Lawnik, Ioannis P. Antoniades, Ioannis Kafetzis, Murilo S. Baptista and Christos Volos

Symmetry 2023, 15(3), 726; https://doi.org/10.3390/sym15030726 - 14 Mar 2023

Cited by 4 | Viewed by 1320

Abstract

In this work, a chaotification technique is proposed for increasing the complexity of chaotic maps. The technique consists of adding the remainder of multiple scalings of the map’s value for the next iteration, so that the most- and least-significant digits are combined. By [...] Read more.

In this work, a chaotification technique is proposed for increasing the complexity of chaotic maps. The technique consists of adding the remainder of multiple scalings of the map’s value for the next iteration, so that the most- and least-significant digits are combined. By appropriate parameter tuning, the resulting map can achieve a higher Lyapunov exponent value, a result that was first proven theoretically and then showcased through numerical simulations for a collection of chaotic maps. As a proposed application of the transformed maps, the encryption of B-spline curves and patches was considered. The symmetric encryption consisted of two steps: a shuffling of the control point coordinates and an additive modulation. A transformed chaotic map was utilised to perform both steps. The resulting ciphertext curves and patches were visually unrecognisable compared to the plaintext ones and performed well on several statistical tests. The proposed work gives an insight into the potential of the remainder operator for chaotification, as well as the chaos-based encryption of curves and computer graphics. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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13 pages, 2305 KiB

Open AccessArticle

Towards a Holographic-Type Perspective in the Analysis of Complex-System Dynamics

by Ștefana Agop, Dumitru Filipeanu, Claudiu-Gabriel Țigănaș, Claudia-Elena Grigoraș-Ichim, Lucia Moroșan-Dănilă, Alina Gavriluț, Maricel Agop and Gavril Ștefan

Symmetry 2023, 15(3), 681; https://doi.org/10.3390/sym15030681 - 08 Mar 2023

Viewed by 1039

Abstract

By operating with the Scale Relativity Theory by means of two scenarios (Schrӧdinger and Madelung-type scenarios) in the dynamics of complex systems, we can achieve a description of these complex systems through a holographic-type perspective. Then, a gauge invariance of the Riccati type [...] Read more.

By operating with the Scale Relativity Theory by means of two scenarios (Schrӧdinger and Madelung-type scenarios) in the dynamics of complex systems, we can achieve a description of these complex systems through a holographic-type perspective. Then, a gauge invariance of the Riccati type becomes functional in complex-system dynamics, which implies several consequences: conservation laws (in particular, for dynamics, the kinetic momentum conservation law), simultaneity and synchronization among the structural units’ (belonging to a complex system) dynamics, and temporal patterns through harmonic mappings. Finally, an economic case analysis is highlighted. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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19 pages, 758 KiB

Open AccessArticle

Analysis of the Type V Intermittency Using the Perron-Frobenius Operator

by Sergio Elaskar, Ezequiel del Rio and Walkiria Schulz

Symmetry 2022, 14(12), 2519; https://doi.org/10.3390/sym14122519 - 29 Nov 2022

Cited by 4 | Viewed by 862

Abstract

A methodology to study the reinjection process in type V intermittency is introduced. The reinjection probability density function (RPD), and the probability density of the laminar lengths (RPDL) for type V intermittency are calculated. A family of maps with discontinuous and continuous RPD [...] Read more.

A methodology to study the reinjection process in type V intermittency is introduced. The reinjection probability density function (RPD), and the probability density of the laminar lengths (RPDL) for type V intermittency are calculated. A family of maps with discontinuous and continuous RPD functions is analyzed. Several tests were performed, in which the proposed technique was compared with the classical theory of intermittency, the M function methodology, and numerical data. The analysis exposed that the new technique can accurately capture the numerical data. Therefore, the scheme presented herein is a useful tool to theoretically evaluate the statistical variables for type V intermittency. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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Review

Jump to: Research

54 pages, 778 KiB

Open AccessFeature PaperReview

Review of Chaotic Intermittency

by Sergio Elaskar and Ezequiel del Río

Symmetry 2023, 15(6), 1195; https://doi.org/10.3390/sym15061195 - 02 Jun 2023

Cited by 2 | Viewed by 3788

Abstract

Chaotic intermittency is characterized by a signal that alternates aleatory between long regular (pseudo-laminar) phases and irregular bursts (pseudo-turbulent or chaotic phases). This phenomenon has been found in physics, chemistry, engineering, medicine, neuroscience, economy, etc. As a control parameter increases, the number of [...] Read more.

Chaotic intermittency is characterized by a signal that alternates aleatory between long regular (pseudo-laminar) phases and irregular bursts (pseudo-turbulent or chaotic phases). This phenomenon has been found in physics, chemistry, engineering, medicine, neuroscience, economy, etc. As a control parameter increases, the number of chaotic phases also increases. Therefore, intermittency presents a continuous route from regular behavior to chaotic motion. In this paper, a review of different types of intermittency is carried out. In addition, the description of two recent formulations to evaluate the reinjection processes is developed. The new theoretical formulations have allowed us to explain several tests previously called pathological. The theoretical background also includes the noise effects in the reinjection mechanism. Full article

(This article belongs to the Special Issue Symmetry in Nonlinear Dynamics and Chaos II)

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