Time-delay Chaotic Systems Research Articles

Color image encryption algorithm based on Mackey–Glass time-delay chaotic system and quantum random walk

To ensure the confidentiality and integrity of image data and prevent unauthorized data tampering and privacy leaks. This study proposes a new color image encryption scheme based on the Mackey–Glass time-delay chaotic system and quantum random walk. This approach fully leverages the unpredictability of quantum random walks to generate random values. It combines the differences in Hamming distance between the three RGB channels of color images to create a highly complex and random key. The overall image and the three independent RGB channels are arranged in ascending order using Logistic-tent chaotic mapping and the Mackey–Glass time-delay chaotic system to obfuscate the image data. The deformed fractional-order Lorenz chaotic system is introduced, integrated with DNA encoding and decoding technology, and XOR operations are performed to achieve encryption at the spatial and pixel levels, thereby increasing the complexity of decryption. Through extensive experimental research, this solution has demonstrated excellent results in tests such as adjacent pixel correlation, information entropy, and key sensitivity. It has an excellent ability to protect the privacy of images and provides a reliable guarantee for the security of image data.

A new image encryption based on hybrid heterogeneous time-delay chaotic systems

<abstract><p>Chaos theory has been widely utilized in password design, resulting in an encryption algorithm that exhibits strong security and high efficiency. However, rapid advancements in cryptanalysis technology have rendered single system generated sequences susceptible to tracking and simulation, compromising encryption algorithm security. To address this issue, we propose an image encryption algorithm based on hybrid heterogeneous time-delay chaotic systems. Our algorithm utilizes a collection of sequences generated by multiple heterogeneous time-delay chaotic systems, rather than sequences from a single chaotic system. Specifically, three sequences are randomly assigned to image pixel scrambling and diffusion operations. Furthermore, the time-delay chaotic system comprises multiple hyperchaotic systems with positive Lyapunov exponents, exhibiting a more complex dynamic behavior than non-delay chaotic systems. Our encryption algorithm is developed by a plurality of time-delay chaotic systems, thereby increasing the key space, enhancing security, and making the encrypted image more difficult to crack. Simulation experiment results verify that our algorithm exhibits superior encryption efficiency and security compared to other encryption algorithms.</p></abstract>

Interpretable Design of Reservoir Computing Networks Using Realization Theory.

The reservoir computing networks (RCNs) have been successfully employed as a tool in learning and complex decision-making tasks. Despite their efficiency and low training cost, practical applications of RCNs rely heavily on empirical design. In this article, we develop an algorithm to design RCNs using the realization theory of linear dynamical systems. In particular, we introduce the notion of α -stable realization and provide an efficient approach to prune the size of a linear RCN without deteriorating the training accuracy. Furthermore, we derive a necessary and sufficient condition on the irreducibility of the number of hidden nodes in linear RCNs based on the concepts of controllability and observability from systems theory. Leveraging the linear RCN design, we provide a tractable procedure to realize RCNs with nonlinear activation functions. We present numerical experiments on forecasting time-delay systems and chaotic systems to validate the proposed RCN design methods and demonstrate their efficacy.

Adaptive Neural Network Synchronization Control for Uncertain Fractional-Order Time-Delay Chaotic Systems

We propose an adaptive radial basis (RBF) neural network controller based on Lyapunov stability theory for uncertain fractional-order time-delay chaotic systems (FOTDCSs) with different time delays. The controller does not depend on the system model and can achieve synchronous control under the condition that nonlinear uncertainties and external disturbances are completely unknown. Stability analysis showed that the error system asymptotically tended to zero in combination with the relevant lemma. Numerical simulation results show the effectiveness of the controller.

Matrix projective synchronization between time delay chaotic systems with disturbances and nonlinearity

Matrix projective synchronization between time delay chaotic systems with disturbances and nonlinearity

The Synchronization of a Class of Time-Delayed Chaotic Systems Using Sliding Mode Control Based on a Fractional-Order Nonlinear PID Sliding Surface and Its Application in Secure Communication

A novel approach for the synchronization of a class of chaotic systems with uncertainty, unknown time delays, and external disturbances is presented. The control method given here is expressed by combining sliding mode control approaches with adaptive rules. A sliding surface of fractional order has been developed to construct the control strategy of the abovementioned sliding mode by employing the structure of nonlinear fractional PID (NLPID) controllers. The suggested control mechanism using Lyapunov’s theorem developed robust adaptive rules in such a way that the estimation error of the system’s unknown parameters and time delays tends to be zero. Furthermore, the proposed robust control approach’s stability has been demonstrated using Lyapunov stability criteria and Lipschitz conditions. Then, in order to assess the performance of the proposed mechanism, the presented control approach was used to simulate the synchronization of two chaotic jerk systems with uncertainty, unknown time delays, and external distortion. The results of the simulation confirm the robust and desirable synchronization performance. Finally, a secure communications mechanism based on the proposed technique is shown as a practical implementation of the introduced control strategy, in which the message signal is disguised in the transmitter with high security and well recovered in the receiver with high quality, according to the mean squared error (MES) criteria.

OBSERVER-BASED SYNCHRONIZATION OF TIME-DELAY COMPLEX-VARIABLE CHAOTIC SYSTEMS WITH COMPLEX PARAMETERS

Time-delay and unavailability of system states bring extra difficulties to synchronization of chaotic systems. This paper investigates observer-based synchronization of time-delay complex-variable chaotic systems (CVCSs) with complex parameters. Differing from the existing works that considered time delay only in the linear term of some special CVCSs, the two cases of time delay in and out of nonlinear terms in general CVCSs are considered here, respectively. In addition, for the case that system states are not available for control, an observer-based output feedback control approach is developed to estimate the system states and guarantee the time-delay CVCSs to achieve synchronization. Moreover, by combining Lyapunov–Krasovskii function with linear matrix inequality in the complex field, two sufficient criteria are derived to ensure complete synchronization of time-delay CVCSs. Finally, a numerical example is presented to illustrate the theoretical results.

Synchronization Analysis of a New Four-Dimensional Time-Delay Lorenz System and Its Circuit Experiments

Time-delay chaotic systems with multiple positive Lyapunov exponents have been extensively studied in the field of information security. This paper proposes a new four-dimensional time-delay Lorenz system and its chaotic synchronization through the Lyapunov–Krasovskii theory. The sufficient conditions for the stability of the new chaotic system are obtained by the Routh–Hurwitz criterion, and the control parameters are found to have a significant impact on the speed of synchronous convergence. Furthermore, oscillation circuit simulation is essential for contributing the chaotic system to practical applications. Accordingly, the circuits of the chaotic time-delay system and its coupled synchronous control circuit are innovatively designed by Multisim. Experiment results illustrate the behaviors of various attractors in the new time-delay Lorenz system and the effectiveness of the proposed asymptotic synchronous method.

Secure Spread Spectrum Communication Using Super-Orthogonal Optical Chaos Signals

This paper proposes an analog spread spectrum secure communication system by designing a super-orthogonal two-dimensional electro-optic time delayed chaotic system. Two negatively correlated (called “super-orthogonal”) broadband optical chaotic signals can be generated. According to the binary symbol to be transmitted, the two signals are coupled into channel in turn. Benefiting from the wide spectrum and super-orthogonal characteristics of the two optical chaotic signals, the scheme shows excellent anti-noise performance which is verified by numerical simulations. Moreover, by introducing the modulation signal time delay feedback, the security of the system can be improved. The system is capable of resisting time delay signature extraction and return map attack, which are inherent vulnerabilities of time-delayed chaotic system and chaotic shift keying mechanism.

Adaptive Memoryless Sliding Mode Control of Uncertain Rössler Systems with Unknown Time Delays

In this paper, by adopting sliding mode control, an adaptive memoryless control scheme has been developed for uncertain Rössler chaotic systems with unknown time delays. Firstly, the proposed adaptive control can force the trajectories of controlled Rössler time-delayed chaotic systems into the specified sliding manifold. Then, the Riemann sum is introduced to analyze the stability of the equivalent dynamics in the sliding manifold. The control performance can be predicted even if the controlled systems have unmatched uncertainties and unknown time delays, which have not been well addressed in the literature. Numerical simulations are included to demonstrate the feasibility of the proposed scheme.

Memristor-based time-delay hyperchaotic system with circuit simulation and image encryption

In present context, researchers have only found that memristor-based time-delay (MBTD) chaotic systems have rich dynamic behavior, but there are still many shortcomings in analyses and applications of MBTD chaotic systems. In this paper, relying on a memristor-based 4D chaotic system, by introducing delay parameters into this system, we not only find that the Lyapunov exponents (LEs) of the system change from one positive to multiple positive, i.e., the system changes from chaotic to hyperchaotic. And the coexistence of the system becomes more obvious. Although the circuit design of the delayed chaotic system is complex, the circuit implementation in Multisim is still given in this work. Ultimately, a novel digital image encryption algorithm based on this system is explained in details by introducing a hash algorithm. The proposed encryption algorithm, with strong robustness to external interference, good key sensitivity and good plaintext sensitivity, can handle images of different shapes, can effectively implement plaintext image encryption, and can resist differential attacks. All of the above further illustrate the advantages of the encryption algorithm and good randomness of the 4D MBTD hyperchaotic system.

A Novel Quasi-Periodically Forced System with Time-delay Exhibiting Megastability and Hyperchaos

Megastability and multistability in a time-delay chaotic system is a relatively recent topic of study. A novel quasi-periodically forced system with time-delay is addressed in this paper. Various layered attractors, as well as hyperchaos, coexist in the new system. The phase plot, bifurcation diagram, and Lyapunov spectrum are used to investigate the dynamical behaviour of the quasi-periodically forced system with time delay. The existence of a time-delay chaotic system of this kind is extremely unusual in the literature.

FPGA Realization of Two Different Fractional- Order Time-Delay Chaotic System With Predefined Synchronization Time

FPGA Realization of Two Different Fractional- Order Time-Delay Chaotic System With Predefined Synchronization Time

Fixed-Time Adaptive Time-Varying Matrix Projective Synchronization of Time-Delayed Chaotic Systems with Different Dimensions

This paper deals with the fixed-time adaptive time-varying matrix projective synchronization (ATVMPS) of different dimensional chaotic systems (DDCSs) with time delays and unknown parameters. Firstly, to estimate the unknown parameters, adaptive parameter updated laws are designed. Secondly, to realize the fixed-time ATVMPS of the time-delayed DDCSs, an adaptive delay-unrelated controller is designed, where time delays of chaotic systems are known or unknown. Thirdly, some simple fixed-time ATVMPS criteria are deduced, and the rigorous proof is provided by employing the inequality technique and Lyapunov theory. Furthermore, the settling time of fixed-time synchronization (Fix-TS) is obtained, which depends only on controller parameters and system parameters and is independent of the system’s initial states. Finally, simulation examples are presented to validate the theoretical analysis.

Results for Chaos Synchronization with New Multi-Fractional Order of Neural Networks by Multi-Time Delay

A new finding is proposed for multi-fractional order of neural networks by multi-time delay (MFNNMD) to obtain stable chaotic synchronization. Moreover, our new result proved that chaos synchronization of two MFNNMDs could occur with fixed parameters and initial conditions with the proposed control scheme called sliding mode control (SMC) based on the time-delay chaotic systems. In comparison, the fractional-order Lyapunov direct method (FLDM) is proposed and is implemented to SMC to maintain the systems’ sturdiness and assure the global convergence of the error dynamics. An extensive literature survey has been conducted, and we found that many researchers focus only on fractional order of neural networks (FNNs) without delay in different systems. Furthermore, the proposed method has been tested with different multi-fractional orders and time-delay values to find the most stable MFNNMD. Finally, numerical simulations are presented by taking two MFNNMDs as an example to confirm the effectiveness of our control scheme.

A New Synchronization Method for Time-Delay Fractional Complex Chaotic System and Its Application

This paper proposes a class of time-delay fractional complex Lu¨ system and utilizes the adomian decomposition algorithm to study the dynamics of the system. Firstly, the time chaotic attractor, coexistence attractor and parameter space are studied. The bifurcation diagram and complexity are used to analyze the dynamic characteristics of the system. Secondly, the definition of modified fractional projective difference function synchronization (MFPDFS) is introduced. The corresponding synchronous controller is designed to realize the MFPDFS of the time-delay fractional complex Lu¨ system. Thirdly, based on the background of wireless speech communication system (WSCs), the MFPDFS controller is used to realize the secure speech transmission. Finally, the effectiveness of the controller is verified by numerical simulation. The signal-noise ratio (SNR) analysis of speech transmission is given. The performance of secure communication is verified by numerical simulation.

Memristor-based time-delay chaotic system with hidden extreme multi-stability and pseudo-random sequence generator

Memristor-based time-delay chaotic system with hidden extreme multi-stability and pseudo-random sequence generator

Adaptive Modulus Hybrid Projective Combination Synchronization of Time-Delay Chaotic Systems with Uncertainty and Disturbance and its Application in Secure Communication

Adaptive Modulus Hybrid Projective Combination Synchronization of Time-Delay Chaotic Systems with Uncertainty and Disturbance and its Application in Secure Communication

Delay-range-dependent ℋ∞ synchronization approaches for time-delay chaotic systems

The purpose of this paper is to investigate the problem of controller design for synchronization of the chaotic drive and response systems accompanied with bounded time delays limited by delay rates, uncertain parameters and admissible destabilization. An adequate condition is obtained employing the linear matrix inequality and a neoteric Lyapunov–Krasovskii functional to guarantee the global exponential synchronization of the studied chaotic systems. A control law is applied which would be added to the studied systems such that we can ensure the synchronization of the objective systems. The approach is proposed to solve the problem of the asymptotic synchronization of the studied systems. Numerical examples are illustrated to prove the feasibility and effectiveness of the current approach in this paper.

Adaptive Predefined-Time Synchronization of Two Different Fractional-Order Chaotic Systems With Time-Delay

This paper devotes to the adaptive globally synchronization within predefined-time of two time-delayed fractional-order chaotic systems. Firstly, through fractional calculus, two novel different fractional-order systems with time-delay are proposed, whose convergence is guaranteed and phase trajectory is given. Secondly, by exploiting the non-negative Lyapunov function and inequality theorem, a novel global predefined-time stability theorem is proposed, which can ensure the settling time tunable. And the upper bound of the settling time estimation is more accurate compared with the classical results. With the help of novel predefined-time stability theorem, two active controllers are designed, namely the fixed-time synchronization controller and predefined-time synchronization controller, to achieve the fixed-time synchronization and the predefined-time synchronization of two different time-delayed fractional-order chaotic systems respectively. Finally, several numerical simulations are presented in order to show the effectiveness of the proposed methods.