Six-dimensional System Research Articles

AI-Lorenz: A physics-data-driven framework for Black-Box and Gray-Box identification of chaotic systems with symbolic regression

Discovering mathematical models that characterize the observed behavior of dynamical systems remains a major challenge, especially for systems in a chaotic regime, due to their sensitive dependence on initial conditions and the complex non-linear interactions within the system. The challenge is even greater when the physics underlying such systems is not yet understood, and scientific inquiry must solely rely on empirical data. Despite advancements such as the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm, which has shown success in identifying chaotic systems with reasonable accuracy, challenges remain in dealing with noise, sparse data, and the need for models that generalize well across different chaotic systems. Driven by the need to fill this gap, we develop a framework named AI-Lorenz that learns mathematical expressions modeling complex dynamical behaviors by identifying differential equations from noisy and sparse observable data. We train a physics-informed neural network (PINN) with the eXtreme Theory of Functional Connections (X-TFC) algorithm by using data and known physics (when available) to learn the dynamics of a system, its rate of change in time, and missing model terms, which are used as input for a symbolic regression algorithm (PySR) to autonomously distill the explicit mathematical terms. This, in turn, enables us to predict the future evolution of the dynamical behavior. The performance of this framework is validated by recovering the right-hand sides and unknown terms of certain complex, chaotic systems, such as the well-known Lorenz system, a six-dimensional hyperchaotic system, the non-autonomous Sprott chaotic system, and the slow-fast Duffing system, and comparing them with their known analytical expressions and state-of-the-art regression and system identification methods.

Qualitative Properties of a Physically Extended Six-Dimensional Lorenz System

In this paper, the qualitative properties of a physically extended six-dimensional Lorenz system, with additional physical terms describing rotation and density, which was proposed in [Moon et al., 2019] have been investigated. The dissipation, invariance, Lyapunov exponents, Kaplan–Yorke dimension, ultimate boundedness and global attractivity of this six-dimensional Lorenz system have been discussed in detail according to the chaotic systems theory. We find that this system exhibits chaos phenomena for a new set of parameters. It is well known that the general method for studying the bounds of a chaotic system is to construct a suitable Lyapunov-like function (or the generalized positive definite and radically unbounded Lyapunov function). However, the higher the dimension of a chaotic system, the more difficult it is to construct the Lyapunov-like function. The innovation of this paper is that we first construct the suitable Lyapunov-like function for this six-dimensional Lorenz system, and then we prove that this system is not only globally bounded for varying parameters, but it also gives a collection of global absorbing sets for this system with respect to all parameters of this system according to Lyapunov’s direct method and the optimization method. Furthermore, we obtain the rate of the trajectories going from the exterior to the global absorbing set. Some numerical simulations are presented to validate our research results. Finally, we give a direct application of the results obtained in this paper. According to the results of this paper, we can conclude that the equilibrium point [Formula: see text] of this system is globally exponentially stable.

Feminization of poverty: an analysis of multidimensional poverty among rural women in China

Few studies from an individual perspective have analyzed the multidimensional poverty of rural women in China. Therefore, based on the CFPS data from 2010 to 2020 and the Alkire-Foster approach, this study built a six-dimensional system to portray the status of multidimensional poverty among rural women. The overall comparisons found that rural women were more likely to be multidimensional poor than other subgroups. And the results of rural women showed significant demographic and spatio-temporal differences. That is, older rural women were more deprived than younger rural women. Rural women with spouses or confidence were less deprived than those without spouses or confidence, respectively. From the spatial perspective, the censored headcount ratios of rural women in descending order were Western Region, Central Region and Eastern Region. From the temporal perspective, the risk of rural women’s multidimensional poverty decreased significantly from 2010 to 2020. The importance of non-material indicators was gradually becoming prominent, including education, health and subjective wellbeing. The conclusions can contribute to the development of policies, even if some limitations need to be further improved.

Chaotic image encryption algorithm based on dynamic Hachimoji DNA coding and computing

With the increasing awareness of privacy protection, people pay more and more attention to strengthening the security of image data transmitted over the network. Therefore, this paper designs a chaotic image encrypting algorithm based on dynamic Hachimoji DNA coding and computing to protect images. The Hachimoji DNA coding method provides richer coding rules to dynamically encode images than the traditional DNA coding method, improving the complexity and security of the encryption algorithm. First, the original image is rearranged and encoded with the dynamic Hachimoji DNA coding method according to the sorting and encoding controller sequence generated by a six-dimensional hyperchaotic system. Second, various DNA operations are performed on the encoded image. Among these operations, we not only use the common operations but also propose a new DNA operation called bitwise inversion. Finally, the DNA image is decoded using the dynamic decoding method to obtain the encrypted image. Experiments demonstrated that the image encryption algorithm has a good security effect and can effectively resist common attacks.

An <i>n</i>-dimensional discrete hyperchaotic system and its application in audio encryption

Discrete chaotic system, as a pseudo-random signal source, plays a very important role in realizing secure communication. However, many low-dimensional chaotic systems are prone to chaos degradation. Therefore, many scholars have studied the construction of high-dimensional chaotic systems. However, many existing algorithms for constructing high-dimensional chaotic systems have relatively high time complexity and relatively complex structures. To solve this problem, this paper explores an <i>n</i>-dimensional discrete hyperchaotic system with a simple structure. Firstly, the <i>n</i>-dimensional discrete hyperchaotic system is constructed by using sine function and power function and simple operations. Then, it is theoretically analyzed based on Jacobian matrix method that the system can have the positive Lyapunov exponents. Next, the algorithm time complexity, sample entropy, correlation dimension and other indexes are compared with those of the existing methods. The experimental results show that our system has a simple structure, high complexity and good algorithm time complexity. Therewith, a six-dimensional chaotic system is chosen as an example, and the phase diagram, bifurcation diagram, Lyapunov expnonents, complexity and other characteristics of the system are analyzed. The results show that the proposed system has good chaotic characteristics. Moreover, to show the application of the proposed system, we apply it to audio encryption. According to this system, we combine it with the XOR operation and true random numbers to explore a novel method of one-cipher audio encryption. Through experimental simulation, compared with some existing audio encryption algorithms, this algorithm can satisfy various statistical tests and resist various common attacks. It is also validated that the proposed system can be effectively applied to the field of audio encryption.

6D Hyperchaotic Encryption Model for Ensuring Security to 3D Printed Models and Medical Images

In the 6G era, where ultra-fast and reliable communication is expected to be ubiquitous, encryption shall continue to play a crucial role in ensuring the security and privacy of data. Encryption and decryption of medical images and 3D printed models using 6D hyperchaotic function is proposed in this research work for ensuring security in data transfer. Here we envisage using a six-dimensional hyperchaotic system for encryption purposes which shall offer a high level of security due to its complex and unpredictable dynamics with multiple positive Lyapunov exponents. This system can potentially enhance the encryption process for 3D objects and medical images, ensuring the protection of sensitive data and preventing unauthorized access. A hyperchaotic system is a type of dynamical system characterized by exhibiting more than one positive Lyapunov exponent, which indicates strong sensitivity to initial conditions. These systems have more degrees of freedom and complex and intricate dynamics compared to standard chaotic systems. The security of the encryption scheme depends on the complexity of the hyperchaotic system and the randomness of the secret key. The parameters of a 6D hyperchaotic system shall be used as an encryption key with six dimensions, each with its range of values, and shall provide many possible keys. In this work, we implemented a 6D hyperchaotic system for the encryption of the 3D printed model and medical images. The performance evaluation was done by metrics entropy, correlation, Number of Pixels Change Rate (NPCR), and Unified Averaged Changed Intensity (UACI) which revealed the robustness of the encryption model in ensuring security. Hyperchaotic systems can be efficiently implemented in parallel computing architectures, which allow faster encryption and decryption processes.

A New 6D Hyperchaos-Based Medical Image Encryption Scheme Using k-Fibonacci Numbers

In the era of information technology, hospitals, medical centers, doctors and health specialists share millions of images. Securing these images is essential to maintaining patients’ privacy. Several cryptographic algorithms are implemented to meet the increasing demands of transmission systems. As a contribution to securing these data, we propose a novel robust and efficient algorithm. We randomly generate numbers from a six-dimensional hyperchaotic system and Arnold’s cat map to confuse the original images. The confused images are then diffused using a k-Fibonacci matrix. The performance of the proposed algorithm was evaluated using several parameters tested on a set of reference images of different sizes and modalities. The obtained results are conclusive and the performances are excellent, sometimes exceeding those found by other recent algorithms proposed in the literature.

Circuit simulation and image encryption based on a six-dimensional cellular neural network hyperchaotic system

Circuit simulation and image encryption based on a six-dimensional cellular neural network hyperchaotic system

On a dynamical system linked to the BKL scenario

We consider the six-dimensional dynamical system in three components introduced by Ryan to describe the scenario of Belinskii, Khalatnikov and Lifshitz to the cosmological singularity when the spatial metric tensor is not diagonal. Despite its nonintegrability, recently proven by Goldstein and Piechocki, the three four-dimensional systems defined by canceling one of the three components happen to be integrable. We express their general solution as a rational function of, respectively, two exponential functions, a third Painlevé function, two exponential functions.

Mathematical Modelling of Lumpy Skin Disease in Dairy Cow

Lumpy skin disease (LSD) is a threatnening disease caused by virus that affects large ruminant animals mainly cattle. LSD is native in many African and Asian countries, having existed for over one hundred years in Africa where it originally started and has recently expanded to the Middle East region. A disease model consisting of six-dimensional system of ordinary differential equations was formulated to properly understand the spread and control of LSD. The classes considered includes the susceptible cow, vaccinated cow, infectious cow, recovered cow, susceptible mosquito and infectious mosquito. This paper captures two modes of transmission of the disease, direct contact and contact through mosquito bites. Inclusion of vaccination parameter makes this model different from existing literature. We show the model is well-posed through positivity and boundedness of the state variables. The basic reproduction number R0 of the model was derived and the disease-free equilibrium points was shown to be locally asymptotically stable when R0 < 1. Sensitivity analysis to identify the most sensitive parameters was conducted and the vaccination parameter was varied using matplotlib module in Python to check the effect of vaccination on both vaccinated and infectious cow. Increase in the vaccination parameter cause the number of vaccinated cow to increase and hence leads to reduction in the number of infected cow.

Hyperchaos, constraints and its stability control in a 6D hyperchaotic particle motion system

Firstly, a novel six-dimensional (6D) hyperchaotic particle motion system is formulated. The equilibrium points and their characteristics, Poincaré sections, Lyapunov exponents, bifurcations and multi-periodic windows are studied. Secondly, we present two nonholonomic constrained systems. In order to analyze the particle motion trajectories under constraints, the explicit equations for constrained systems are given. Based on Lyapunov exponents, Poincaré maps and bifurcations, we can see that the different hyperchaotic phenomena of the particle motion can be generated by introducing nonholonomic constraints. Finally, the stability control of the 6D hyperchaotic particle motion system is realized by separately using constraint control method and linear feedback control method. Numerical simulations of the dynamical behaviors of the six-dimensional hyperchaotic particle motion system are carried out in order to illustrate the complex phenomena of the systems and verify the analysis results.

Nonintegrability of the SEIR epidemic model

We prove the nonintegrability of the susceptible–exposed–infected–removed (SEIR) epidemic model in the Bogoyavlenskij sense. This property of the SEIR model is different from the more fundamental susceptible–infected–removed (SIR) model, which is Bogoyavlenskij-integrable. Our basic tool for the proof is an extension of the Morales–Ramis theory due to Ayoul and Zung. Moreover, we introduce three new state variables and extend the system to a six-dimensional system to treat transcendental first integrals and commutative vector fields. We also use the fact that the incomplete gamma function Γ(α,x) is not elementary for α⁄∈N, of which a proof is included.

Existence of hidden attractors in nonlinear hydro-turbine governing systems and its stability analysis

This work studies the stability and hidden dynamics of the nonlinear hydro-turbine governing system with an output limiting link, and propose a new six-dimensional system, which exhibits some hidden attractors. The parameter switching algorithm is used to numerically study the dynamic behaviors of the system. Moreover, it is investigated that for some parameters the system with a stable equilibrium point can generate strange hidden attractors. A self-excited attractor with the change of its parameters is also recognized. In addition, numerical simulations are carried out to analyze the dynamic behaviors of the proposed system by using the Lyapunov exponent spectra, Lyapunov dimensions, bifurcation diagrams, phase space orbits, and basins of attraction. Consequently, the findings in this work show that the basins of hidden attractors are tiny for which the standard computational procedure for localization is unavailable. These simulation results are conducive to better understanding of hidden chaotic attractors in higher-dimensional dynamical systems, and are also of great significance in revealing chaotic oscillations such as uncontrolled speed adjustment in the operation of hydropower station due to small changes of initial values.

On a six-dimensional Artificial Neural Network Model

This work introduces a new six-dimensional system with chaotic and periodic solutions. For special values of parameters, we calculate the Kaplan-Yorke dimension and we show the dynamics of Lyapunov exponents. Some definitions and propositions are given. Visualizations where possible, are provided.

Chaos of the Six-Dimensional Non-Autonomous System for the Circular Mesh Antenna

In the process of aerospace service, circular mesh antennas generate large nonlinear vibrations under an alternating thermal load. In this paper, the Smale horseshoe and Shilnikov-type multi-pulse chaotic motions of the six-dimensional non-autonomous system for circular mesh antennas are first investigated. The Poincare map is generalized and applied to the six-dimensional non-autonomous system to analyze the existence of Smale horseshoe chaos. Based on the topological horseshoe theory, the three-dimensional solid torus structure is mapped into a logarithmic spiral structure, and the original structure appears to expand in two directions and contract in one direction. There exists chaos in the sense of a Smale horseshoe. The nonlinear equations of the circular mesh antenna under the conditions of the unperturbed and perturbed situations are analyzed, respectively. For the perturbation analysis of the six-dimensional non-autonomous system, the energy difference function is calculated. The transverse zero point of the energy difference function satisfies the non-degenerate conditions, which indicates that the system exists Shilnikov-type multi-pulse chaotic motions. In summary, the researches have verified the existence of chaotic motion in the six-dimensional non-autonomous system for the circular mesh antenna.

Genetic engineering – construction of a network of arbitrary dimension with periodic attractor

It is shown, how to construct a system of ordinary differential equations of arbitrary order, which has the periodic attractor and models some genetic network of arbitrary size. The construction is carried out by combining of multiple systems of lower dimensions with known periodic attractors. In our example the six-dimensional system is constructed, using two identical three-dimensional systems, which have stable periodic solutions.

Associations between Lifestyle Changes, Risk Perception and Anxiety during COVID-19 Lockdowns: A Case Study in Xi'an.

The outbreak of COVID-19 dramatically changed individuals’ lifestyles, which in turn triggered psychological stress and anxiety. Many previous studies have discussed the relationships between lifestyle changes and anxiety and risk perception and anxiety independently. However, few papers have discussed these factors in a comprehensive and systematic manner. We established a six-dimensional system to assess changes in individuals’ lifestyles, which include dietary habits, physical activity (PA), sleep, screen time, smoking and alcohol consumption, and interaction with neighbors. Then, we collected information relating to socio-demographics, lifestyle changes, risk perception, and anxiety, and discussed their associations using multilinear and stepwise logistic regressions. The results show that not all lifestyle changes had an influence on anxiety. Changes in PA and interaction with neighbors were not significantly associated with anxiety. Risk perception was found to be inversely related to anxiety. Changes in dietary habits, family harmony, and net income were negatively related to anxiety among the group with higher risk perception. As individuals perceived a higher severity of COVID-19, the impact of their financial status on anxiety increased. These findings provide a valuable resource for local governments seeking to refine their pandemic strategies by including approaches such as advocating healthy lifestyles and stabilizing the job market to improve individuals’ mental health during lockdowns.

On the Dynamics of New 4D and 6D Hyperchaotic Systems

One of the most interesting problems is the investigation of the boundaries of chaotic or hyperchaotic systems. In addition to estimating the Lyapunov and Hausdorff dimensions, it can be applied in chaos control and chaos synchronization. In this paper, by means of the analytical optimization, comparison principle, and generalized Lyapunov function theory, we find the ultimate bound set for a new six-dimensional hyperchaotic system and the globally exponentially attractive set for a new four-dimensional Lorenz- type hyperchaotic system. The novelty of this paper is that it not only shows the 4D hyperchaotic system is globally confined but also presents a collection of global trapping regions of this system. Furthermore, it demonstrates that the trajectories of the 4D hyperchaotic system move at an exponential rate from outside the trapping zone to its inside. Finally, some numerical simulations are shown to demonstrate the efficacy of the findings.

Chaotic RF Generator for Sub-1-GHz Chaos-Based Communication Systems: Mathematical Modeling and Experimental Validation

In this paper, we have studied, designed and realized a single-transistor chaotic generator with a smooth power spectrum of about −35 dBm in frequency bandwidth up to 1 GHz. The chaos generator model is described by a continuous-time six-dimensional autonomous system assuming an exponential nonlinearity. Chaotic behavior is characterized by bifurcation diagrams, Lyapunov exponents, phase portraits of the attractors and spectra of the oscillations, using both numerical and circuit simulations. Advanced Design System (ADS)-based simulations were carried out to support the theoretical analysis, and to validate the mathematical model. The simulations are carried out using real transistor parameters and taking into account the properties of the substrate, the influence of the board topology and the characteristics of the layout material. This simulation method known as EM/Circuit Co-Simulation allows the simulation results to approach as closely as possible to those of the experiments. In the light of the positive simulation results, the proposed structure is realized to demonstrate its feasibility, and to confirm the numerical results. The prototype is manufactured and mounted on a breadboard using the surface-mount devices and BFP193 bipolar junction transistor. After realizing the generator, we pushed it further towards perfection, through the proposal and realization of a broadband amplifier, in order to gain more bandwidth to improve the spectral characteristic of the generator, which makes it promising for many communication applications such as spread spectrum communication, direct chaotic communication, etc.

Stability Analysis and Nonlinear Vibrations of the Ring Truss Antenna with the Six-Dimensional System

Stability Analysis and Nonlinear Vibrations of the Ring Truss Antenna with the Six-Dimensional System