Pub Date : 2023-12-21DOI: 10.3390/fractalfract8010010
Abdelkader Moumen, F. Z. Ladrani, Mohamed Ferhat, Amin Benaissa Cherif, Mohamed Bouye, Keltoum Bouhali
In this paper, we consider a system of random impulsive differential equations with infinite delay. When utilizing the nonlinear variation of Leray–Schauder’s fixed-point principles together with a technique based on separable vector-valued metrics to establish sufficient conditions for the existence of solutions, under suitable assumptions on Y1, Y2 and ϖ1, ϖ2, which greatly enriched the existence literature on this system, there is, however, no hope to discuss the uniqueness result in a convex case. In the present study, we analyzed the influence of the impulsive and infinite delay on the solutions to our system. In addition, to the best of our acknowledge, there is no result concerning coupled random system in the presence of impulsive and infinite delay.
{"title":"Existence Result for Coupled Random First-Order Impulsive Differential Equations with Infinite Delay","authors":"Abdelkader Moumen, F. Z. Ladrani, Mohamed Ferhat, Amin Benaissa Cherif, Mohamed Bouye, Keltoum Bouhali","doi":"10.3390/fractalfract8010010","DOIUrl":"https://doi.org/10.3390/fractalfract8010010","url":null,"abstract":"In this paper, we consider a system of random impulsive differential equations with infinite delay. When utilizing the nonlinear variation of Leray–Schauder’s fixed-point principles together with a technique based on separable vector-valued metrics to establish sufficient conditions for the existence of solutions, under suitable assumptions on Y1, Y2 and ϖ1, ϖ2, which greatly enriched the existence literature on this system, there is, however, no hope to discuss the uniqueness result in a convex case. In the present study, we analyzed the influence of the impulsive and infinite delay on the solutions to our system. In addition, to the best of our acknowledge, there is no result concerning coupled random system in the presence of impulsive and infinite delay.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"133 40","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138953612","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.3390/fractalfract8010004
A. Ben Makhlouf, Lassaad Mchiri, Mohamed Rhaima
In this study, we delve into the examination of Finite Time Stability (FTS) within a specific class of Fractional-Order Systems (FOS) with time delays. By applying a fixed-point theorem, we establish novel sufficient conditions to ensure FTS for time-delayed FOS within 1<σ<2. Moreover, we investigate the existence and uniqueness of global solutions for this particular system. To demonstrate the credibility of our results, we substantiate our findings through the presentation of two illustrative examples.
{"title":"Stability Analysis of Finite Time for a Class of Nonlinear Time-Delay Fractional-Order Systems","authors":"A. Ben Makhlouf, Lassaad Mchiri, Mohamed Rhaima","doi":"10.3390/fractalfract8010004","DOIUrl":"https://doi.org/10.3390/fractalfract8010004","url":null,"abstract":"In this study, we delve into the examination of Finite Time Stability (FTS) within a specific class of Fractional-Order Systems (FOS) with time delays. By applying a fixed-point theorem, we establish novel sufficient conditions to ensure FTS for time-delayed FOS within 1<σ<2. Moreover, we investigate the existence and uniqueness of global solutions for this particular system. To demonstrate the credibility of our results, we substantiate our findings through the presentation of two illustrative examples.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"119 7","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138958341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.3390/fractalfract8010008
A. Baazeem, Y. Nawaz, M. Arif
This contribution addresses a fractal numerical scheme that can be employed for handling fractal time-dependent parabolic equations. The numerical scheme presented in this contribution can be used to discretize integer order and fractal derivatives in a given differential equation. Therefore, the scheme and results can be used for both cases. The proposed finite difference scheme is based on two stages. Fractal time derivatives are discretized by employing the proposed approach. For the scalar convection–diffusion equation, we derive the stability condition of the proposed fractal scheme. Using a nonlinear chemical reaction, the approach is also used to solve the Quantum Calculus model of a Williamson nanofluid’s unsteady Darcy–Forchheimer flow over flat and oscillatory sheets. The findings indicate a negative correlation between the velocity profile and the porosity parameter and inertia coefficient, with an increase in these factors resulting in a drop in the velocity profile. Additionally, the fractal scheme under consideration is being compared to the fractal Crank–Nicolson method, revealing that the proposed scheme exhibits a superior convergence speed compared to the fractal Crank–Nicolson method. Several problems involving the motion of non-Newtonian nanofluids through magnetic fields and porous media can be investigated with the help of the proposed numerical scheme. This research has implications for developing more efficient heat transfer and energy conversion devices based on nanofluids.
{"title":"Finite Difference Modeling of Time Fractal Impact on Unsteady Magneto-hydrodynamic Darcy–Forchheimer Flow in Non-Newtonian Nanofluids with the q-Derivative","authors":"A. Baazeem, Y. Nawaz, M. Arif","doi":"10.3390/fractalfract8010008","DOIUrl":"https://doi.org/10.3390/fractalfract8010008","url":null,"abstract":"This contribution addresses a fractal numerical scheme that can be employed for handling fractal time-dependent parabolic equations. The numerical scheme presented in this contribution can be used to discretize integer order and fractal derivatives in a given differential equation. Therefore, the scheme and results can be used for both cases. The proposed finite difference scheme is based on two stages. Fractal time derivatives are discretized by employing the proposed approach. For the scalar convection–diffusion equation, we derive the stability condition of the proposed fractal scheme. Using a nonlinear chemical reaction, the approach is also used to solve the Quantum Calculus model of a Williamson nanofluid’s unsteady Darcy–Forchheimer flow over flat and oscillatory sheets. The findings indicate a negative correlation between the velocity profile and the porosity parameter and inertia coefficient, with an increase in these factors resulting in a drop in the velocity profile. Additionally, the fractal scheme under consideration is being compared to the fractal Crank–Nicolson method, revealing that the proposed scheme exhibits a superior convergence speed compared to the fractal Crank–Nicolson method. Several problems involving the motion of non-Newtonian nanofluids through magnetic fields and porous media can be investigated with the help of the proposed numerical scheme. This research has implications for developing more efficient heat transfer and energy conversion devices based on nanofluids.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"115 14","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138958378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.3390/fractalfract8010005
Mohsan Raza, D. Breaz, Saima Mushtaq, Luminița-Ioana Cotîrlă, F. Tawfiq, Eleonora Rapeanu
The aim of this study is to investigate a certain sufficiency criterion for uniform convexity, strong starlikeness, and strong convexity of Rabtonov fractional exponential functions. We also study the starlikeness and convexity of order γ. Moreover, we find conditions so that the Rabotnov functions belong to the class of bounded analytic functions and Hardy spaces. Various consequences of these results are also presented.
{"title":"Geometric Properties and Hardy Spaces of Rabotnov Fractional Exponential Functions","authors":"Mohsan Raza, D. Breaz, Saima Mushtaq, Luminița-Ioana Cotîrlă, F. Tawfiq, Eleonora Rapeanu","doi":"10.3390/fractalfract8010005","DOIUrl":"https://doi.org/10.3390/fractalfract8010005","url":null,"abstract":"The aim of this study is to investigate a certain sufficiency criterion for uniform convexity, strong starlikeness, and strong convexity of Rabtonov fractional exponential functions. We also study the starlikeness and convexity of order γ. Moreover, we find conditions so that the Rabotnov functions belong to the class of bounded analytic functions and Hardy spaces. Various consequences of these results are also presented.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"81 11","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138956858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.3390/fractalfract8010006
Ziad M. Ali, Ahmed Mahdy Ahmed, H. Hasanien, S. Aleem
In this study, a nonlinear Archimedes wave swing (AWS) energy conversion system was employed to enable the use of irregular sea waves to provide useful electricity. Instead of the conventional PI controllers used in prior research, this study employed fractional-order PID (FOPID) controllers to control the back-to-back configuration of AWS. The aim was to maximize the energy yield from waves and maintain the grid voltage and the capacitor DC link voltage at predetermined values. In this study, six FOPID controllers were used to accomplish the control goals, leading to an array of thirty parameters required to be fine-tuned. In this regard, a hybrid jellyfish search optimizer and particle swarm optimization (HJSPSO) algorithm was adopted to select the optimal control gains. Verification of the performance of the proposed FOPID control system was achieved by comparing the system results to two conventional PID controllers and one FOPID controller. The conventional PID controllers were tuned using a recently presented metaheuristic algorithm called the Coot optimization algorithm (COOT) and the classical particle swarm optimization algorithm (PSO). Moreover, the FOPID was also tuned using the well-known genetic algorithm (GA). The system investigated in this study was subjected to various unsymmetrical and symmetrical fault disturbances. When compared with the standard COOT-PID, PSO-PID, and GA-FOPID controllers, the HJSPSO-FOPID results show a significant improvement in terms of performance and preserving control goals during system instability
{"title":"Optimal Design of Fractional-Order PID Controllers for a Nonlinear AWS Wave Energy Converter Using Hybrid Jellyfish Search and Particle Swarm Optimization","authors":"Ziad M. Ali, Ahmed Mahdy Ahmed, H. Hasanien, S. Aleem","doi":"10.3390/fractalfract8010006","DOIUrl":"https://doi.org/10.3390/fractalfract8010006","url":null,"abstract":"In this study, a nonlinear Archimedes wave swing (AWS) energy conversion system was employed to enable the use of irregular sea waves to provide useful electricity. Instead of the conventional PI controllers used in prior research, this study employed fractional-order PID (FOPID) controllers to control the back-to-back configuration of AWS. The aim was to maximize the energy yield from waves and maintain the grid voltage and the capacitor DC link voltage at predetermined values. In this study, six FOPID controllers were used to accomplish the control goals, leading to an array of thirty parameters required to be fine-tuned. In this regard, a hybrid jellyfish search optimizer and particle swarm optimization (HJSPSO) algorithm was adopted to select the optimal control gains. Verification of the performance of the proposed FOPID control system was achieved by comparing the system results to two conventional PID controllers and one FOPID controller. The conventional PID controllers were tuned using a recently presented metaheuristic algorithm called the Coot optimization algorithm (COOT) and the classical particle swarm optimization algorithm (PSO). Moreover, the FOPID was also tuned using the well-known genetic algorithm (GA). The system investigated in this study was subjected to various unsymmetrical and symmetrical fault disturbances. When compared with the standard COOT-PID, PSO-PID, and GA-FOPID controllers, the HJSPSO-FOPID results show a significant improvement in terms of performance and preserving control goals during system instability","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"71 18","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138956706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-19DOI: 10.3390/fractalfract8010003
Yifan Xu, Ying Luo, Xin Luo, Yangquan Chen, Wei Liu
A novel fractional-order model, incorporating coupled hysteresis and creep effects, is proposed for typical piezoelectric actuators in this study. Throughout the actuation process, various nonlinear behaviors such as piezoelectric hysteresis, non-local memory, peak transition, and creep nonlinearity are accurately characterized by the model. Offering a simpler structure and superior tracking performance compared to conventional models, the proposed fractional-order model parameters are identified using a method that integrates actuator dynamics and employs the particle swarm optimization algorithm. Experimental validation on a piezoelectric actuation platform confirms the model’s superior accuracy and simplified structure, contributing to a deeper understanding of piezoelectric actuation mechanisms and providing an efficient modeling tool for enhanced system performance.
{"title":"Fractional-Order Modeling of Piezoelectric Actuators with Coupled Hysteresis and Creep Effects","authors":"Yifan Xu, Ying Luo, Xin Luo, Yangquan Chen, Wei Liu","doi":"10.3390/fractalfract8010003","DOIUrl":"https://doi.org/10.3390/fractalfract8010003","url":null,"abstract":"A novel fractional-order model, incorporating coupled hysteresis and creep effects, is proposed for typical piezoelectric actuators in this study. Throughout the actuation process, various nonlinear behaviors such as piezoelectric hysteresis, non-local memory, peak transition, and creep nonlinearity are accurately characterized by the model. Offering a simpler structure and superior tracking performance compared to conventional models, the proposed fractional-order model parameters are identified using a method that integrates actuator dynamics and employs the particle swarm optimization algorithm. Experimental validation on a piezoelectric actuation platform confirms the model’s superior accuracy and simplified structure, contributing to a deeper understanding of piezoelectric actuation mechanisms and providing an efficient modeling tool for enhanced system performance.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":" 7","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138961732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-18DOI: 10.3390/fractalfract7120890
B. Shevtsov, O. Sheremetyeva
To understand how the temporal non-locality («memory») properties of a process affect its critical regimes, the power-law compound and time-fractional Poisson process is presented as a universal hereditary model of criticality. Seismicity is considered as an application of the theory of criticality. On the basis of the proposed hereditarian criticality model, the critical regimes of seismicity are investigated. It is shown that the seismic process has the property of «memory» (non-locality over time) and statistical time-dependence of events. With a decrease in the fractional exponent of the Poisson process, the relaxation slows down, which can be associated with the hardening of the medium and the accumulation of elastic energy. Delayed relaxation is accompanied by an abnormal increase in fluctuations, which is caused by the non-local correlations of random events over time. According to the found criticality indices, the seismic process is in subcritical regimes for the zero and first moments and in supercritical regimes for the second statistical moment of events’ reoccurrence frequencies distribution. The supercritical regimes indicate the instability of the deformation changes that can go into a non-stationary regime of a seismic process.
{"title":"Fractional Criticality Theory and Its Application in Seismology","authors":"B. Shevtsov, O. Sheremetyeva","doi":"10.3390/fractalfract7120890","DOIUrl":"https://doi.org/10.3390/fractalfract7120890","url":null,"abstract":"To understand how the temporal non-locality («memory») properties of a process affect its critical regimes, the power-law compound and time-fractional Poisson process is presented as a universal hereditary model of criticality. Seismicity is considered as an application of the theory of criticality. On the basis of the proposed hereditarian criticality model, the critical regimes of seismicity are investigated. It is shown that the seismic process has the property of «memory» (non-locality over time) and statistical time-dependence of events. With a decrease in the fractional exponent of the Poisson process, the relaxation slows down, which can be associated with the hardening of the medium and the accumulation of elastic energy. Delayed relaxation is accompanied by an abnormal increase in fluctuations, which is caused by the non-local correlations of random events over time. According to the found criticality indices, the seismic process is in subcritical regimes for the zero and first moments and in supercritical regimes for the second statistical moment of events’ reoccurrence frequencies distribution. The supercritical regimes indicate the instability of the deformation changes that can go into a non-stationary regime of a seismic process.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"43 S203","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138965295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-18DOI: 10.3390/fractalfract7120889
M. M. Al-Sawalha, S. Mukhtar, Rasool Shah, A. Ganie, Khaled Moaddy
The primary goal of this study is to create and characterise solitary wave solutions for the conformable Fractional Coupled Boussinesq-Whitham-Broer-Kaup Equations (FCBWBKEs), a model that governs shallow water waves. Through wave transformations and the chain rule, the authors used the modified Extended Direct Algebraic Method (mEDAM) for transforming FCBWBKEs into a more manageable Nonlinear Ordinary Differential Equation (NODE). This accomplishment is particularly noteworthy because it surpasses the drawbacks linked to both the Caputo and Riemann–Liouville definitions in complying to the chain rule. The study uses visual representations such as 3D, 2D, and contour graphs to demonstrate the dynamic nature of solitary wave solutions. Furthermore, the investigation of diverse wave phenomena such as kinks, shock waves, periodic waves, and bell-shaped kink waves highlights the range of knowledge obtained in the study of shallow water wave behavior. Overall, this study introduces novel methodologies that produce valuable and consistent results for the problem under consideration.
{"title":"Solitary Waves Propagation Analysis in Nonlinear Dynamical System of Fractional Coupled Boussinesq-Whitham-Broer-Kaup Equation","authors":"M. M. Al-Sawalha, S. Mukhtar, Rasool Shah, A. Ganie, Khaled Moaddy","doi":"10.3390/fractalfract7120889","DOIUrl":"https://doi.org/10.3390/fractalfract7120889","url":null,"abstract":"The primary goal of this study is to create and characterise solitary wave solutions for the conformable Fractional Coupled Boussinesq-Whitham-Broer-Kaup Equations (FCBWBKEs), a model that governs shallow water waves. Through wave transformations and the chain rule, the authors used the modified Extended Direct Algebraic Method (mEDAM) for transforming FCBWBKEs into a more manageable Nonlinear Ordinary Differential Equation (NODE). This accomplishment is particularly noteworthy because it surpasses the drawbacks linked to both the Caputo and Riemann–Liouville definitions in complying to the chain rule. The study uses visual representations such as 3D, 2D, and contour graphs to demonstrate the dynamic nature of solitary wave solutions. Furthermore, the investigation of diverse wave phenomena such as kinks, shock waves, periodic waves, and bell-shaped kink waves highlights the range of knowledge obtained in the study of shallow water wave behavior. Overall, this study introduces novel methodologies that produce valuable and consistent results for the problem under consideration.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"106 s415","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138965103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Chaos-based image encryption has become a prominent area of research in recent years. In comparison to ordinary chaotic systems, fractional-order chaotic systems tend to have a greater number of control parameters and more complex dynamical characteristics. Thus, an increasing number of researchers are introducing fractional-order chaotic systems to enhance the security of chaos-based image encryption. However, their suggested algorithms still suffer from some security, practicality, and efficiency problems. To address these problems, we first constructed a new fractional-order 3D Lorenz chaotic system and a 2D sinusoidally constrained polynomial hyper-chaotic map (2D-SCPM). Then, we elaborately developed a multi-image encryption algorithm based on the new fractional-order 3D Lorenz chaotic system and 2D-SCPM (MIEA-FCSM). The introduction of the fractional-order 3D Lorenz chaotic system with the fourth parameter not only enables MIEA-FCSM to have a significantly large key space but also enhances its overall security. Compared with recent alternatives, the structure of 2D-SCPM is simpler and more conducive to application implementation. In our proposed MIEA-FCSM, multi-channel fusion initially reduces the number of pixels to one-sixth of the original. Next, after two rounds of plaintext-related chaotic random substitution, dynamic diffusion, and fast scrambling, the fused 2D pixel matrix is eventually encrypted into the ciphertext one. According to numerous experiments and analyses, MIEA-FCSM obtained excellent scores for key space (2541), correlation coefficients (<0.004), information entropy (7.9994), NPCR (99.6098%), and UACI (33.4659%). Significantly, MIEA-FCSM also attained an average encryption rate as high as 168.5608 Mbps. Due to the superiority of the new fractional-order chaotic system, 2D-SCPM, and targeted designs, MIEA-FCSM outperforms many recently reported leading image encryption algorithms.
{"title":"Exploiting Newly Designed Fractional-Order 3D Lorenz Chaotic System and 2D Discrete Polynomial Hyper-Chaotic Map for High-Performance Multi-Image Encryption","authors":"Wei Feng, Quanwen Wang, Hui Liu, Yu Ren, Junhao Zhang, Shubo Zhang, Kun Qian, Heping Wen","doi":"10.3390/fractalfract7120887","DOIUrl":"https://doi.org/10.3390/fractalfract7120887","url":null,"abstract":"Chaos-based image encryption has become a prominent area of research in recent years. In comparison to ordinary chaotic systems, fractional-order chaotic systems tend to have a greater number of control parameters and more complex dynamical characteristics. Thus, an increasing number of researchers are introducing fractional-order chaotic systems to enhance the security of chaos-based image encryption. However, their suggested algorithms still suffer from some security, practicality, and efficiency problems. To address these problems, we first constructed a new fractional-order 3D Lorenz chaotic system and a 2D sinusoidally constrained polynomial hyper-chaotic map (2D-SCPM). Then, we elaborately developed a multi-image encryption algorithm based on the new fractional-order 3D Lorenz chaotic system and 2D-SCPM (MIEA-FCSM). The introduction of the fractional-order 3D Lorenz chaotic system with the fourth parameter not only enables MIEA-FCSM to have a significantly large key space but also enhances its overall security. Compared with recent alternatives, the structure of 2D-SCPM is simpler and more conducive to application implementation. In our proposed MIEA-FCSM, multi-channel fusion initially reduces the number of pixels to one-sixth of the original. Next, after two rounds of plaintext-related chaotic random substitution, dynamic diffusion, and fast scrambling, the fused 2D pixel matrix is eventually encrypted into the ciphertext one. According to numerous experiments and analyses, MIEA-FCSM obtained excellent scores for key space (2541), correlation coefficients (<0.004), information entropy (7.9994), NPCR (99.6098%), and UACI (33.4659%). Significantly, MIEA-FCSM also attained an average encryption rate as high as 168.5608 Mbps. Due to the superiority of the new fractional-order chaotic system, 2D-SCPM, and targeted designs, MIEA-FCSM outperforms many recently reported leading image encryption algorithms.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"14 8","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138967546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-15DOI: 10.3390/fractalfract7120886
A. Al Themairi, G. Mahmoud, A. Farghaly, T. Abed-Elhameed
This paper introduces the complex Rayleigh–van-der- Pol–Duffing oscillators (RVDOs), which are hyperchaotic and can be autonomous or nonautonomous. The fundamental dynamics of the autonomous and nonautonomous complex RVDOs, including dissipation, symmetry, fixed points, and stability, are studied. These oscillators are found in various necessary fields of physics and engineering. The paper proposes a scheme to achieve phase synchronization (PS) and antiphase synchronization (APS) for different dimensional models. These kinds of synchronization are considered a generalization of several other types of synchronization. We use the active control method based on Lyapunov’s stability theory for this scheme. By analytically determining the control functions, the scheme achieved PS and APS. Our scheme is applied to study the PS of hyperchaotic behaviors for two distinct hyperchaotic nonautonomous and autonomous complex RVDOs. Additionally, the scheme is employed to achieve the APS of a chaotic real nonautonomous RVDO and a hyperchaotic complex autonomous RVDO, including those with different dimensions. Our work presents numerical results that plot the amplitudes and phases of these hyperchaotic behaviors, demonstrating the achievement of the PS and APS. The encryption and decryption of grayscale images are researched based on APS. The experimental results of image encryption and decryption are computed with information entropy, visual analysis, and histograms.
{"title":"Complex Rayleigh–van-der-Pol–Duffing Oscillators: Dynamics, Phase, Antiphase Synchronization, and Image Encryption","authors":"A. Al Themairi, G. Mahmoud, A. Farghaly, T. Abed-Elhameed","doi":"10.3390/fractalfract7120886","DOIUrl":"https://doi.org/10.3390/fractalfract7120886","url":null,"abstract":"This paper introduces the complex Rayleigh–van-der- Pol–Duffing oscillators (RVDOs), which are hyperchaotic and can be autonomous or nonautonomous. The fundamental dynamics of the autonomous and nonautonomous complex RVDOs, including dissipation, symmetry, fixed points, and stability, are studied. These oscillators are found in various necessary fields of physics and engineering. The paper proposes a scheme to achieve phase synchronization (PS) and antiphase synchronization (APS) for different dimensional models. These kinds of synchronization are considered a generalization of several other types of synchronization. We use the active control method based on Lyapunov’s stability theory for this scheme. By analytically determining the control functions, the scheme achieved PS and APS. Our scheme is applied to study the PS of hyperchaotic behaviors for two distinct hyperchaotic nonautonomous and autonomous complex RVDOs. Additionally, the scheme is employed to achieve the APS of a chaotic real nonautonomous RVDO and a hyperchaotic complex autonomous RVDO, including those with different dimensions. Our work presents numerical results that plot the amplitudes and phases of these hyperchaotic behaviors, demonstrating the achievement of the PS and APS. The encryption and decryption of grayscale images are researched based on APS. The experimental results of image encryption and decryption are computed with information entropy, visual analysis, and histograms.","PeriodicalId":12435,"journal":{"name":"Fractal and Fractional","volume":"21 22","pages":""},"PeriodicalIF":5.4,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138999876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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