In this paper,an adaptive polynomial chaos expansion method(PCE)based on the method of moments(MoM)is proposed to construct surrogate models for electromagnetic scattering and further sensitivity analysis.The MoM is a...In this paper,an adaptive polynomial chaos expansion method(PCE)based on the method of moments(MoM)is proposed to construct surrogate models for electromagnetic scattering and further sensitivity analysis.The MoM is applied to accurately solve the electric field integral equation(EFIE)of electromagnetic scattering from homogeneous dielectric targets.Within the bistatic radar cross section(RCS)as the research object,the adaptive PCE algorithm is devoted to selecting the appropriate order to construct the multivariate surrogate model.The corresponding sensitivity results are given by the further derivative operation,which is compared with those of the finite difference method(FDM).Several examples are provided to demonstrate the effectiveness of the proposed algorithm for sensitivity analysis of electromagnetic scattering from homogeneous dielectric targets.展开更多
This paper presents a new computational method for forward uncertainty quantification(UQ)analyses on large-scale structural systems in the presence of arbitrary and dependent random inputs.The method consists of a gen...This paper presents a new computational method for forward uncertainty quantification(UQ)analyses on large-scale structural systems in the presence of arbitrary and dependent random inputs.The method consists of a generalized polynomial chaos expansion(GPCE)for statistical moment and reliability analyses associated with the stochastic output and a static reanalysis method to generate the input-output data set.In the reanalysis,we employ substructuring for a structure to isolate its local regions that vary due to random inputs.This allows for avoiding repeated computations of invariant substructures while generating the input-output data set.Combining substructuring with static condensation further improves the computational efficiency of the reanalysis without losing accuracy.Consequently,the GPCE with the static reanalysis method can achieve significant computational saving,thus mitigating the curse of dimensionality to some degree for UQ under high-dimensional inputs.The numerical results obtained from a simple structure indicate that the proposed method for UQ produces accurate solutions more efficiently than the GPCE using full finite element analyses(FEAs).We also demonstrate the efficiency and scalability of the proposed method by executing UQ for a large-scale wing-box structure under ten-dimensional(all-dependent)random inputs.展开更多
We present a scheme for the anti-control of chaos in the p-Ge photoconductor system by using a chaotic laser to irradiate and disturb this system. The numerical simulations show that this scheme can be effectively use...We present a scheme for the anti-control of chaos in the p-Ge photoconductor system by using a chaotic laser to irradiate and disturb this system. The numerical simulations show that this scheme can be effectively used to control periodic states in this t^Ge system into chaotic states. Moreover, the different chaos states with different chaotic orbits can be obtained by appropriately adjusting the disturbance intensity and disturbance frequency, and by increasing this intensity or reducing this frequency, this p-Ge system gradually evolves to fully developed chaotic states.展开更多
A novel semiconductor laser system is presented based on a twisted fiber.To study the period-control and chaos-anticontrol of the laser system,we design a type of optic path as a control setup using the combination of...A novel semiconductor laser system is presented based on a twisted fiber.To study the period-control and chaos-anticontrol of the laser system,we design a type of optic path as a control setup using the combination of the twisted fiber and the polarization controller while we present a physical dynamics model of the delayed dual-feedback laser containing the twisted fiber effect.We give an analysis of the effect of the twisted fiber on the laser.We use the effects of the delayed phase and the rotation angle of the twisted fiber and the characteristics of the system to achieve control of the laser.The laser is deduced to a stable state,a double-periodic state,a period-6 state,a period-8 state,a period-9 state,a multi-period state,beat phenomenon,and so on.The periodic laser can be anti-controlled to chaos.Some chaos-anti-control area is found.The laser system is very useful for the study of chaos-control of the laser setup and the applications of some physics effects.展开更多
The utilization of thin plate systems based on acoustic vibration holds significant importance in micro-nano manipulation and the exploration of nonlinear science. This paper focuses on the analysis of an actual thin ...The utilization of thin plate systems based on acoustic vibration holds significant importance in micro-nano manipulation and the exploration of nonlinear science. This paper focuses on the analysis of an actual thin plate system driven by acoustic wave signals. By combining the mechanical analysis of thin plate microelements with the Bubnov–Galerkin integral method, the governing equation for the forced vibration of a square thin plate is derived. Notably,the reaction force of the thin plate vibration system is defined as f=α|w|, resembling Hooke’s law. The energy function and energy level curve of the system are also analyzed. Subsequently, the amplitude–frequency response function of the thin plate oscillator is solved using the harmonic balance method. Through numerical simulations, the amplitude–frequency curves are analyzed for different vibration modes under the influence of various parameters. Furthermore, the paper demonstrates the occurrence of conservative chaotic motions in the thin plate oscillator using theoretical and numerical methods. Dynamics maps illustrating the system’s states are presented to reveal the evolution laws of the system. By exploring the effects of force fields and system energy, the underlying mechanism of chaos is interpreted. Additionally, the phenomenon of chaos in the oscillator can be controlled through the method of velocity and displacement states feedback, which holds significance for engineering applications.展开更多
This study proposes a novel fractional discrete-time macroeconomic system with incommensurate order.The dynamical behavior of the proposed macroeconomic model is investigated analytically and numerically.In particular...This study proposes a novel fractional discrete-time macroeconomic system with incommensurate order.The dynamical behavior of the proposed macroeconomic model is investigated analytically and numerically.In particular,the zero equilibrium point stability is investigated to demonstrate that the discrete macroeconomic system exhibits chaotic behavior.Through using bifurcation diagrams,phase attractors,the maximum Lyapunov exponent and the 0–1 test,we verified that chaos exists in the new model with incommensurate fractional orders.Additionally,a complexity analysis is carried out utilizing the approximation entropy(ApEn)and C_(0)complexity to prove that chaos exists.Finally,the main findings of this study are presented using numerical simulations.展开更多
Cyberattack detection has become an important research domain owing to increasing number of cybercrimes in recent years.Both Machine Learning(ML)and Deep Learning(DL)classification models are useful in effective ident...Cyberattack detection has become an important research domain owing to increasing number of cybercrimes in recent years.Both Machine Learning(ML)and Deep Learning(DL)classification models are useful in effective identification and classification of cyberattacks.In addition,the involvement of hyper parameters in DL models has a significantly influence upon the overall performance of the classification models.In this background,the current study develops Intelligent Cybersecurity Classification using Chaos Game Optimization with Deep Learning(ICC-CGODL)Model.The goal of the proposed ICC-CGODL model is to recognize and categorize different kinds of attacks made upon data.Besides,ICC-CGODL model primarily performs min-max normalization process to normalize the data into uniform format.In addition,Bidirectional Gated Recurrent Unit(BiGRU)model is utilized for detection and classification of cyberattacks.Moreover,CGO algorithm is also exploited to adjust the hyper parameters involved in BiGRU model which is the novelty of current work.A wide-range of simulation analysis was conducted on benchmark dataset and the results obtained confirmed the significant performance of ICC-CGODL technique than the recent approaches.展开更多
The advancements in technology have substantially grown the size of image data.Traditional image encryption algorithms have limited capabilities to deal with the emerging challenges in big data,including compression a...The advancements in technology have substantially grown the size of image data.Traditional image encryption algorithms have limited capabilities to deal with the emerging challenges in big data,including compression and noise toleration.An image encryption method that is based on chaotic maps and orthogonal matrix is proposed in this study.The proposed scheme is built on the intriguing characteristics of an orthogonal matrix.Gram Schmidt disperses the values of pixels in a plaintext image by generating a random orthogonal matrix using logistic chaotic map.Following the diffusion process,a block-wise random permutation of the data is performed using multi-chaos.The proposed scheme provides sufficient security and resilience to JPEG compression and channel noise through a series of experiments and security evaluations.It enables Partial Encryption(PE)for faster processing as well as complete encryption for increased security.The higher values of the number of pixels change rates and unified average change intensity confirm the security of the encryption scheme.In contrast to other schemes,the proposed approach can perform full and partial encryption depending on security requirements.展开更多
Optical chaos has attracted widespread attention owing to its complex dynamic behaviors.However,the time delay signature(TDS)caused by the external cavity mode reduces the complexity of optical chaos.We propose and nu...Optical chaos has attracted widespread attention owing to its complex dynamic behaviors.However,the time delay signature(TDS)caused by the external cavity mode reduces the complexity of optical chaos.We propose and numerically demonstrate the critical dispersion of chirped fiber Bragg grating(CFBG)for eliminating the TDS of laser chaos in this work.The critical dispersion,as a function of relaxation frequency and bandwidth of the optical spectrum,is found through extensive dynamics simulations.It is shown that the TDS can be eliminated when the dispersion of CFBG is above this critical dispersion.In addition,the influence of dispersive feedback light and output light from a laser is investigated.These results provide important quantitative guidance for designing chaotic semiconductor lasers without TDS.展开更多
The security of digital images transmitted via the Internet or other public media is of the utmost importance.Image encryption is a method of keeping an image secure while it travels across a non-secure communication ...The security of digital images transmitted via the Internet or other public media is of the utmost importance.Image encryption is a method of keeping an image secure while it travels across a non-secure communication medium where it could be intercepted by unauthorized entities.This study provides an approach to color image encryption that could find practical use in various contexts.The proposed method,which combines four chaotic systems,employs singular value decomposition and a chaotic sequence,making it both secure and compression-friendly.The unified average change intensity,the number of pixels’change rate,information entropy analysis,correlation coefficient analysis,compression friendliness,and security against brute force,statistical analysis and differential attacks are all used to evaluate the algorithm’s performance.Following a thorough investigation of the experimental data,it is concluded that the proposed image encryption approach is secure against a wide range of attacks and provides superior compression friendliness when compared to chaos-based alternatives.展开更多
In recent years,fractional-order chaotic maps have been paid more attention in publications because of the memory effect.This paper presents a novel variable-order fractional sine map(VFSM)based on the discrete fracti...In recent years,fractional-order chaotic maps have been paid more attention in publications because of the memory effect.This paper presents a novel variable-order fractional sine map(VFSM)based on the discrete fractional calculus.Specially,the order is defined as an iterative function that incorporates the current state of the system.By analyzing phase diagrams,time sequences,bifurcations,Lyapunov exponents and fuzzy entropy complexity,the dynamics of the proposed map are investigated comparing with the constant-order fractional sine map.The results reveal that the variable order has a good effect on improving the chaotic performance,and it enlarges the range of available parameter values as well as reduces non-chaotic windows.Multiple coexisting attractors also enrich the dynamics of VFSM and prove its sensitivity to initial values.Moreover,the sequence generated by the proposed map passes the statistical test for pseudorandom number and shows strong robustness to parameter estimation,which proves the potential applications in the field of information security.展开更多
In previous works, the theoretical and experimental deterministic scalar kinematic structures, the theoretical and experimental deterministic vector kinematic structures, the theoretical and experimental deterministic...In previous works, the theoretical and experimental deterministic scalar kinematic structures, the theoretical and experimental deterministic vector kinematic structures, the theoretical and experimental deterministic scalar dynamic structures, and the theoretical and experimental deterministic vector dynamic structures have been developed to compute the exact solution for deterministic chaos of the exponential pulsons and oscillons that is governed by the nonstationary three-dimensional Navier-Stokes equations. To explore properties of the kinetic energy, rectangular, diagonal, and triangular summations of a matrix of the kinetic energy and general terms of various sums have been used in the current paper to develop quantization of the kinetic energy of deterministic chaos. Nested structures of a cumulative energy pulson, an energy pulson of propagation, an internal energy oscillon, a diagonal energy oscillon, and an external energy oscillon have been established. In turn, the energy pulsons and oscillons include group pulsons of propagation, internal group oscillons, diagonal group oscillons, and external group oscillons. Sequentially, the group pulsons and oscillons contain wave pulsons of propagation, internal wave oscillons, diagonal wave oscillons, and external wave oscillons. Consecutively, the wave pulsons and oscillons are composed of elementary pulsons of propagation, internal elementary oscillons, diagonal elementary oscillons, and external elementary oscillons. Topology, periodicity, and integral properties of the exponential pulsons and oscillons have been studied using the novel method of the inhomogeneous Fourier expansions via eigenfunctions in coordinates and time. Symbolic computations of the exact expansions have been performed using the experimental and theoretical programming in Maple. Results of the symbolic computations have been justified by probe visualizations.展开更多
An exact three-dimensional solution for stochastic chaos of I wave groups of M random internal waves governed by the Navier-Stokes equations is developed. The Helmholtz decomposition is used to expand the Dirichlet pr...An exact three-dimensional solution for stochastic chaos of I wave groups of M random internal waves governed by the Navier-Stokes equations is developed. The Helmholtz decomposition is used to expand the Dirichlet problem for the Navier-Stokes equations into the Archimedean, Stokes, and Navier problems. The exact solution is obtained with the help of the method of decomposition in invariant structures. Differential algebra is constructed for six families of random invariant structures: random scalar kinematic structures, time-complementary random scalar kinematic structures, random vector kinematic structures, time-complementary random vector kinematic structures, random scalar dynamic structures, and random vector dynamic structures. Tedious computations are performed using the experimental and theoretical programming in Maple. The random scalar and vector kinematic structures and the time-complementary random scalar and vector kinematic structures are applied to solve the Stokes problem. The random scalar and vector dynamic structures are employed to expand scalar and vector variables of the Navier problem. Potentialization of the Navier field becomes available since vortex forces, which are expressed via the vector potentials of the Helmholtz decomposition, counterbalance each other. On the contrary, potential forces, which are described by the scalar potentials of the Helmholtz decomposition, superimpose to generate the gradient of a dynamic random pressure. Various constituents of the kinetic energy are ascribed to diverse interactions of random, three-dimensional, nonlinear, internal waves with a two-fold topology, which are termed random exponential oscillons and pulsons. Quantization of the kinetic energy of stochastic chaos is developed in terms of wave structures of random elementary oscillons, random elementary pulsons, random internal, diagonal, and external elementary oscillons, random wave pulsons, random internal, diagonal, and external wave oscillons, random group pulsons, random internal, diagonal, and external group oscillons, a random energy pulson, random internal, diagonal, and external energy oscillons, and a random cumulative energy pulson.展开更多
基金supported by the Young Scientists Fund of the National Natural Science Foundation of China(No.62102444)a Major Research Project in Higher Education Institutions in Henan Province(No.23A560015).
文摘In this paper,an adaptive polynomial chaos expansion method(PCE)based on the method of moments(MoM)is proposed to construct surrogate models for electromagnetic scattering and further sensitivity analysis.The MoM is applied to accurately solve the electric field integral equation(EFIE)of electromagnetic scattering from homogeneous dielectric targets.Within the bistatic radar cross section(RCS)as the research object,the adaptive PCE algorithm is devoted to selecting the appropriate order to construct the multivariate surrogate model.The corresponding sensitivity results are given by the further derivative operation,which is compared with those of the finite difference method(FDM).Several examples are provided to demonstrate the effectiveness of the proposed algorithm for sensitivity analysis of electromagnetic scattering from homogeneous dielectric targets.
基金Project supported by the National Research Foundation of Korea(Nos.NRF-2020R1C1C1011970 and NRF-2018R1A5A7023490)。
文摘This paper presents a new computational method for forward uncertainty quantification(UQ)analyses on large-scale structural systems in the presence of arbitrary and dependent random inputs.The method consists of a generalized polynomial chaos expansion(GPCE)for statistical moment and reliability analyses associated with the stochastic output and a static reanalysis method to generate the input-output data set.In the reanalysis,we employ substructuring for a structure to isolate its local regions that vary due to random inputs.This allows for avoiding repeated computations of invariant substructures while generating the input-output data set.Combining substructuring with static condensation further improves the computational efficiency of the reanalysis without losing accuracy.Consequently,the GPCE with the static reanalysis method can achieve significant computational saving,thus mitigating the curse of dimensionality to some degree for UQ under high-dimensional inputs.The numerical results obtained from a simple structure indicate that the proposed method for UQ produces accurate solutions more efficiently than the GPCE using full finite element analyses(FEAs).We also demonstrate the efficiency and scalability of the proposed method by executing UQ for a large-scale wing-box structure under ten-dimensional(all-dependent)random inputs.
文摘We present a scheme for the anti-control of chaos in the p-Ge photoconductor system by using a chaotic laser to irradiate and disturb this system. The numerical simulations show that this scheme can be effectively used to control periodic states in this t^Ge system into chaotic states. Moreover, the different chaos states with different chaotic orbits can be obtained by appropriately adjusting the disturbance intensity and disturbance frequency, and by increasing this intensity or reducing this frequency, this p-Ge system gradually evolves to fully developed chaotic states.
文摘A novel semiconductor laser system is presented based on a twisted fiber.To study the period-control and chaos-anticontrol of the laser system,we design a type of optic path as a control setup using the combination of the twisted fiber and the polarization controller while we present a physical dynamics model of the delayed dual-feedback laser containing the twisted fiber effect.We give an analysis of the effect of the twisted fiber on the laser.We use the effects of the delayed phase and the rotation angle of the twisted fiber and the characteristics of the system to achieve control of the laser.The laser is deduced to a stable state,a double-periodic state,a period-6 state,a period-8 state,a period-9 state,a multi-period state,beat phenomenon,and so on.The periodic laser can be anti-controlled to chaos.Some chaos-anti-control area is found.The laser system is very useful for the study of chaos-control of the laser setup and the applications of some physics effects.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 61973172, 62003177, 62103204, 62003175, and 61973175)the Joint Fund of the Ministry of Education for Equipment Pre-research (Grant No. 8091B022133)General Terminal IC Interdisciplinary Science Center of Nankai University。
文摘The utilization of thin plate systems based on acoustic vibration holds significant importance in micro-nano manipulation and the exploration of nonlinear science. This paper focuses on the analysis of an actual thin plate system driven by acoustic wave signals. By combining the mechanical analysis of thin plate microelements with the Bubnov–Galerkin integral method, the governing equation for the forced vibration of a square thin plate is derived. Notably,the reaction force of the thin plate vibration system is defined as f=α|w|, resembling Hooke’s law. The energy function and energy level curve of the system are also analyzed. Subsequently, the amplitude–frequency response function of the thin plate oscillator is solved using the harmonic balance method. Through numerical simulations, the amplitude–frequency curves are analyzed for different vibration modes under the influence of various parameters. Furthermore, the paper demonstrates the occurrence of conservative chaotic motions in the thin plate oscillator using theoretical and numerical methods. Dynamics maps illustrating the system’s states are presented to reveal the evolution laws of the system. By exploring the effects of force fields and system energy, the underlying mechanism of chaos is interpreted. Additionally, the phenomenon of chaos in the oscillator can be controlled through the method of velocity and displacement states feedback, which holds significance for engineering applications.
文摘This study proposes a novel fractional discrete-time macroeconomic system with incommensurate order.The dynamical behavior of the proposed macroeconomic model is investigated analytically and numerically.In particular,the zero equilibrium point stability is investigated to demonstrate that the discrete macroeconomic system exhibits chaotic behavior.Through using bifurcation diagrams,phase attractors,the maximum Lyapunov exponent and the 0–1 test,we verified that chaos exists in the new model with incommensurate fractional orders.Additionally,a complexity analysis is carried out utilizing the approximation entropy(ApEn)and C_(0)complexity to prove that chaos exists.Finally,the main findings of this study are presented using numerical simulations.
基金The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work under Grant Number(RGP 2/180/43)Princess Nourah bint Abdulrahman University Researchers Supporting Project Number(PNURSP2022R161)+1 种基金Princess Nourah bint Abdulrahman University,Riyadh,Saudi ArabiaThe authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code:(22UQU4210118DSR07).
文摘Cyberattack detection has become an important research domain owing to increasing number of cybercrimes in recent years.Both Machine Learning(ML)and Deep Learning(DL)classification models are useful in effective identification and classification of cyberattacks.In addition,the involvement of hyper parameters in DL models has a significantly influence upon the overall performance of the classification models.In this background,the current study develops Intelligent Cybersecurity Classification using Chaos Game Optimization with Deep Learning(ICC-CGODL)Model.The goal of the proposed ICC-CGODL model is to recognize and categorize different kinds of attacks made upon data.Besides,ICC-CGODL model primarily performs min-max normalization process to normalize the data into uniform format.In addition,Bidirectional Gated Recurrent Unit(BiGRU)model is utilized for detection and classification of cyberattacks.Moreover,CGO algorithm is also exploited to adjust the hyper parameters involved in BiGRU model which is the novelty of current work.A wide-range of simulation analysis was conducted on benchmark dataset and the results obtained confirmed the significant performance of ICC-CGODL technique than the recent approaches.
文摘The advancements in technology have substantially grown the size of image data.Traditional image encryption algorithms have limited capabilities to deal with the emerging challenges in big data,including compression and noise toleration.An image encryption method that is based on chaotic maps and orthogonal matrix is proposed in this study.The proposed scheme is built on the intriguing characteristics of an orthogonal matrix.Gram Schmidt disperses the values of pixels in a plaintext image by generating a random orthogonal matrix using logistic chaotic map.Following the diffusion process,a block-wise random permutation of the data is performed using multi-chaos.The proposed scheme provides sufficient security and resilience to JPEG compression and channel noise through a series of experiments and security evaluations.It enables Partial Encryption(PE)for faster processing as well as complete encryption for increased security.The higher values of the number of pixels change rates and unified average change intensity confirm the security of the encryption scheme.In contrast to other schemes,the proposed approach can perform full and partial encryption depending on security requirements.
基金funded by NERC grant NE/M012190/1, China Scholarship Council (No. 201708320317)National Natural Science Foundation of China (Grant No. 41974073, Grant No. 41404053, Grant No. 41604134, Grant No. 41174165, Grant No. 41974073)+3 种基金Special Project for Meteo-scientific Research in the Public Interest (Grant No. GYHY201306073-2)Macao Foundation and the pre-research project on Civil Aerospace Technologies No. D020308 and D020303 funded by China’s National Space Administrationopening fund of State Key Laboratory of Lunar and Planetary Sciences (Macao University of Science and Technology) (Macao FDCT grant No. 119/2017/A3)the Specialized Research Fund for State Key Laboratory of Space Weather, Graduate Research and Practice Innovation Program (KYCX17_0896),the Specialized Research Fund for State Key Laboratory of Space Weather
基金the National Natural Science Foundation of China(Grant No.62105190)the Natural Science Foundation of Shanxi Province of China(Grant No.20210302124268)+1 种基金the Scientific and Technological Innovation Programs of Higher Education Institutions of Shanxi Province of China(Grant No.2021L285)the Youth Researchof Shanxi University of Finance and Economics(Grant No.QN-202015)。
文摘Optical chaos has attracted widespread attention owing to its complex dynamic behaviors.However,the time delay signature(TDS)caused by the external cavity mode reduces the complexity of optical chaos.We propose and numerically demonstrate the critical dispersion of chirped fiber Bragg grating(CFBG)for eliminating the TDS of laser chaos in this work.The critical dispersion,as a function of relaxation frequency and bandwidth of the optical spectrum,is found through extensive dynamics simulations.It is shown that the TDS can be eliminated when the dispersion of CFBG is above this critical dispersion.In addition,the influence of dispersive feedback light and output light from a laser is investigated.These results provide important quantitative guidance for designing chaotic semiconductor lasers without TDS.
基金funded by Deanship of Scientific Research at King Khalid University under Grant Number R.G.P.2/86/43.
文摘The security of digital images transmitted via the Internet or other public media is of the utmost importance.Image encryption is a method of keeping an image secure while it travels across a non-secure communication medium where it could be intercepted by unauthorized entities.This study provides an approach to color image encryption that could find practical use in various contexts.The proposed method,which combines four chaotic systems,employs singular value decomposition and a chaotic sequence,making it both secure and compression-friendly.The unified average change intensity,the number of pixels’change rate,information entropy analysis,correlation coefficient analysis,compression friendliness,and security against brute force,statistical analysis and differential attacks are all used to evaluate the algorithm’s performance.Following a thorough investigation of the experimental data,it is concluded that the proposed image encryption approach is secure against a wide range of attacks and provides superior compression friendliness when compared to chaos-based alternatives.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.62071496,61901530,and 62061008)the Natural Science Foundation of Hunan Province of China(Grant No.2020JJ5767).
文摘In recent years,fractional-order chaotic maps have been paid more attention in publications because of the memory effect.This paper presents a novel variable-order fractional sine map(VFSM)based on the discrete fractional calculus.Specially,the order is defined as an iterative function that incorporates the current state of the system.By analyzing phase diagrams,time sequences,bifurcations,Lyapunov exponents and fuzzy entropy complexity,the dynamics of the proposed map are investigated comparing with the constant-order fractional sine map.The results reveal that the variable order has a good effect on improving the chaotic performance,and it enlarges the range of available parameter values as well as reduces non-chaotic windows.Multiple coexisting attractors also enrich the dynamics of VFSM and prove its sensitivity to initial values.Moreover,the sequence generated by the proposed map passes the statistical test for pseudorandom number and shows strong robustness to parameter estimation,which proves the potential applications in the field of information security.
文摘In previous works, the theoretical and experimental deterministic scalar kinematic structures, the theoretical and experimental deterministic vector kinematic structures, the theoretical and experimental deterministic scalar dynamic structures, and the theoretical and experimental deterministic vector dynamic structures have been developed to compute the exact solution for deterministic chaos of the exponential pulsons and oscillons that is governed by the nonstationary three-dimensional Navier-Stokes equations. To explore properties of the kinetic energy, rectangular, diagonal, and triangular summations of a matrix of the kinetic energy and general terms of various sums have been used in the current paper to develop quantization of the kinetic energy of deterministic chaos. Nested structures of a cumulative energy pulson, an energy pulson of propagation, an internal energy oscillon, a diagonal energy oscillon, and an external energy oscillon have been established. In turn, the energy pulsons and oscillons include group pulsons of propagation, internal group oscillons, diagonal group oscillons, and external group oscillons. Sequentially, the group pulsons and oscillons contain wave pulsons of propagation, internal wave oscillons, diagonal wave oscillons, and external wave oscillons. Consecutively, the wave pulsons and oscillons are composed of elementary pulsons of propagation, internal elementary oscillons, diagonal elementary oscillons, and external elementary oscillons. Topology, periodicity, and integral properties of the exponential pulsons and oscillons have been studied using the novel method of the inhomogeneous Fourier expansions via eigenfunctions in coordinates and time. Symbolic computations of the exact expansions have been performed using the experimental and theoretical programming in Maple. Results of the symbolic computations have been justified by probe visualizations.
文摘An exact three-dimensional solution for stochastic chaos of I wave groups of M random internal waves governed by the Navier-Stokes equations is developed. The Helmholtz decomposition is used to expand the Dirichlet problem for the Navier-Stokes equations into the Archimedean, Stokes, and Navier problems. The exact solution is obtained with the help of the method of decomposition in invariant structures. Differential algebra is constructed for six families of random invariant structures: random scalar kinematic structures, time-complementary random scalar kinematic structures, random vector kinematic structures, time-complementary random vector kinematic structures, random scalar dynamic structures, and random vector dynamic structures. Tedious computations are performed using the experimental and theoretical programming in Maple. The random scalar and vector kinematic structures and the time-complementary random scalar and vector kinematic structures are applied to solve the Stokes problem. The random scalar and vector dynamic structures are employed to expand scalar and vector variables of the Navier problem. Potentialization of the Navier field becomes available since vortex forces, which are expressed via the vector potentials of the Helmholtz decomposition, counterbalance each other. On the contrary, potential forces, which are described by the scalar potentials of the Helmholtz decomposition, superimpose to generate the gradient of a dynamic random pressure. Various constituents of the kinetic energy are ascribed to diverse interactions of random, three-dimensional, nonlinear, internal waves with a two-fold topology, which are termed random exponential oscillons and pulsons. Quantization of the kinetic energy of stochastic chaos is developed in terms of wave structures of random elementary oscillons, random elementary pulsons, random internal, diagonal, and external elementary oscillons, random wave pulsons, random internal, diagonal, and external wave oscillons, random group pulsons, random internal, diagonal, and external group oscillons, a random energy pulson, random internal, diagonal, and external energy oscillons, and a random cumulative energy pulson.
