Pub Date : 2024-12-30DOI: 10.1016/j.chaos.2024.115943
Ivan A. Moloshnikov, Alexander G. Sboev, Aleksandr A. Kutukov, Roman B. Rybka, Mikhail S. Kuvakin, Oleg O. Fedorov, Saveliy V. Zavertyaev
The paper addresses the practically significant problem of transmitting signals through nonlinear optical media by solving generalized nonlinear Schrödinger equations using various modifications of Physics-Informed Neural Networks (PINNs). The study provides numerical soliton solutions for Schrödinger equations of the order as high as four. To tackle this problem, the paper compares segmental modifications of PINNs, including BC-PINNs, FB-PINNs, and MoE-PINNs. Additionally, an adaptive option for selecting collocation points is proposed and explored. The efficiency of the numerical solutions is evaluated using three approaches: comparison with the precise analytical solutions, and two metrics based on conservation laws. The results show that the modified segmentation approach, combined with the developed adaptive selection of collocation points, greatly improves the accuracy and the convergence of PINNs compared to the initial version of the method. On such example problems as the interaction of a soliton with a Gaussian function, two solitons interaction, and the solution of a 4th-order equation, the proposed method demonstrates improved convergence of the numerical solution.
{"title":"Analysis of neural network methods for obtaining soliton solutions of the nonlinear Schrödinger equation","authors":"Ivan A. Moloshnikov, Alexander G. Sboev, Aleksandr A. Kutukov, Roman B. Rybka, Mikhail S. Kuvakin, Oleg O. Fedorov, Saveliy V. Zavertyaev","doi":"10.1016/j.chaos.2024.115943","DOIUrl":"https://doi.org/10.1016/j.chaos.2024.115943","url":null,"abstract":"The paper addresses the practically significant problem of transmitting signals through nonlinear optical media by solving generalized nonlinear Schrödinger equations using various modifications of Physics-Informed Neural Networks (PINNs). The study provides numerical soliton solutions for Schrödinger equations of the order as high as four. To tackle this problem, the paper compares segmental modifications of PINNs, including BC-PINNs, FB-PINNs, and MoE-PINNs. Additionally, an adaptive option for selecting collocation points is proposed and explored. The efficiency of the numerical solutions is evaluated using three approaches: comparison with the precise analytical solutions, and two metrics based on conservation laws. The results show that the modified segmentation approach, combined with the developed adaptive selection of collocation points, greatly improves the accuracy and the convergence of PINNs compared to the initial version of the method. On such example problems as the interaction of a soliton with a Gaussian function, two solitons interaction, and the solution of a 4th-order equation, the proposed method demonstrates improved convergence of the numerical solution.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"40 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912300","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-30DOI: 10.1016/j.chaos.2024.115960
D. Zhevnenko, F. Meshchaninov, V. Shmakov, E. Kharchenko, V. Kozhevnikov, A. Chernova, A. Belov, A. Mikhaylov, E. Gornev
Modeling the switching dynamics of memristive devices poses significant challenges for real-world applications, particularly in achieving long-term operational stability. While conventional compact models are effective for short-term simulations, they fail to capture the degradation effects and complexities associated with extended switching behavior. In this work, we propose a novel framework for forecasting memristor switching series using state-of-the-art deep learning architectures. Experimental data from Au/Ta/ZrO₂(Y)/TaOx/TiN/Ti-based memristors were used to compare a classical compact model—featuring a linear drift model with ARIMA corrections—against advanced neural networks, including TimesNet, FredFormer, ATFNet, and SparseTSF. Our results demonstrate that deep learning models, particularly TimesNet, significantly improve predictive accuracy and robustness over long-term switching series. This study provides a foundation for integrating deep learning into memristor modeling, paving the way for more reliable and scalable simulations.
{"title":"Introducing a neural network approach to memristor dynamics: A comparative study with traditional compact models","authors":"D. Zhevnenko, F. Meshchaninov, V. Shmakov, E. Kharchenko, V. Kozhevnikov, A. Chernova, A. Belov, A. Mikhaylov, E. Gornev","doi":"10.1016/j.chaos.2024.115960","DOIUrl":"https://doi.org/10.1016/j.chaos.2024.115960","url":null,"abstract":"Modeling the switching dynamics of memristive devices poses significant challenges for real-world applications, particularly in achieving long-term operational stability. While conventional compact models are effective for short-term simulations, they fail to capture the degradation effects and complexities associated with extended switching behavior. In this work, we propose a novel framework for forecasting memristor switching series using state-of-the-art deep learning architectures. Experimental data from Au/Ta/ZrO₂(Y)/TaOx/TiN/Ti-based memristors were used to compare a classical compact model—featuring a linear drift model with ARIMA corrections—against advanced neural networks, including TimesNet, FredFormer, ATFNet, and SparseTSF. Our results demonstrate that deep learning models, particularly TimesNet, significantly improve predictive accuracy and robustness over long-term switching series. This study provides a foundation for integrating deep learning into memristor modeling, paving the way for more reliable and scalable simulations.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"3 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912295","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-30DOI: 10.1016/j.chaos.2024.115936
Zehui Zhang, Kangci Zhu, Fang Wang
Epidemics pose a significant threat to humanity. During the early stages of an outbreak, individuals often lack comprehensive or timely access to disease-related information. The primary mode of information propagation is indirect, primarily originating from friends or their extended networks. The primary mode of information propagation is indirect, primarily originating from friends or their extended networks. Additionally, the spread of information is influenced by the incubation period of infected individuals. In this study, we develop a novel information–disease coupled propagation model, integrating both indirect information transmission and individual disease incubation periods into the dynamics of information–disease interaction on multiplex networks. It is called time-delay ID-CIP. We derive the epidemic outbreak threshold using a microscopic Markov chain approach and compare our model with classical pairwise interaction propagation and recent higher-order models. The findings suggest that the proposed information propagation mechanism is more effective in suppressing disease spread. Numerical simulations reveal that prior to an outbreak, awareness density converges to zero in the steady state, helping prevent epidemic-related rumor propagation. The disease’s incubation period has no effect on the density of the infected population in the steady state; however, it significantly impacts the density of individual’s epidemic-related awareness.
{"title":"Indirect information propagation model with time-delay effect on multiplex networks","authors":"Zehui Zhang, Kangci Zhu, Fang Wang","doi":"10.1016/j.chaos.2024.115936","DOIUrl":"https://doi.org/10.1016/j.chaos.2024.115936","url":null,"abstract":"Epidemics pose a significant threat to humanity. During the early stages of an outbreak, individuals often lack comprehensive or timely access to disease-related information. The primary mode of information propagation is indirect, primarily originating from friends or their extended networks. The primary mode of information propagation is indirect, primarily originating from friends or their extended networks. Additionally, the spread of information is influenced by the incubation period of infected individuals. In this study, we develop a novel information–disease coupled propagation model, integrating both indirect information transmission and individual disease incubation periods into the dynamics of information–disease interaction on multiplex networks. It is called time-delay ID-CIP. We derive the epidemic outbreak threshold using a microscopic Markov chain approach and compare our model with classical pairwise interaction propagation and recent higher-order models. The findings suggest that the proposed information propagation mechanism is more effective in suppressing disease spread. Numerical simulations reveal that prior to an outbreak, awareness density converges to zero in the steady state, helping prevent epidemic-related rumor propagation. The disease’s incubation period has no effect on the density of the infected population in the steady state; however, it significantly impacts the density of individual’s epidemic-related awareness.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"160 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-30DOI: 10.1016/j.chaos.2024.115950
D. Bazeia, M.A. Liao, M.A. Marques
This work deals with systems of two real scalar fields coupled to impurity functions, meant to model inhomogeneities often encountered in real physical applications. We investigate the theoretical properties of these systems and some of the consequences of impurity doping. We show that the theory may be modified in a way that preserves some BPS sectors, while also greatly impacting the behavior and internal structure of the solution, and exemplify those results with an investigation of a few interesting models in which impurities are coupled to a theory with a quartic potential. It is shown that, in the presence of impurities, the asymptotic behavior of field configurations may be changed, leading to solutions with different long-range properties, which are relevant to several physical applications. Our examples also highlight other important consequences that may follow from the addition of impurities, such as the presence of zero-modes that can significantly change the internal structure of a given solution without altering its energy, the creation of new topological sectors that did not exist in the impurity-free theory, and the possibility of stable, nontrivial configurations generated by topologically trivial boundary conditions. We have also shown that it is sometimes possible to find energy minimizers in BPS sectors which were unpopulated in the canonical theory. These features show that impurities allow for significant flexibility in both the form of energy minimizers and the boundary conditions used to generate them, which may potentially broaden the range of applicability of the theory.
{"title":"Two-field models in the presence of impurities","authors":"D. Bazeia, M.A. Liao, M.A. Marques","doi":"10.1016/j.chaos.2024.115950","DOIUrl":"https://doi.org/10.1016/j.chaos.2024.115950","url":null,"abstract":"This work deals with systems of two real scalar fields coupled to impurity functions, meant to model inhomogeneities often encountered in real physical applications. We investigate the theoretical properties of these systems and some of the consequences of impurity doping. We show that the theory may be modified in a way that preserves some BPS sectors, while also greatly impacting the behavior and internal structure of the solution, and exemplify those results with an investigation of a few interesting models in which impurities are coupled to a theory with a quartic potential. It is shown that, in the presence of impurities, the asymptotic behavior of field configurations may be changed, leading to solutions with different long-range properties, which are relevant to several physical applications. Our examples also highlight other important consequences that may follow from the addition of impurities, such as the presence of zero-modes that can significantly change the internal structure of a given solution without altering its energy, the creation of new topological sectors that did not exist in the impurity-free theory, and the possibility of stable, nontrivial configurations generated by topologically trivial boundary conditions. We have also shown that it is sometimes possible to find energy minimizers in BPS sectors which were unpopulated in the canonical theory. These features show that impurities allow for significant flexibility in both the form of energy minimizers and the boundary conditions used to generate them, which may potentially broaden the range of applicability of the theory.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"17 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-30DOI: 10.1016/j.chaos.2024.115945
Jinlong Ma, Hongfei Zhao
Hypergraphs provide a precise framework for capturing higher-order interactions in complex social systems, as well as synergy and discounting effects describe the nonlinear accumulation of benefits in social dilemmas initially. Inspired by the accurate representation of environmental evaluation trends by higher-order reputation, we propose a novel model called third-order reputation-based dynamic assessment to adjust synergy and discounting effects dynamically. Specifically, behavioral shifts are evaluated using third-order reputation, considering both personal and opponent reputations, alongside the dynamic reputation threshold adapting based on global average reputation and local conditions. Synergy and discounting effects dynamically adjust based on the gap between group reputation and the established threshold. Numerical simulations reveal that the third-order reputation-based dynamic assessment effectively promotes the evolution of cooperation in spatial public goods games on uniform random hypergraphs. An increase in the environmental reputation-adjusted investment factor α, the reputation gap factor β, and the reputation change value w all contribute to enhancing cooperation. The local–global reputation weighting factor θ indicates that global reputation has a more significant impact on promoting cooperation than local reputation. All four reputation rules promote cooperation, with the Shunning rule resulting in the clearest distinction between full cooperation and defection. Image Scoring is particularly effective in reducing defection. Furthermore, Simple Standing and Stern Judging similarly reduce defection, but they achieve lower levels of cooperation than Shunning.
{"title":"Synergy and discounting effects in spatial public goods games on hypergraphs: The role of third-order reputation-based dynamic assessment","authors":"Jinlong Ma, Hongfei Zhao","doi":"10.1016/j.chaos.2024.115945","DOIUrl":"https://doi.org/10.1016/j.chaos.2024.115945","url":null,"abstract":"Hypergraphs provide a precise framework for capturing higher-order interactions in complex social systems, as well as synergy and discounting effects describe the nonlinear accumulation of benefits in social dilemmas initially. Inspired by the accurate representation of environmental evaluation trends by higher-order reputation, we propose a novel model called third-order reputation-based dynamic assessment to adjust synergy and discounting effects dynamically. Specifically, behavioral shifts are evaluated using third-order reputation, considering both personal and opponent reputations, alongside the dynamic reputation threshold adapting based on global average reputation and local conditions. Synergy and discounting effects dynamically adjust based on the gap between group reputation and the established threshold. Numerical simulations reveal that the third-order reputation-based dynamic assessment effectively promotes the evolution of cooperation in spatial public goods games on uniform random hypergraphs. An increase in the environmental reputation-adjusted investment factor <mml:math altimg=\"si138.svg\" display=\"inline\"><mml:mi>α</mml:mi></mml:math>, the reputation gap factor <mml:math altimg=\"si145.svg\" display=\"inline\"><mml:mi>β</mml:mi></mml:math>, and the reputation change value <mml:math altimg=\"si159.svg\" display=\"inline\"><mml:mi>w</mml:mi></mml:math> all contribute to enhancing cooperation. The local–global reputation weighting factor <mml:math altimg=\"si152.svg\" display=\"inline\"><mml:mi>θ</mml:mi></mml:math> indicates that global reputation has a more significant impact on promoting cooperation than local reputation. All four reputation rules promote cooperation, with the Shunning rule resulting in the clearest distinction between full cooperation and defection. Image Scoring is particularly effective in reducing defection. Furthermore, Simple Standing and Stern Judging similarly reduce defection, but they achieve lower levels of cooperation than Shunning.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"70 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-30DOI: 10.1016/j.chaos.2024.115948
Kiran Asma, Muhammad Asif Zahoor Raja, Chuan-Yu Chang, Muhammad Junaid Ali Asif Raja, Muhammad Shoaib
A vulnerable mobile device remains a critical concern for the sustainable development of information security infrastructure, and the massive increase in mobile malware propagation further amplifies the need for heightened cybersecurity awareness among mobile users. In this paper, a novel framework is presented to explore the machine learning solutions for nonlinear fractional cybersecurity awareness on mobile malware propagation (NFCSA-MMP) model by constructing multilayer autoregressive exogenous networks (MARXNs) trained iteratively by the Levenberg-Marquardt (MARXNs-LM) algorithm. The NFCSA-MMP system represented with Unaware-Susceptible, Aware-Susceptible, Latent, Breakout, Quarantine and Recovery fractional compartments models the different stages of mobile devices states during malware propagation and recovery. To scrutinize the propagation mechanism of mobile malware, the simulation data generated by utilizing Grünwald–Letnikov (GL) fractional finite difference-based computing procedure for NFCSA-MMP model for both integer and fractional ordered values corresponding to variation in the rate of security-aware mobile devices connected to a network, the rate of latent mobile devices becomes breakout, and the recovery rates of latent, breakout, and quarantined devices due to treatment. The proposed methodology MARXNs-LM is executed on acquired datasets randomly sectioned into training, testing and validation samples by achieving the minimum value of the mean square error (MSE) to determine the machine predictive solution of NFCSA-MMP for each scenario. The vigorousness of proposed MARXNs-LM scheme proven by comparative analysis on convergence trends on reduction of MSE, magnitude of absolute deviation, input-output correlation, error histograms and error autocorrelation statistics for solving stiff NFCSA-MMP model.
{"title":"Machine learning-driven exogenous neural architecture for nonlinear fractional cybersecurity awareness model in mobile malware propagation","authors":"Kiran Asma, Muhammad Asif Zahoor Raja, Chuan-Yu Chang, Muhammad Junaid Ali Asif Raja, Muhammad Shoaib","doi":"10.1016/j.chaos.2024.115948","DOIUrl":"https://doi.org/10.1016/j.chaos.2024.115948","url":null,"abstract":"A vulnerable mobile device remains a critical concern for the sustainable development of information security infrastructure, and the massive increase in mobile malware propagation further amplifies the need for heightened cybersecurity awareness among mobile users. In this paper, a novel framework is presented to explore the machine learning solutions for nonlinear fractional cybersecurity awareness on mobile malware propagation (NFCSA-MMP) model by constructing multilayer autoregressive exogenous networks (MARXNs) trained iteratively by the Levenberg-Marquardt (MARXNs-LM) algorithm. The NFCSA-MMP system represented with Unaware-Susceptible, Aware-Susceptible, Latent, Breakout, Quarantine and Recovery fractional compartments models the different stages of mobile devices states during malware propagation and recovery. To scrutinize the propagation mechanism of mobile malware, the simulation data generated by utilizing Grünwald–Letnikov (GL) fractional finite difference-based computing procedure for NFCSA-MMP model for both integer and fractional ordered values corresponding to variation in the rate of security-aware mobile devices connected to a network, the rate of latent mobile devices becomes breakout, and the recovery rates of latent, breakout, and quarantined devices due to treatment. The proposed methodology MARXNs-LM is executed on acquired datasets randomly sectioned into training, testing and validation samples by achieving the minimum value of the mean square error (MSE) to determine the machine predictive solution of NFCSA-MMP for each scenario. The vigorousness of proposed MARXNs-LM scheme proven by comparative analysis on convergence trends on reduction of MSE, magnitude of absolute deviation, input-output correlation, error histograms and error autocorrelation statistics for solving stiff NFCSA-MMP model.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"20 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider a probability space <mml:math altimg="si1.svg" display="inline"><mml:mrow><mml:mo>(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>ℱ</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. This paper primarily investigates a general multifractal formalism within the probability space <mml:math altimg="si1.svg" display="inline"><mml:mrow><mml:mo>(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>ℱ</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. Our first objective is to introduce a multifractal generalization of the Hausdorff and packing measures. We then explore the properties of the general multifractal Hausdorff measure and the multifractal packing measure within <mml:math altimg="si1.svg" display="inline"><mml:mrow><mml:mo>(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>ℱ</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, examining their implications for the general multifractal spectrum functions. We investigate the relationship between the general multifractal measures and the nature of general multifractal dimensions within this framework. Additionally, we obtain an analogue of Frostman’s lemma for the general multifractal Hausdorff and packing measures in probability spaces. Using this analogue, we derive representations for the functions <mml:math altimg="si4.svg" display="inline"><mml:msubsup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ℋ</mml:mi></mml:mrow><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>̃</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:msubsup></mml:math> and <mml:math altimg="si5.svg" display="inline"><mml:msubsup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="script">P</mml:mi></mml:mrow><mml:mrow><mml:mover accent="true"><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>̃</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:msubsup></mml:math>. Furthermore, we provide a technique to demonstrate that <mml:math altimg="si6.svg" display="inline"><mml:mi mathvariant="normal">E</mml:mi></mml:math> is an <mml:math altimg="si7.svg" display="inline"><mml:mrow><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>π</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-fractal with respect to <mml:math altimg="si8.svg" display="inline"><mml:mi>τ</mml:mi></mml:math>, leading to density theorems for the multifractal Hausdorff and packing measures in these probability spaces. Finally, we present a general theorem for multifractal formalism on probability spaces, deriving results for general multifractal Hausdorff and packing functions that vary with respect to arbitrary probability measures at points <mml:math altimg="si9.svg" display="inline"><mml:mi>α</mml:mi></mml:math> where the multifractal functions <mml:math altimg="si10.svg" display="inline"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mm
考虑一个概率空间（Z,,τ）。本文主要研究概率空间（Z,,τ）内的一般多重分形形式。我们的第一个目标是介绍Hausdorff和packing测度的多重分形推广。然后，我们探讨了一般多重分形Hausdorff测度和多重分形包装测度在（Z,,τ）内的性质，研究了它们对一般多重分形谱函数的含义。我们在这个框架内研究了一般多重分形测度与一般多重分形维数性质之间的关系。此外，我们还得到了概率空间中一般多重分形Hausdorff和packing测度的Frostman引理的一个类比。利用这种类比，我们导出了函数b h h π n和bPπ n的表示。此外，我们提供了一种技术来证明E是一个关于τ的（α,π）分形，从而得到了多重分形Hausdorff的密度定理和这些概率空间中的填充测度。最后，我们给出了概率空间上多重分形形式的一个一般定理，得到了在多重分形函数b h π π (α)和bpi π π (α)不同的点α处随任意概率测度变化的一般多重分形Hausdorff函数和packing函数的结果。
{"title":"Probabilistic spaces and generalized dimensions: A multifractal approach","authors":"Lixin Guo, Bilel Selmi, Zhiming Li, Haythem Zyoudi","doi":"10.1016/j.chaos.2024.115953","DOIUrl":"https://doi.org/10.1016/j.chaos.2024.115953","url":null,"abstract":"Consider a probability space <mml:math altimg=\"si1.svg\" display=\"inline\"><mml:mrow><mml:mo>(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>ℱ</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. This paper primarily investigates a general multifractal formalism within the probability space <mml:math altimg=\"si1.svg\" display=\"inline\"><mml:mrow><mml:mo>(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>ℱ</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>. Our first objective is to introduce a multifractal generalization of the Hausdorff and packing measures. We then explore the properties of the general multifractal Hausdorff measure and the multifractal packing measure within <mml:math altimg=\"si1.svg\" display=\"inline\"><mml:mrow><mml:mo>(</mml:mo><mml:mi>Z</mml:mi><mml:mo>,</mml:mo><mml:mi>ℱ</mml:mi><mml:mo>,</mml:mo><mml:mi>τ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>, examining their implications for the general multifractal spectrum functions. We investigate the relationship between the general multifractal measures and the nature of general multifractal dimensions within this framework. Additionally, we obtain an analogue of Frostman’s lemma for the general multifractal Hausdorff and packing measures in probability spaces. Using this analogue, we derive representations for the functions <mml:math altimg=\"si4.svg\" display=\"inline\"><mml:msubsup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>ℋ</mml:mi></mml:mrow><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>̃</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:msubsup></mml:math> and <mml:math altimg=\"si5.svg\" display=\"inline\"><mml:msubsup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant=\"script\">P</mml:mi></mml:mrow><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:mi>π</mml:mi></mml:mrow><mml:mrow><mml:mo>̃</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:msubsup></mml:math>. Furthermore, we provide a technique to demonstrate that <mml:math altimg=\"si6.svg\" display=\"inline\"><mml:mi mathvariant=\"normal\">E</mml:mi></mml:math> is an <mml:math altimg=\"si7.svg\" display=\"inline\"><mml:mrow><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>π</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-fractal with respect to <mml:math altimg=\"si8.svg\" display=\"inline\"><mml:mi>τ</mml:mi></mml:math>, leading to density theorems for the multifractal Hausdorff and packing measures in these probability spaces. Finally, we present a general theorem for multifractal formalism on probability spaces, deriving results for general multifractal Hausdorff and packing functions that vary with respect to arbitrary probability measures at points <mml:math altimg=\"si9.svg\" display=\"inline\"><mml:mi>α</mml:mi></mml:math> where the multifractal functions <mml:math altimg=\"si10.svg\" display=\"inline\"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mm","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"67 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2024-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-28DOI: 10.1016/j.chaos.2024.115952
Ali Murat Garipcan, Yılmaz Aydin, Fatih Özkaynak
In this study, an innovative substitution box (s-box) method is proposed, which combines the complexity and unpredictability of chaotic systems with the natural randomness and security features of Deoxyribonucleic Acid (DNA) encoding, in response to the increasing security requirements in the field of modern cryptography. This method, based on the integration of a two-dimensional (2D) hyper-chaotic Vincent map (VM) and DNA encoding techniques, aims to produce secure and high-performance s-boxes. To evaluate the reliability of the s-boxes, fundamental cryptographic criteria such as nonlinearity (NL), strict avalanche criterion (SAC), differential probability (DP), linear approximation probability (LAP), and bit independence criterion (BIC) are considered. Additionally, a novel chaos-based evolutionary optimization algorithm is proposed to optimize the NL criterion of the s-boxes. This algorithm offers a cost-effective and highly efficient alternative compared to classical metaheuristic methods, providing a balanced structure between security and performance. Experimental findings and comparison results demonstrate that the proposed s-boxes offer effective and reliable solutions for modern cryptographic applications, such as secure data transmission and storage. From a broader perspective, the study makes a meaningful contribution to the literature in the field of data security by presenting a high-performance s-box design meeting the security requirements of cryptographic systems and is resilient against current threats.
{"title":"An efficient 2D hyper chaos and DNA encoding-based s-box generation method using chaotic evolutionary improvement algorithm for nonlinearity","authors":"Ali Murat Garipcan, Yılmaz Aydin, Fatih Özkaynak","doi":"10.1016/j.chaos.2024.115952","DOIUrl":"https://doi.org/10.1016/j.chaos.2024.115952","url":null,"abstract":"In this study, an innovative substitution box (s-box) method is proposed, which combines the complexity and unpredictability of chaotic systems with the natural randomness and security features of Deoxyribonucleic Acid (DNA) encoding, in response to the increasing security requirements in the field of modern cryptography. This method, based on the integration of a two-dimensional (2D) hyper-chaotic Vincent map (VM) and DNA encoding techniques, aims to produce secure and high-performance s-boxes. To evaluate the reliability of the s-boxes, fundamental cryptographic criteria such as nonlinearity (NL), strict avalanche criterion (SAC), differential probability (DP), linear approximation probability (LAP), and bit independence criterion (BIC) are considered. Additionally, a novel chaos-based evolutionary optimization algorithm is proposed to optimize the NL criterion of the s-boxes. This algorithm offers a cost-effective and highly efficient alternative compared to classical metaheuristic methods, providing a balanced structure between security and performance. Experimental findings and comparison results demonstrate that the proposed s-boxes offer effective and reliable solutions for modern cryptographic applications, such as secure data transmission and storage. From a broader perspective, the study makes a meaningful contribution to the literature in the field of data security by presenting a high-performance s-box design meeting the security requirements of cryptographic systems and is resilient against current threats.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"46 12 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-28DOI: 10.1016/j.chaos.2024.115939
Yakub Kayode Saheed, Sanjay Misra
The integration of Cyber-Physical Systems (CPS) within the Internet of Things (IoT) ecosystem has transformed various sectors, enabling intelligent, interconnected environments that blend computational and physical processes. However, the security and privacy vulnerabilities within CPS-IoT networks remain critical, as anomalies can lead to severe, system-wide consequences. To address these challenges, this research introduces a novel, explainable, privacy-preserving Deep Neural Network (DNN) framework for anomaly detection in CPS-enabled IoT networks. While deep learning models are widely used in Intrusion Detection Systems (IDSs) for their capability to analyze vast data sources, their high false-positive rates and lack of interpretability present limitations. Our framework, therefore, employs a deep SHpley Additive exPlanations (SHAP) technique to clarify the DNN's decision-making process, aiding users and cybersecurity experts in validating and reinforcing the system's resilience. This approach was tested on two state-of-the-art datasets—Edge-IIoTset and X-IIoTID—demonstrating outstanding results. For binary classification, both datasets achieved 100 % accuracy, precision, recall, and F1-score, while multi-class scenarios reached nearly perfect metrics, with Edge-IIoTset achieving 99.98 % accuracy and X-IIoTID achieving 99.99 %. Additionally, our model showed significantly faster training times without compromising testing efficiency. The results confirm that this proposed explainable DNN framework offers robust, real-time, and privacy-preserving intrusion detection, enhancing CPS-IoT networks' defenses against advanced cyber threats.
{"title":"CPS-IoT-PPDNN: A new explainable privacy preserving DNN for resilient anomaly detection in Cyber-Physical Systems-enabled IoT networks","authors":"Yakub Kayode Saheed, Sanjay Misra","doi":"10.1016/j.chaos.2024.115939","DOIUrl":"https://doi.org/10.1016/j.chaos.2024.115939","url":null,"abstract":"The integration of Cyber-Physical Systems (CPS) within the Internet of Things (IoT) ecosystem has transformed various sectors, enabling intelligent, interconnected environments that blend computational and physical processes. However, the security and privacy vulnerabilities within CPS-IoT networks remain critical, as anomalies can lead to severe, system-wide consequences. To address these challenges, this research introduces a novel, explainable, privacy-preserving Deep Neural Network (DNN) framework for anomaly detection in CPS-enabled IoT networks. While deep learning models are widely used in Intrusion Detection Systems (IDSs) for their capability to analyze vast data sources, their high false-positive rates and lack of interpretability present limitations. Our framework, therefore, employs a deep SHpley Additive exPlanations (SHAP) technique to clarify the DNN's decision-making process, aiding users and cybersecurity experts in validating and reinforcing the system's resilience. This approach was tested on two state-of-the-art datasets—Edge-IIoTset and X-IIoTID—demonstrating outstanding results. For binary classification, both datasets achieved 100 % accuracy, precision, recall, and F1-score, while multi-class scenarios reached nearly perfect metrics, with Edge-IIoTset achieving 99.98 % accuracy and X-IIoTID achieving 99.99 %. Additionally, our model showed significantly faster training times without compromising testing efficiency. The results confirm that this proposed explainable DNN framework offers robust, real-time, and privacy-preserving intrusion detection, enhancing CPS-IoT networks' defenses against advanced cyber threats.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"36 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The influence of the epileptiform neuronal activity on the response of a CMOS-integrated ZrO2(Y)-based memristive crossbar and its conductivity was studied. Epileptiform neuronal activity was obtained in vitro in the hippocampal slices of laboratory mice using 4-aminopyridine experimental model. Synaptic plasticity of the memristive crossbar induced by epileptiform neuronal activity pulses was detected. Qualitatively, the results obtained in the case of normal (without pathologies) and epileptiform neuronal activity with and without noise coincide. For quantitative analysis, the value of the relative change in synaptic weight has been calculated for such important biological mechanisms of synapses as paired-pulse facilitation/depression, post-tetanic potentiation/depression, and long-term potentiation/depression. It has been shown that average value of the relative change in synaptic weight and its scatter are smaller mainly in the case of epileptiform neuronal activity pulses. An effect of the influence of noise included in the neuronal activity was found, which consists in the fact that the current response of the memristive crossbar is smaller in the presence of noise. The results of this study can be used in the development of new generation hardware-implemented computing devices with high performance and energy efficiency for the tasks of restorative medicine and robotics. In particular, using these results, neurohybrid devices can be developed for processing epileptiform activity in real time and for its suppression.
{"title":"Investigation of in vitro neuronal activity processing using a CMOS-integrated ZrO2(Y)-based memristive crossbar","authors":"M.N. Koryazhkina, A.V. Lebedeva, D.D. Pakhomova, I.N. Antonov, V.V. Razin, E.D. Budylina, A.I. Belov, A.N. Mikhaylov, A.A. Konakov","doi":"10.1016/j.chaos.2024.115959","DOIUrl":"https://doi.org/10.1016/j.chaos.2024.115959","url":null,"abstract":"The influence of the epileptiform neuronal activity on the response of a CMOS-integrated ZrO<ce:inf loc=\"post\">2</ce:inf>(Y)-based memristive crossbar and its conductivity was studied. Epileptiform neuronal activity was obtained <ce:italic>in vitro</ce:italic> in the hippocampal slices of laboratory mice using 4-aminopyridine experimental model. Synaptic plasticity of the memristive crossbar induced by epileptiform neuronal activity pulses was detected. Qualitatively, the results obtained in the case of normal (without pathologies) and epileptiform neuronal activity with and without noise coincide. For quantitative analysis, the value of the relative change in synaptic weight has been calculated for such important biological mechanisms of synapses as paired-pulse facilitation/depression, post-tetanic potentiation/depression, and long-term potentiation/depression. It has been shown that average value of the relative change in synaptic weight and its scatter are smaller mainly in the case of epileptiform neuronal activity pulses. An effect of the influence of noise included in the neuronal activity was found, which consists in the fact that the current response of the memristive crossbar is smaller in the presence of noise. The results of this study can be used in the development of new generation hardware-implemented computing devices with high performance and energy efficiency for the tasks of restorative medicine and robotics. In particular, using these results, neurohybrid devices can be developed for processing epileptiform activity in real time and for its suppression.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"54 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2024-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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