Three-dimensional Chaotic System Research Articles

Analysis of a new three-dimensional Jerk chaotic system with transient chaos and its adaptive backstepping synchronous control

A new three-dimensional Jerk chaotic system with line equilibrium points is proposed. The system is researched in detail by the Lyapunov exponent graph, bifurcation diagram, phase diagram, and time domain waveform diagram, which show that the system has rich dynamical behaviors, such as eight types of coexisting attractors, extreme multistability of four different attractor states, and offset boosting in two directions. In addition, the system also has six types of transient chaos, which greatly increase the complexity of the system. We study the variation of the spectral entropy (SE) and C0 complexity when the system takes different initial values. Also, in this paper, the initial conditions under which the system is in a synchronized state are determined by initial values with higher complexity. The correctness of the theoretical analysis and numerical simulation is verified by circuit simulation and hardware experiments. Finally, the new system achieves synchronization control utilizing a designed adaptive backstepping controller, laying the foundation for its subsequent use in secure communications.

A novel underwater weak signal detection method based on parameter optimized VMD and 3D chaotic system

Based on chaos theory, combined with improved variational mode decomposition, a novel method for underwater weak signal detection and frequency extraction is proposed. The aim of this research is to address the challenge of detecting underwater signals and extracting their frequencies under ultra-low Signal-to-Noise Ratio (SNR). A novel three-dimensional (3D) chaotic system with infinitely shrinking attractors is proposed, which incorporates a cosine function to achieve chaos control and enhancement. The nonlinear characteristics of the system, such as Lyapunov exponent and bifurcation, are analyzed in detail. Then, we design a frequency array and perform scale transformation on the chaotic system to construct a detection model that covering all frequencies. The model can transition from critical chaos to large period or intermittent chaos in the presence of external periodic signals by adjusting the parameters of the cosine function. Considering the relationship between the generation of intermittent chaos and the frequency of target signal, a frequency extraction algorithm based on chaos sparrow search algorithm optimization variational mode decomposition (CSSA-VMD) and Hilbert transform is designed. The algorithm can estimate the real frequency of underwater signals by performing noise reduction and envelope analysis on intermittent chaotic signals. The experimental results demonstrate that the proposed system can detect weak signals in complex marine backgrounds and extract their frequency features, with a detection SNR of -48 dB and a frequency error of less than 0.01%. Importantly, the circuit of the detection system has been implemented, verifying its feasibility for detecting and tracking underwater signals.

Analysis of a time-dependent memristor-based chaotic system and its application in image encryption

Abstract Compared with ordinary chaotic systems, memristor-based chaotic systems have more complex dynamic behaviors and are more suitable for image encryption algorithms. In this paper, a four-dimensional chaotic system is constructed by introducing a cubic nonlinear memristor into a three-dimensional chaotic system. Firstly, the dynamic characteristics of the constructed memristor-based chaotic system are analyzed in detail, and the simulation results show that the system has different attractors with different topological structures at different simulation times. Within a fixed simulation time, the system has 15 attractors with different topological structures under different parameter values, and there is a phenomenon of multiple stability in the system, indicating high complexity. Based on the above discoveries, a color image encryption algorithm including scrambling and diffusion is designed. Experimental results show that this algorithm can perfectly hide the information of the plaintext image, and the decrypted image is consistent with the plaintext image. Finally, the security of the algorithm is analyzed by using key space and so on. The analysis results indicate that the encryption algorithm designed in this paper can effectively resist external attacks and has high security.

A Deterministic Chaos-Model-Based Gaussian Noise Generator

The abilities of quantitative description of noise are restricted due to its origin, and only statistical and spectral analysis methods can be applied, while an exact time evolution cannot be defined or predicted. This emphasizes the challenges faced in many applications, including communication systems, where noise can play, on the one hand, a vital role in impacting the signal-to-noise ratio, but possesses, on the other hand, unique properties such as an infinite entropy (infinite information capacity), an exponentially decaying correlation function, and so on. Despite the deterministic nature of chaotic systems, the predictability of chaotic signals is limited for a short time window, putting them close to random noise. In this article, we propose and experimentally verify an approach to achieve Gaussian-distributed chaotic signals by processing the outputs of chaotic systems. The mathematical criterion on which the main idea of this study is based on is the central limit theorem, which states that the sum of a large number of independent random variables with similar variances approaches a Gaussian distribution. This study involves more than 40 mostly three-dimensional continuous-time chaotic systems (Chua’s, Lorenz’s, Sprott’s, memristor-based, etc.), whose output signals are analyzed according to criteria that encompass the probability density functions of the chaotic signal itself, its envelope, and its phase and statistical and entropy-based metrics such as skewness, kurtosis, and entropy power. We found that two chaotic signals of Chua’s and Lorenz’s systems exhibited superior performance across the chosen metrics. Furthermore, our focus extended to determining the minimum number of independent chaotic signals necessary to yield a Gaussian-distributed combined signal. Thus, a statistical-characteristic-based algorithm, which includes a series of tests, was developed for a Gaussian-like signal assessment. Following the algorithm, the analytic and experimental results indicate that the sum of at least three non-Gaussian chaotic signals closely approximates a Gaussian distribution. This allows for the generation of reproducible Gaussian-distributed deterministic chaos by modeling simple chaotic systems.

A 5-D memristive hyperchaotic system with extreme multistability and its application in image encryption

By coupling memristors to nonlinear circuits, more complex dynamical behaviors can be induced. However, to date, there has been insufficient attention given to high-dimensional chaotic systems based on memristors. In this paper, a magnetic-controlled memristor is combined with a three-dimensional chaotic system, resulting in a five-dimensional memristive chaotic system. Through dynamic analysis and numerical simulations, the chaotic nature of the system is elucidated based on fundamental system behaviors, including Lyapunov dimension, dissipativity, stability of equilibrium points, 0–1 test, and Poincaré mapping. During the complex dynamical analysis of this system, unique dynamical behaviors are discovered, including intermittent chaos, transient chaos, extreme multistability, and offset-boosting. Moreover, the consistency between numerical calculations and the physical implementation of the actual system is verified through equivalent circuit design. Finally, this system is applied to image encryption, leading to the design of an efficient and secure hyper-chaotic image encryption algorithm, whose effectiveness is confirmed through several security tests.

Dynamical analysis of a new 3D chaotic system: non-equilibrium point, conservative, rotationally hidden attractor

A novel three-dimensional chaotic system is proposed in this paper. The nonlinear features are enhanced by adding e-exponent function to make the system behavior more complex and diverse. Through calculation and experimental verification, it is found that the system is a conservative system without equilibrium point, and this kind of system may produce hidden dynamic behavior. Therefore, four different dynamic analysis tools are chosen to simulate the possible hidden dynamics. The rotation factor is used to create some new models on the original system equation. With constantly adjustments of the rotation angle, the different resulting rotational hidden attractors are used to further prove that such chaos exists in the new proposed system. In the end, the realization of DSP shows the feasibility of this system in practical applications.

Design of Pseudo-random Sequence Generator based on Three-dimensional Discrete Chaotic System

In this paper, a pseudo-random sequence generator with better randomness is constructed through the targeted improvement of chaotic system. Firstly, a new three-dimensional discrete chaotic system is proposed on the basis of the existing one-dimensional logistic chaotic system by improving and optimizing the dimensionality and coupling mode, and its Lyapunov exponent and other characteristics are analyzed; then, based on the chaotic system, a pseudo-random sequence generator is designed by using post-processing and heterodyne, etc.; after that, the performance of the pseudo-random sequence is discussed by using the evaluation methods of relevance, homogeneity, and NIST test. After that, the performance of the pseudo-random sequence is discussed using correlation, uniformity, NIST test and other evaluation methods. The results show that the three-dimensional discrete chaotic system has good chaotic properties, and the pseudo-random sequence generator based on this three-dimensional discrete chaotic system has good random performance and can pass the NIST test. The work in this paper provides an idea for the improvement of logistic mapping chaotic system and enriches the application form of pseudorandom sequence generator in information security.

Two Independent Offset Controllers in a Three-Dimensional Chaotic System

Offset boosting is a crucial passage for chaotic signal modification in chaos-based engineering. Searching an offset controller for a 3D chaotic system is usually complex, let alone two independent nonbifurcating offset constants. This paper constructs a class of chaotic systems, providing a single constant posing direct offset boosting for two dimensions. This offset boostable chaotic system regime has multiple typical control modes, including a system variable single control, synchronous common control, reverse control, and differential control. This new type of chaotic systems also finds two-dimensional offset boosting combined with amplitude control.

Dynamic Analysis of A Three Dimension Chaotic System

Memristor is a newly realized physical element, and it is a non-linear circuit element with memory function. We constructed a new three-dimensional chaotic system is constructed which contains four parameters and five non-linear terms. The system is analyzed by nonlinear dynamic analysis methods such as theoretical deduction analysis, numerical simulation, lyapunov exponent spectrum, and bifurcation diagram. The dynamic behavior of the system. This paper proposes a new three-dimensional chaotic system. The system contains four parameters, and each equation contains a non-linear product term. Based on theoretical derivation, numerical simulation, Lyapunov exponential spectrum, bifurcation diagram, the basic dynamic characteristics of the chaotic system are analyzed.

Dynamical analysis and finite-time synchronization of grid-scroll memristive chaotic system without equilibrium

The study of multi-scroll memristive chaotic system has attracted great interest due to their unique dynamics. In this paper, a novel five-dimensional chaotic system devoid of equilibria is constructed by introducing two memristors into a three-dimensional chaotic system, resulting in the emergence of a grid-scroll attractor. The evolution of the chaotic system is analyzed using techniques such as phase diagrams, Poincare maps, bifurcation diagrams, and Lyapunov exponents. The basin of attraction is delineated to investigate the coexisting attractors in the system. By manipulating the parameters, the partial amplitude of the system can be modulated. To experimentally confirm the existence of the proposed system, a circuit is meticulously designed and implemented. At last, an adaptive controller is designed, and it is observed from the simulation results that a synchronization state can be reached by the two chaotic systems within a finite time.

Dynamical analysis and circuit implementation of a memristive chaotic system with infinite coexisting attractors

In order to obtain a chaotic system with more complex dynamic characteristics and more suitable for engineering applications, this paper combines a general memristor containing a hyperbolic tangent function with a simple three-dimensional chaotic system to construct a four-dimensional memristive chaotic system with infinite coexisting attractors. The memristive chaotic system is thoroughly studied through numerical simulations of various nonlinear systems, including the Lyapunov exponent spectra, bifurcation diagram, C0 complexity, two-parameter bifurcation diagram and basins of attraction. The analysis reveals that this system has complex dynamical behavior. It includes not only periodic limit loops and chaotic attractors that depend on the variation of system parameters, but also the extreme multi-stability phenomenon of infinite coexisting attractors that depend on the variation of the initial conditions of the system. In addition, the chaos degradation and offset boosting control of the system are also studied and analyzed. Finally, the correctness and realizability of the memristive chaotic system are verified by circuit simulation and hardware circuit fabrication.The experimental results show that this memristive chaotic system can lay the foundation for practical engineering fields such as secure communication and image encryption.

Optimizing the combination of data-driven and model-based elements in hybrid reservoir computing.

Hybrid reservoir computing combines purely data-driven machine learning predictions with a physical model to improve the forecasting of complex systems. In this study, we investigate in detail the predictive capabilities of three different architectures for hybrid reservoir computing: the input hybrid (IH), output hybrid (OH), and full hybrid (FH), which combines IH and OH. By using nine different three-dimensional chaotic model systems and the high-dimensional spatiotemporal chaotic Kuramoto-Sivashinsky system, we demonstrate that all hybrid reservoir computing approaches significantly improve the prediction results, provided that the model is sufficiently accurate. For accurate models, we find that the OH and FH results are equivalent and significantly outperform the IH results, especially for smaller reservoir sizes. For totally inaccurate models, the predictive capabilities of IH and FH may decrease drastically, while the OH architecture remains as accurate as the purely data-driven results. Furthermore, OH allows for the separation of the reservoir and the model contributions to the output predictions. This enables an interpretation of the roles played by the data-driven and model-based elements in output hybrid reservoir computing, resulting in higher explainability of the prediction results. Overall, our findings suggest that the OH approach is the most favorable architecture for hybrid reservoir computing, when taking accuracy, interpretability, robustness to model error, and simplicity into account.

Projection Synchronization of Three-Dimensional Chaotic Systems With Active Control Based on DNA Strand Displacement.

The emergence of biological computing based on DNA strand displacement has allowed chaotic systems to have more abundant dynamic behaviors. So far, the synchronization of chaotic systems based on DNA strand displacement has been mainly realized by coupling control and PID control. In this paper, the projection synchronization of chaotic systems based on DNA strand displacement is achieved using an active control method. First, some basic catalytic reaction modules and annihilation reaction modules are constructed based on the theoretical knowledge of DNA strand displacement. Second, the chaotic system and the controller are designed according to the above mentioned modules. On the basis of chaotic dynamics, the complex dynamic behavior of the system is verified by the lyapunov exponents spectrum and the bifurcation diagram. Third, the active controller based on DNA strand displacement is used to realize the projection synchronization between the drive system and the response system, where the projection can be adjusted within a certain range by changing the value of the scale factor. The result of projection synchronization of chaotic system is more flexible, which is realized by active controller. Our control method provides an efficient way to achieve synchronization of chaotic systems based on DNA strand displacement. The designed projection synchronization is verified to have excellent timeliness and robustness by the results Visual DSD simulation.

Adaptive Fast Image Encryption Algorithm Based on Three-Dimensional Chaotic System.

This paper introduces a novel three-dimensional chaotic system that exhibits diverse dynamic behaviors as parameters vary, including phase trajectory offset behaviors and expansion-contraction phenomena. This model encompasses a broad chaotic range and proves suitable for integration within image encryption. Building upon this chaotic system, the study devised a fast image encryption algorithm with an adaptive mechanism, capable of autonomously determining optimal encryption strategies to enhance algorithm security. In pursuit of heightened encryption speed, an FPGA-based chaotic sequence generator was developed for the image encryption algorithm, leveraging the proposed chaotic system. Furthermore, a more efficient scrambling algorithm was devised. Experimental results underscore the superior performance of this algorithm in terms of both encryption duration and security.

Synchronization of chaotic satellite systems with an unknown parameter

Spacecraft (satellite) formation has several applications in space exploration. Synchronization controllers are essential in satellite formation systems. This article proposes a novel robust adaptive synchronization control (RASC) technique for two identical three-dimensional chaotic satellite systems attitude synchronization with an unknown system parameter, unknown bounded model uncertainties, unknown external disturbances, and nonlinear terms. This controller achieves smooth error vector convergence to zero. The closed-loop has a robust performance as the controller avoids plant nonlinear terms cancellation. This controller has two parts. (i) A nonlinear controller for keeping the closed-loop stable; it flourishes robust and faster error vector convergence to zero with reduced fluctuations. (ii) An adaptive nonlinear controller that compensates exogenous disturbances and model uncertainties using online estimations of model uncertainties and bounds. The Lyapunov stability theory assures the closed-loop's global asymptotic robust stability. Computer simulation results-based analysis verifies the theoretical findings. The article includes a comparative study and discusses the control effort saturation effects.

A memristive chaotic system and its application in weak signal detection

In this paper, a novel four-dimensional memristive chaotic system is constructed by incorporating a memristor model into a three-dimensional chaotic system. Through the analysis of the Lyapunov exponent, bifurcation diagram, and Poincaré cross-section of the system, it has been observed that the system is capable of exhibiting a stable chaotic state, as well as complex dynamic behaviors, such as attractor coexistence, transient chaos, and offset boosting. To validate the actual existence of the system, a real circuit has been built based on Multisim simulation, and the numerical simulation results, along with the actual simulation results, are in agreement, thereby confirming the practical feasibility of the circuit. To achieve weak signal detection, a backstepping synchronization controller has been designed, which can detect the frequency and amplitude of unknown signals. It is obvious that this method does not require the determination of the critical threshold. Instead of, it relies on the system being in a chaotic state. The proposed detection method provides a new perspective for weak signal detection.

Physics-Informed Long-Short-Term Memory Neural Network for Parameters Estimation of Nonlinear Systems

This paper proposes a physics-informed long-short-term memory neural network optimisation method to estimate all parameters of an arbitrary nonlinear system initialised using any initial conditions. Although the technique is applicable for parameters' estimation of any nonlinear system, in this paper, two well-known chaotic systems are considered with different initial conditions. The three-dimensional Lorenz chaotic system and the four-dimensional Rössler hyperchaotic system are considered. The Lorenz system exhibits stable, chaotic and periodic behaviours for some sets of parameters. Thus, different sets of parameters of the Lorenz system are individually estimated, leading to stable, periodic and chaotic behaviours. In 10000 epochs, the stable and periodic parameters' estimation accuracy is more than 99%; in 15000 epochs, chaotic parameters' estimation accuracy is more than 99%. Upto same epochs, physics-informed neural network is employed for parameter estimation of stable, periodic and chaotic behaviours of the Lorenz system. The comparison results of estimated parameters are shown with dataset obtained using ode45. Next, Circuit realisation of the Lorenz system is done and the dataset is obtained using multisim circuit simulation for stable, chaotic and periodic behaviour. The proposed physics-informed long-short-term memory neural network is used to obtain parameters from the practical data. Then, the estimation of the Lorenz system with a continuum variation of parameters' sets from stable to chaotic to stable behaviours is successfully achieved using the proposed physics-informed long-short-term memory neural network. Finally, the hyperchaotic Rössler system parameters are successfully estimated using the proposed technique with more than 99% accuracy in 20000 epochs.

Image compression-hiding algorithm based on compressive sensing and integer wavelet transformation

In this paper, a three-dimensional chaotic system is proposed. Based on this chaotic system and two-dimensional compressive sensing, an asymmetric visually meaningful image compression-hiding algorithm is presented. Firstly, in the keystream generation stage, a novel parameter transformation model is constructed to pick up the feature information from the plain image as the plaintext key. Then, Rivest-Shamir-Adleman algorithm is employed to encrypt the plaintext key into the ciphertext key seen as public key. Before generating the initial values for the chaotic system, a new initial value getting model is designed to transform both the plaintext and the ciphertext keys. After solving the chaotic system, the keystream is produced which is then used in the image encryption process. Secondly, in the compression and encryption phase, a novel keystream pre-processing model is built to generate new sequences with a confusion performed on the plain image. Then, a newly constructed measurement matrix is designed to do two-dimensional compressive sensing on confusing the image to get measurements. Before obtaining the cipher image, a double diffusion operation is applied on these measurements. Thirdly, in the image hiding stage, the carrier image is performed by integer wavelet transformation to obtain coefficient matrices. Then, the cipher image is decomposed in decimal, getting the ones, tens and hundreds of pixels to form three bit matrices, of which are embedded into the three medium-high coefficient matrices of integer wavelet transformation, respectively. Finally, after performing inverse integer wavelet transformation, the carrier image containing secrets, i.e., visually meaningful encrypted image, is obtained. Experimental results also show that at a low compression ratio of 0.25, the normalized correlation coefficient between the original plain image and the recovered image is almost equal to one, while the peak signal-to-noise ratio between the carrier image containing secrets and its original carrier can reach as high as 42 dB. In addition, the proposed image compression-hiding algorithm performs good ability consideing the brute force attack and the cropping attack.

A Novel Conservative Chaotic System Involved in Hyperbolic Functions and Its Application to Design an Efficient Colour Image Encryption Scheme

It is well known that, compared to low-dimension chaotic systems, three-dimensional chaotic systems have a wider parameter range, more complicated behaviour, and better unpredictability. This fact motivated us to introduce a novel image encryption method that employs a three-dimensional chaotic system. We proposed a novel three-dimensional conservative system that can exhibit chaotic behaviour involving hyperbolic functions. The dynamical behaviours of the proposed system are discovered by calculating Lyapunov exponents and bifurcation diagrams. Thereafter, we designed an image encryption method based on the proposed system and a 4×4 self-invertible matrix. A modified Diffie–Hellman key exchange protocol was utilised to generate the self-invertible key matrix Km employed in the diffusion stage. Our approach has three main stages. In the first stage, the proposed three-dimensional system utilises the original image to create three sequences, two of which are chosen for confusion and diffusion processes. The next stage involves confusing the image’s pixels by changing the positions of pixels using these sequences. In the third stage, the confused image is split into sub-blocks of size 4×4, and each block is encrypted by multiplying it with Km. Simulation findings demonstrated that the proposed image scheme has a high level of security and is resistant to statistical analysis, noise, and other attacks.

A non-autonomous mega-extreme multistable chaotic system

Megastable and extreme multistable systems comprise two major new branches of multistable systems. So far, they have been studied separately in various chaotic systems. Nevertheless, to the best of our knowledge, no chaotic system has so far been reported that possesses both types of multistability. This paper introduces the first three-dimensional non-autonomous chaotic system that displays megastability and extreme multistability, jointly called mega-extreme multistability. Our model shows extreme multistability for a variation of an initial condition associated with one system variable and megastability concerning another variable. The different types of coexisting attractors are characterized by the corresponding phase portraits and first return maps, as well as by constructing the appropriate bifurcation diagrams, calculating the Lyapunov spectra, the Kaplan-Yorke dimension and the connecting curves, and by determining the corresponding basins of attraction. The system is explicitly shown to be dissipative, with the dissipation being state-dependent. We demonstrate the feasibility and applicability of our model by designing and simulating an appropriate analog circuit.