Complex Chaotic Systems Research Articles

A novel adaptive parameter zeroing neural network for the synchronization of complex chaotic systems and its field programmable gate array implementation

This article proposes a novel adaptive-parameter zeroing neural network (NAP-ZNN) to control the synchronization of complex chaotic systems. Different from previous zeroing neural networks (ZNNs), the NAP-ZNN model incorporates adaptive-parameter, a unique blend of time-varying convergence factor and fuzzy control parameter, which is adjusted continuously according to the real-time synchronizing error, offering greater flexibility. The NAP-ZNN model aims to address the time-varying problem in complex-valued domain, which is scarcely explored, it rapidly impels two complex chaotic systems to synchronize within predetermined time and effectively mitigates noise interference. Moreover, the upper bound of the convergence time is theoretically calculated in noiseless environments, and the disturbance-resistant property is demonstrated under various noisy disturbances. The control performance of NAP-ZNN model has been separately compared to other general ZNNs (NAP-ZNN synchronizes at 0.097 s while three general ZNNs spend 0.43 s, 0.78 s and 0.84 s, and only NAP-ZNN complete synchronization within 0.1 s under noisy interferences while general ZNNs fail) and the traditional adaptive control method through two synchronization simulation (RMSE of NAP-ZNN under sinusoidal noise is 3.165 × 10-3 while traditional is 3.383 at t = 1 s). The numerical outcomes clearly illustrate the superior performance of the NAP-ZNN model in comparison to other synchronization strategies. The hardware implementation of the NAP-ZNN model has been successfully realized through the Field Programmable Gate Array (FPGA), and the chaos synchronization process has been vividly captured via an oscilloscope, further reinforcing the practicality and effectiveness of the NAP-ZNN model.

Two-dimensional hyperchaos-based encryption and compression algorithm for agricultural UAV-captured planar images.

This study presents an approach that integrates compressed sensing technology with two-dimensional hyperchaotic coupled Fourier oscillator systems (2D-HCFOS) to address the challenge of slow encryption speeds in agricultural unmanned aerial vehicles (UAVs). The primary challenge in enhancing encryption speed lies in the limited capacity inherent in traditional chaotic-based systems and the computational complexity of their processes. The 2D-HCFOS utilizes a complex two-dimensional hybrid chaotic system, which significantly enhances the security of agricultural UAV image data. Notably, the image encryption process is performed on a personal computer connected to the drone, ensuring efficient processing. By integrating advanced Fourier series and nonlinear coupled oscillators, the model surpasses existing chaotic-based methods, improving both the pseudo-randomness and robustness of encryption. Additionally, incorporating Bonouille functions into the discrete cosine transform (DCT) domain results in a sparser measurement matrix, which is essential for efficient encryption on personal computers. The effectiveness of 2D-HCFOS in securely encrypting agricultural drone images has been rigorously validated through simulations and analytical evaluations using sophisticated row, rotation, and matrix encryption techniques. The improved security performance is further verified by comparative analysis. Compared with other models, the Lyapunov index of 2D-HCFOS is 15.1039, and the sample entropy is 2.4987, indicating that it possesses superior chaotic performance and encryption reliability.

Specified time dual-group synchronization of uncertain complex chaotic systems

Aiming at the specified time dual-group synchronization problem of multi-wing complex chaotic systems containing uncertain terms and external disturbances, a new specified-time sliding mode control scheme is proposed, which directly synchronizes the complex chaotic system without separating the real and imaginary parts of the complex chaotic system. First, a new specified time stability criterion is used to construct the integral sliding mode surface of the synchronous error system to ensure stable sliding motion within the specified time. Subsequently, a proximity controller is designed to drive the error system to reach and remain on the sliding surface within another specified time, thereby achieving specified-time synchronization. In order to realize the proposed stability concept, this paper introduces a new sliding surface and defines the corresponding control law and adaptive rate. The effectiveness of this scheme is proved through Lyapunov stability theory and specified time stability theory. Numerical simulation results show that the scheme has strong robustness to uncertainties and external disturbances, and the controller is not affected by internal uncertainties and external disturbances. Compared to other stabilization time control schemes, this scheme has a shorter synchronization time. In general, this study introduces complex variables and adopts a scheme in which sliding mode surface parameters and controller parameters can be preset to simultaneously achieve dual-group synchronization of two groups of complex chaotic systems within the complex domain. This study offers greater flexibility, presenting novel ideas and approaches for the synchronization control of complex systems. It holds significant theoretical and practical value, providing valuable references and insights for research and applications in related fields.

Analytic solutions of alpha-beta -time- derivatives complex finance chaotic dynamical system: synchronization and extended center manifold. An explicit approach

The present study focuses on a real finance nonlinear dynamic system (FNLDS), which has been shown to exhibit chaotic behavior. The solutions for such nonlinear dynamical systems (NLDSs) have typically been derived using numerical techniques. The objective of this study aims to; firstly, derive approximate analytical solutions for the complex FNLDS (CFNLDS) by constructing the Picard iterative scheme. The convergence of this scheme is proven, and the error analysis shows good tolerance, indicating the efficiency of the technique. Second, a novel criterion for synchronizing the real and imaginary parts of the system is presented, based on a necessary condition. Thirdly, a new method for constructing the extended center manifold is introduced. The 3D portrait reveals a feedback scroll pattern, while the 2D portrait, representing the mutual components, shows multiple pools. The synchronization of the real and imaginary parts of the system is demonstrated graphically. The FNLDS is tested for sensitivity dependence against tiny variations in the initial conditions, and it is found that the system components are moderately sensitive. Furthermore, the Hamiltonian and the extended center manifold establish a two-fold structure. It is observed that the effect of the α-β derivative leads to a delay in the behavior of the solutions.

Hamilton energy of a complex chaotic system and offset boosting

The complex differential system can be obtained by introducing complex variable in the real differential system. Complex variables can be decomposed into real component and imaginary component, which makes the complex differential systems have more complex dynamic behaviors. Complex chaotic system is used in secure communications to increase the security of cryptographic systems. In this study, we designed a complex differential system by incorporating a complex variable into a 3D differential system. Dynamics of this complex differential system are investigated by applying typical nonlinear analysis tools. Furthermore, Hamilton energy function for complex differential system is obtained based on Helmholtz’s theorem. The values of Hamilton energy with different oscillations of complex differential system are calculated. In addition, offset boosting control for the complex chaotic signal is realized by adding a constant to variable of complex system. Simulation shows that the position of the chaotic attractor in phase space can be flexibly shifted by applying the offset parameter.

New Ways to Modelling and Predicting Ionosphere Variables

The new way of thinking science from Newtonian determinism to nonlinear unpredictability and the dawn of advanced computer science and technology can be summarized in the words of the theoretical physicist Michel Baranger, who, in 2000, said in a conference: “Twenty-first-century theoretical physics is coming out of the chaos revolution; it will be about complexity and its principal tool will be the computer.”. This can be extended to natural sciences in general. Modelling and predicting ionosphere variables have been considered since many decades as a paramount objective of research by scientists and engineers. The new approach to natural sciences influenced also ionosphere research. Ionosphere as a part of the solar–terrestrial environment is recognized to be a complex chaotic system, and its study under this new way of thinking should become an important area of ionospheric research. After discussing the new context, this paper will try to review recent advances in the exploration of ionosphere parameter time series in terms of chaos theory and the use of machine-learning algorithms.

Satellite image encryption based on RNA and 7D complex chaotic system

Satellite image encryption based on RNA and 7D complex chaotic system

The Primacy of Doubt: From Quantum Physics to Climate Change, How the Science of Uncertainty Can Help Us Understand Our Chaotic World

The Primacy of Doubt: From Quantum Physics to Climate Change, How the Science of Uncertainty Can Help Us Understand Our Chaotic World

Multistable dynamics and attractors self-reproducing in a new hyperchaotic complex Lü system.

Multistable dynamics analysis of complex chaotic systems is an important problem in the field of chaotic communication security. In this paper, a new hyperchaotic complex Lü system is proposed and its basic dynamics are analyzed. Owing to the introduction of complex variables, the new system has some structurally distinctive attractors, such as flower-shaped and airfoil-shaped attractors. In addition, the evolution process of the limit cycle is also investigated. Next, the multistable coexistence behavior of the system is researched by the method of attraction basins, and the coexistence behavior of two types of hyperchaotic attractors and one type of chaotic and periodic attractors of the system are analyzed. The coexisting hyperchaotic attractors also show flower and airfoil shapes, and four types of coexistence flower-shaped attractors with different structures are perfectly explored. Moreover, the variation of coexistence attractors in the plane and space with parameters is discussed. Then, by introducing a specific piecewise function determined by a two-element method into the new high-dimensional system, the self-reproduction of the attractor can be realized to generate the multistability, and the general steps of attractors self-reproducing in the higher dimensional system are given. Finally, the circuit design of the new system is implemented, which lays a foundation for the application of complex chaotic systems.

A new fractional-order complex chaotic system with extreme multistability and its implementation

In this paper, a new fractional-order complex chaotic system (FOCCS) is proposed and studied. Firstly, the dissipativity and stability are discussed. Secondly, the dynamical characteristics of the system with parameters and order changes are analyzed by using phase diagrams, Lyapunov exponent (LEs) and bifurcation diagrams, respectively. In addition, the dynamical behavior is discussed for q of integer and fractional orders. In particular, the attractor coexistence is found, such as the coexistence of chaotic attractor and chaotic attractor, and chaotic attractor and periodic attractor. Interestingly, the multiple attractors coexistence is found by changing the initial conditions with fixed parameters. Finally, it is implemented on the analog circuit and DSP platform. The study provide a reference for the research and application of chaos.

Special attractors and dynamic transport of the hybrid-order complex Lorenz system

Combining the advantages of both integer-order and fractional-order complex chaotic systems, we propose a hybrid-order complex Lorenz system. We demonstrate its abundant chaotic characteristics, including symmetry and dissipation, fixed points and their stability and Lyapunov exponents, with 0–1 test. Then we show that, as the initial value, parameters and the order are varying, the system exhibits diverse dynamical behaviors, with fixed points, limit cycles and chaotic attractors. We further show that the system has coexisting attractors and parametric attractors. In addition, we find that the system generates different chaotic attractors as the system hybrid order varies, referred to as order attractors. Finally, we examine the dynamic transport of the hybrid-order complex Lorenz system and design a piecewise continuous controller to realize offset boosting control. By varying the initial value, parameters or orders, we realize the dynamic transport of the system. Our simulation results confirm the dynamic transport of the hybrid-order complex Lorenz system.

Data encryption based on 7D complex chaotic system with cubic memristor for smart grid

The information security has an irreplaceable position in the smart grid (SG). In order to avoid the malicious attack and ensure the information security, the cryptographic techniques are essential. This paper focuses on the encryption techniques to ensure the information security of SG. Firstly, an unusual 7-dimensional complex chaotic system (7D-CCS) combined with the cubic memristor is introduced. Besides its phase portraits, Lyapunov exponent, 0–1 test, complexity, and bifurcation diagram are investigated. Then, with the proposed 7D-CCS, we design a data encryption algorithm to ensure the encryption security. Finally, the data and monitoring images in SG are encrypted by the designed encryption scheme. Besides, the encryption performance is given in detailed. The experimental results show that the proposed encryption scheme has quite good encryption performance. Therefore, it can ensure the information security of SG.

FPGA-Based Implementation and Synchronization Design of a New Five-Dimensional Hyperchaotic System.

Considering the security of a communication system, designing a high-dimensional complex chaotic system suitable for chaotic synchronization has become a key problem in chaotic secure communication. In this paper, a new 5-D hyperchaotic system with high order nonlinear terms was constructed and proved to be hyperchaotic by dynamical characterization characteristics, the maximum Lyapunov exponent was close to 2, and there was a better permutation entropy index, while a valid chaotic sequence could be generated in three cycles in the FPGA (Field Programmable Gate Array)-based implementation. A multivariable nonlinear feedback synchronous controller based on FPGA was proposed to design and implement synchronization of high order complex hyperchaotic systems. The results show that the error signal converged to 0 rapidly under the effect of the nonlinear feedback synchronous controller. This lays the foundation for the synchronization of high order complex chaotic systems.

An Extremely Multistable Complex Chaotic System Under Boosting Control

In order to explore the phenomenon of coexisting attractors in complex chaotic systems, a multistable complex chaotic system is designed by using a cosine function and a control parameter to control the Sprott B system in this paper. The basic features of the complex chaotic system are analyzed from the perspectives of symmetry, dissipation, equilibrium points and stability. The analysis results indicate that the complex chaotic system is a symmetrical and dissipative system. It is particularly interesting to note that the system has infinitely many unstable equilibrium points, which generate infinitely many attractors. The dynamical characteristics of the complex chaotic system are evaluated through bifurcation diagrams, spectral entropy complexity and Lyapunov exponents spectrum. The dynamical evaluations illustrate that the system has complex dynamical performances, such as chaotic states, periodic states, especially infinitely many coexisting attractors. Then a chaotic circuit based on the complex chaotic system is designed and simulated by Multisim software. And the hardware circuit is implemented by using digital signal processor (DSP) platform. The hardware experimental results are consistent with the Multisim simulation results.

Data encryption based on a 9D complex chaotic system with quaternion for smart grid

With the development of smart grid, operation and control of a power system can be realized through the power communication network, especially the power production and enterprise management business involve a large amount of sensitive information, and the requirements for data security and real-time transmission are gradually improved. In this paper, a new 9-dimensional (9D) complex chaotic system with quaternion is proposed for the encryption of smart grid data. Firstly, we present the mathematical model of the system, and analyze its attractors, bifurcation diagram, complexity, and 0–1 test. Secondly, the pseudo-random sequences are generated by the new chaotic system to encrypt power data. Finally, the proposed encryption algorithm is verified with power data and images in the smart grid, which can ensure the encryption security and real time. The verification results show that the proposed encryption scheme is technically feasible and available for power data and image encryption in smart grid.

Robust stabilization and synchronization in a network of chaotic systems with time-varying delays

In this paper, a novel controller technique is derived for the network synchronization of chaotic systems with time-varying delay. The stabilization of the complex chaotic systems is achieved by considering an appropriate Krasovskii-Lyapunov functional in order to find the asynchronous robust stabilization law and fast switching topology. In a similar way, the asynchronous technique is based on an appropriate switching topology in order to synchronize the chaotic systems. The time-varying delays occur in different systems which increase the complexity of the synchronization. So for this reason, an asynchronous controller is designed by considering the time-varying delays in order to avoid instability or deteriorate performance of the whole chaotic systems. The remarkable finding of this study is that the robust controller considers the uncertainties on each complex chaotic systems node until an exact synchronization is achieved while switching the action of the controller synchronization and the chaotic node dynamics of the studied complex systems. Finally, numerical examples are considered on chaotic Rössler systems to test and validate the obtained theoretical results.

Several Control Problems of a Class of Complex Nonlinear Systems Based on UDE

This paper mainly studies several control problems of a complex 4D chaotic system. Firstly, the real part and imaginary part of the complex 4D chaotic system are separated, and the system is equivalent to a six-dimensional continuous autonomous real chaotic system. Then, the stabilization, synchronization, and anti-synchronization of the complex four-dimensional chaotic system are realized by using the control method of the combination of dynamic feedback gain control and UDE control, and the corresponding physical controllers are designed respectively. Finally, the correctness and effectiveness of the theoretical results are verified by numerical simulation.

Complex Modified Projective Difference Function Synchronization of Coupled Complex Chaotic Systems for Secure Communication in WSNs

Complex-variable chaotic systems (CVCSs) have numerous advantages over real-variable chaotic systems in chaos communication due to their increased unpredictability, confidentiality, and the ease of implementation. Synchronization between the master and slave systems in CVCSs is key to achieving encryption and decryption. However, existing synchronization schemes for CVCSs require the amplitude of the chaotic signal to be much larger than that of the plaintext. Moreover, traditional chaotic masking of complete synchronization (CS) requires uniformity between the transmitter and receiver ends. Therefore, we propose a complex modified projective difference function synchronization (CMPDFS) of CVCSs to address these issues, where the modified projective matrix helps address the issues with the amplitude. The receiver end is reconstructed without uniformity of the transmitter. We design the CMPDFS controller and propose a new secure communication scheme for wireless sensor networks (WSNs). The basic principle is fundamentally different from traditional chaotic masking. Simulation results and security analysis demonstrate that the CMPDFS communication scheme has a large key space, high sensitivity to encryption keys, high security, and an acceptable encryption speed. Hence, the proposed scheme can improve the security of WSNs. Moreover, it also can be applied to similar communication systems.

Design of intelligent computing networks for nonlinear chaotic fractional Rossler system

In a recent development, the legacy of fractional-order multi-dimensional chaotic systems has attracted numerous researchers owing to their valuable applications in cryptology, control, information security, mathematics, and physical sciences. In this study, strength of artificial intelligence (AI) algorithms are explored for stochastic numerical computing for nonlinear fractional Rossler differential system by introducing state of art transformation in the radial basis neural networks (RNFN). Fractional order complex chaotic dynamical system of Rossler model represented by nonlinear coupled fractional differential equations are numerically solved with state of art fourth order fractional Runge-Kutta solver for different scenarios of control parameter and these outcomes are mensurate as reference solutions to model implementing RBFN with the different initial conditions of the chaotic dynamics. The innovative transformations are introduced to enhance the fast convergence of machine learning RBFN algorithm achieved through bimodal chaotic parameters. Performance of designed computing RBFN is authenticated efficaciously by RMSE metric for fractional order chaotic model of Rossler attractor with outcomes matching of the order of 10 to 15 decimal places in term of accuracy as compared to standard results computed from fractional Runge-Kutta solver. It is believed the proposed AI knacks based research work will open new innovative path in fractional order modelling and analysis of natural chaotic dynamic systems.

The global attractive sets and synchronization of a fractional-order complex dynamical system

<abstract><p>This paper presents a chaotic complex system with a fractional-order derivative. The dynamical behaviors of the proposed system such as phase portraits, bifurcation diagrams, and the Lyapunov exponents are investigated. The main contribution of this effort is an implementation of Mittag-Leffler boundedness. The global attractive sets (GASs) and positive invariant sets (PISs) for the fractional chaotic complex system are derived based on the Lyapunov stability theory and the Mittag-Leffler function. Furthermore, an effective control strategy is also designed to achieve the global synchronization of two fractional chaotic systems. The corresponding boundedness is numerically verified to show the effectiveness of the theoretical analysis.</p></abstract>