Fourth-order Chaotic System Research Articles

Bifurcation, chaos and fixed-time synchronization of memristor cellular neural networks

This study investigates a novel chaotic memristive cellular neural network (CNN) and its synchronization. Firstly, a fourth-order chaotic system with a memristor is established by introducing a CNN and a memristor model. Secondly, the dynamic behavior of the system is analyzed, including its stability, bifurcation, and chaotic attractors. In particular, Hopf bifurcations are investigated in detail. Furthermore, the effects of the memristor’s parameters and initial state on the dynamic behavior of the system are discussed. The conclusions are verified through the use of Lyapunov exponents and bifurcation diagrams. Additionally, the study examines the multistability that arises in memristive CNNs. Moreover, an analog electronic circuit is developed by creating appropriate system parameters to confirm the presence of chaotic attractors. Thirdly, fixed-time synchronization of memristor-based chaotic CNNs is achieved through the use of sliding mode control method. A stability criterion of error system is proposed, and the results are verified through simulation

Dynamic analysis and FPGA implementation of a fourth-order chaotic system with coexisting attractor

To further explore the complex dynamical behaviors in coexisting attractors, a fourth-order chaotic system with four types of coexisting attractors and four unstable equilibrium points is constructed in this paper. The dynamic behavior of the new system is analyzed by means of phase trajectory diagram, time domain waveform diagram, Poincaré map, Lyapunov exponent spectrum and bifurcation diagram. The experimental results show that as the parameters change, the system exhibits rich dynamic behaviors such as stable points, period-doubling bifurcations, coexisting bifurcation modes, and chaotic crises. When the system parameters and memristive parameters change, it is found that the system has different types of coexisting attractors, such as the coexistence of two periodic attractors, the coexistence of two single-scroll chaotic attractors, the coexistence of two double-scroll chaotic attractors, the coexistence of two point attractors. In particular, the system also has the rotation phenomenon of coexisting attractors. Finally, a nonlinear feedback controller is designed, which can make the new system achieve chaos synchronization in a short time. The phase diagram captured by the field-programmable gate array (FPGA) hardware platform is consistent with the numerical simulation results, which proves the feasibility of the system.

A Novel four - Wing chaotic system with multiple attractors based on hyperbolic sine: Application to image encryption*

A novel symmetrical four-wing fourth-order chaotic system is constructed. The basic dynamic characteristics of the system were analyzed by phase diagram, bifurcation diagram, Lyapunov index spectrum, Poincare cross section diagram and 0–1 test. In addition, the chaotic attractors under different parameters in the system are analyzed. In the dynamic analysis of the new system, it is found that the new system has some characteristics, such as multi-stability, offset boosting, multi-state transition phenomenon, transient chaos and intermittent chaos, and coexistence of multiple attractors. These features have the value of in-depth analysis compared to previous systems and can make it promising for more applications. Moreover, after calculating the complexity of the system by C0 algorithm at different initial values, it is found that the complexity of the system has been stable. Due to the existence of many characteristics, the new chaotic system has attracted great attention. At the same time, the circuit design of the system is realized by using Multisim simulation software and FPGA digital hardware circuit. Finally, a novel and efficient image encryption algorithm is designed by combining multi-direction pixel scrambling and DNA dynamic encryption. The NIST test, key space, encryption histogram, adjacent pixel correlation, robustness and information entropy are analyzed by encrypting images using chaotic sequences of the new system. In a word, its plentiful characteristics and intricate phenomena have significant reference value in the field of chaotic image encryption.

A novel fourth order chaotic system and its algorithm for medical image encryption

Today, medical imaging suffers from serious issues such as malicious tampering and privacy leakage. Encryption is an effective way to protect these images from security threats. Chaos has been widely used in image encryption, the majority of these algorithms are based on classical chaotic systems. For now, these systems are easy to analyze and predict, which are not sufficient for image encryption proposes. In this paper, a novel fourth order chaotic system is proposed, accompanied by analysis of Lyapunov exponent and bifurcations. Finally, the application of this system with medical image encryption is proposed. As this system could have six control parameters and four initial conditions, the key space is far greater than 5.1 × 218191, which is large enough to resist brute force attack. Correlation analysis and differential attack analysis further demonstrate that this scheme has a strong resistance against statistical attacks and differential attack.

Analysis, circuit implementation and applications of a novel chaotic system

PurposeLack of optimization and improvement on experimental circuits precludes comprehensive statements. It is a deficiency of the existing chaotic circuit technology. One of the aims of this paper is to solve the above mentioned problems. Another purpose of this paper is to construct a 10 + 4-type chaotic secure communication circuit based on the proposed third-order 4 + 2-type circuit which can output chaotic phase portraits with high accuracy and high stability.Design/methodology/approachIn Section 2 of this paper, a novel third-order 4 + 2 chaotic circuit is constructed and a new third-order Lorenz-like chaotic system is proposed based on the 4 + 2 circuit. Then some simulations are presented to verify that the proposed system is chaotic by using Multisim software. In Section 3, a fourth-order chaotic circuit is proposed on the basis of the third-order 4 + 2 chaotic circuit. In Section 4, the circuit design method of this paper is applied to chaotic synchronization and secure communication. A new 10 + 4-type chaotic secure communication circuit is proposed based on the novel third-order 4 + 2 circuit. In Section 5, the proposed third-order 4 + 2 chaotic circuit and the fourth-order chaotic circuit are implemented in an analog electronic circuit. The analog circuit implementation results match the Multisim results.FindingsThe simulation results show that the proposed fourth-order chaotic circuit can output six phase portraits, and it can output a stable fourth-order double-vortex chaotic signal. A new 10 + 4-type chaotic secure communication circuit is proposed based on the novel third-order 4 + 2 circuit. The scheme has the advantages of clear thinking, efficient and high practicability. The experimental results show that the precision is improved by 2-3 orders of magnitude. Signal-to-noise ratio meets the requirements of engineering design. It provides certain theoretical and technical bases for the realization of a large-scale integrated circuit with a memristor. The proposed circuit design method can also be used in other chaotic systems.Originality/valueIn this paper, a novel third-order 4 + 2 chaotic circuit is constructed and a new chaotic system is proposed on the basis of the 4 + 2 chaotic circuit for the first time. Some simulations are presented to verify its chaotic characteristics by Multisim. Then the novel third-order 4 + 2 chaotic circuit is applied to construct a fourth-order chaotic circuit. Simulation results verify the existence of the new fourth-order chaotic system. Moreover, a new 10 + 4-type chaotic secure communication circuit is proposed based on chaotic synchronization of the novel third-order 4 + 2 circuit. To illustrate the effectiveness of the proposed scheme, the intensity limit and stability of the transmitted signal, the characteristic of broadband and the requirements for accuracy of electronic components are presented by Multisim simulation. Finally, the proposed third-order 4 + 2 chaotic circuit and the fourth-order chaotic circuit are implemented through an analog electronic circuit, which are characterized by their high accuracy and good robustness. The analog circuit implementation results match the Multisim results.

Reduced-Order Antisynchronization of Chaotic Systems via Adaptive Sliding Mode Control

A novel reduced-order adaptive sliding mode controller is developed and experimented in this paper to antisynchronize two different chaotic systems with different order. Based upon the parameters modulation and the adaptive sliding mode control techniques, we show that dynamical evolution of third-order chaotic system can be antisynchronized with the projection of a fourth-order chaotic system even though their parameters are unknown. The techniques are successfully applied to two examples: firstly Lorenz (4th-order) and Lorenz (3rd-order) and secondly the hyperchaotic Lü (4th-order) and Chen (3rd-order). Theoretical analysis and numerical simulations are shown to verify the results.

Adaptive Increasing-Order Synchronization and Anti-Synchronization of Chaotic Systems with Uncertain Parameters

We elaborate the concept of increasing-order synchronization and anti-synchronization of chaotic systems via an adaptive control scheme and modulation parameters. It is shown that the dynamical evolution of a third-order chaotic system can be synchronized and anti-synchronized with a fourth-order chaotic system even though their parameters are unknown. Theoretical analysis and numerical simulations are carried out to verify the results.

Chaos reduced-order anti-synchronization of chaotic systems with fully unknown parameters

In this present paper, we elaborated the concept of the reduced-order anti-synchronization of uncertain chaotic systems. Based upon the parameters modulation and the adaptive control techniques, we show that dynamical evolution of third order chaotic system can be anti-synchronized with the projection of a fourth-order chaotic system even though their parameters are unknown. The techniques are successfully applied to several examples: hyperchaotic Chen system (fourth-order) and Lü system (third-order), theoretical analysis and numerical simulations are shown to verify the results.

Adaptive reduced-order anti-synchronization of chaotic systems with fully unknown parameters

In this paper, we investigate the reduced-order anti-synchronization of uncertain chaotic systems. Based upon the parameters modulation and the adaptive control techniques, we show that dynamical evolution of third-order chaotic system can be anti-synchronized with the canonical projection of a fourth-order chaotic system even though their parameters are unknown. The techniques are successfully applied to two examples: hyperchaotic Lorenz system (fourth-order) and Lorenz system (third-order); Lü hyperchaotic system (fourth-order) and Chen system (third-order). Theoretical analysis and numerical simulations are shown to verify the results.

Generalized reduced-order synchronization of chaotic system based on fast slide mode

A new kind of generalized reduced-order synchronization of different chaotic systems is proposed in this paper. It is shown that dynamical evolution of third-order oscillator can be synchronized with the canonical projection of a fourth-order chaotic system generated through nonsingular states transformation from a cell neural net chaotic system. In this sense, it is said that generalized synchronization is achieved in reduced-order. The synchronization discussed here expands the scope of reduced-order synchronization studied in relevant literatures. In this way, we can achieve generalized reduced-order synchronization between many famous chaotic systems such as the second-order Duffing system and the third-order Lorenz system by designing a fast slide mode controller. Simulation results are provided to verify the operation of the designed synchronization.

An adaptive feedback control for chaos synchronization of nonlinear systems with different order

This paper deals with the chaotic synchronization of fourth-order system and second driven oscillators. Such a problem is related to the synchronization of strictly different chaotic systems. We show that the dynamical evolution of second-order driven oscillators can be synchronized with the projection on canonical planes of a fourth-order chaotic system. In this sense, it is said that the synchronization is achieved in reduced-order. Based on the Lyapunov approach, the adaptation law is determined to tune the controller gain vector in order to track a predetermined linearizing feedback control. An application to secure chaotic communication is also discussed. Numerical simulations are presented to demonstrate the efficiency of the proposed synchronization and secure communication schemes.

Reduced-order synchronization of uncertain chaotic systems via adaptive control

We consider the coupling of two uncertain dynamical systems with different orders using an adaptive feedback linearization controller to achieve reduced-order synchronization between the two systems. Reduced-order synchronization is the problem of synchronization of a slave system with projection of a master system. The synchronization scheme is an exponential linearizing-like controller and a state/uncertainty estimator. As an illustrative example, we show that the dynamical evolution of a second-order driven oscillator can be synchronized with the canonical projection of a fourth-order chaotic system. Simulation results indicated that the proposed control scheme can significantly improve the synchronousness performance. These promising results justify the usefulness of the proposed output feedback controller in the application of secure communication.

Stability analysis for the synchronization of chaotic systems with different order: application to secure communications

In this Letter, the synchronization of different order systems or reduced-order synchronization is studied. We show that dynamical evolution of second-order oscillators can be synchronized with the canonical projection of a fourth-order chaotic system. In this sense, it is said that synchronization is achieved in reduced order. The proposed strategy is an input–output control scheme which comprises an uncertainty estimator and an exponential controller. In this way, the reduced-order synchronization is attained at any prescribed speed in spite of master/slave mismatches. Simulation results are provided to verify the operation of the designed synchronization scheme. The proposed controller is implemented to secure communication.