Function Projective Synchronization between Two Discrete-Time Hyperchaotic Systems Using Backstepping Method

We realize the function projective synchronization (FPS) between two discrete-time hyperchaotic systems, that is, the drive state vectors and the response state vectors can evolve in a proportional scaling function matrix. In this paper, a systematic scheme is explored to investigate the function projective synchronization of two identical discrete-time hyperchaotic systems using the backstepping method. Additionally, FPS of two different hyperchaotic systems is also realized. Numeric simulations are given to verify the effectiveness of our scheme.

Modified Cascade Synchronization Scheme for Discrete-Time Hyperchaotic Systems

In this paper,the modified cascade synchronization scheme is proposed to investigate the synchronizationin discrete-time hyperchaotic systems.By choosing a general kind of proportional scaling error functions and based onrigorous control theory,we take the discrete-time hyperchaotic system due to Wang and 3D generalized Hénon map astwo examples to achieve the modified cascade synchronization,respectively.Numerical simulations are used to verifythe effectiveness of the proposed technique.

Multiple mixed state variable incremental integration for reconstructing extreme multistability in a novel memristive hyperchaotic jerk system with multiple cubic nonlinearity

Memristor-based chaotic systems with infinite equilibria are interesting because they generate extreme multistability.Their initial state-dependent dynamics can be explained in a reduced-dimension model by converting the incremental integration of the state variables into system parameters.However,this approach cannot solve memristive systems in the presence of nonlinear terms other than the memristor term.In addition,the converted state variables may suffer from a degree of divergence.To allow simpler mechanistic analysis and physical implementation of extreme multistability phenomena,this paper uses a multiple mixed state variable incremental integration(MMSVII)method,which successfully reconstructs a four-dimensional hyperchaotic jerk system with multiple cubic nonlinearities except for the memristor term in a three-dimensional model using a clever linear state variable mapping that eliminates the divergence of the state variables.Finally,the simulation circuit of the reduced-dimension system is constructed using Multisim simulation software and the simulation results are consistent with the MATLAB numerical simulation results.The results show that the method of MMSVII proposed in this paper is useful for analyzing extreme multistable systems with multiple higher-order nonlinear terms.

Color Image Compression and Encryption Algorithm Based on 2D Compressed Sensing and Hyperchaotic System

With the advent of the information security era,it is necessary to guarantee the privacy,accuracy,and dependable transfer of pictures.This study presents a new approach to the encryption and compression of color images.It is predicated on 2D compressed sensing(CS)and the hyperchaotic system.First,an optimized Arnold scrambling algorithm is applied to the initial color images to ensure strong security.Then,the processed images are con-currently encrypted and compressed using 2D CS.Among them,chaotic sequences replace traditional random measurement matrices to increase the system’s security.Third,the processed images are re-encrypted using a combination of permutation and diffusion algorithms.In addition,the 2D projected gradient with an embedding decryption(2DPG-ED)algorithm is used to reconstruct images.Compared with the traditional reconstruction algorithm,the 2DPG-ED algorithm can improve security and reduce computational complexity.Furthermore,it has better robustness.The experimental outcome and the performance analysis indicate that this algorithm can withstand malicious attacks and prove the method is effective.

Characteristic analysis of 5D symmetric Hamiltonian conservative hyperchaotic system with hidden multiple stability

Conservative chaotic systems have unique advantages over dissipative chaotic systems in the fields of secure communication and pseudo-random number generator because they do not have attractors but possess good traversal and pseudorandomness. In this work, a novel five-dimensional(5D) Hamiltonian conservative hyperchaotic system is proposed based on the 5D Euler equation. The proposed system can have different types of coordinate transformations and time reversal symmetries. In this work, Hamilton energy and Casimir energy are analyzed firstly, and it is proved that the new system satisfies Hamilton energy conservation and can generate chaos. Then, the complex dynamic characteristics of the system are demonstrated and the conservatism and chaos characteristics of the system are verified through the correlation analysis methods such as phase diagram, equilibrium point, Lyapunov exponent, bifurcation diagram, and SE complexity. In addition, a detailed analysis of the multistable characteristics of the system reveals that many energy-related coexisting orbits exist. Based on the infinite number of center-type and saddle-type equilibrium points, the dynamic characteristics of the hidden multistability of the system are revealed. Then, the National Institute of Standards and Technology(NIST)test of the new system shows that the chaotic sequence generated by the system has strong pseudo-random. Finally, the circuit simulation and hardware circuit experiment of the system are carried out with Multisim simulation software and digital signal processor(DSP) respectively. The experimental results confirm that the new system has good ergodicity and realizability.

A color image encryption algorithm based on hyperchaotic map and DNA mutation

We devise a color image encryption scheme via combining hyperchaotic map,cross-plane operation and gene theory.First,the hyperchaotic map used in the encryption scheme is analyzed and studied.On the basis of the dynamics of hyperchaotic map,a color image encryption scheme is designed.At the end of the encryption process,a DNA mutation operation is used to increase the encoding images’randomness and to improve the encryption algorithm’s security.Finally,simulation experiments,performance analysis,and attack tests are performed to prove the effectiveness and security of the designed algorithm.This work provides the possibility of applying chaos theory and gene theory in image encryption.

Deterministic and Stochastic Fixed-Time Stability of Discrete-time Autonomous Systems

This paper studies deterministic and stochastic fixedtime stability of autonomous nonlinear discrete-time(DT)systems.Lyapunov conditions are first presented under which the fixed-time stability of deterministic DT systems is certified.Extensions to systems under deterministic perturbations as well as stochastic noise are then considered.For the former,sensitivity to perturbations for fixed-time stable DT systems is analyzed,and it is shown that fixed-time attractiveness results from the presented Lyapunov conditions.For the latter,sufficient Lyapunov conditions for fixed-time stability in probability of nonlinear stochastic DT systems are presented.The fixed upper bound of the settling-time function is derived for both fixed-time stable and fixed-time attractive systems,and a stochastic settling-time function fixed upper bound is derived for stochastic DT systems.Illustrative examples are given along with simulation results to verify the introduced results.

Optical image encryption algorithm based on a new four-dimensional memristive hyperchaotic system and compressed sensing

Some existing image encryption schemes use simple low-dimensional chaotic systems, which makes the algorithms insecure and vulnerable to brute force attacks and cracking. Some algorithms have issues such as weak correlation with plaintext images, poor image reconstruction quality, and low efficiency in transmission and storage. To solve these issues,this paper proposes an optical image encryption algorithm based on a new four-dimensional memristive hyperchaotic system(4D MHS) and compressed sensing(CS). Firstly, this paper proposes a new 4D MHS, which has larger key space, richer dynamic behavior, and more complex hyperchaotic characteristics. The introduction of CS can reduce the image size and the transmission burden of hardware devices. The introduction of double random phase encoding(DRPE) enables this algorithm has the ability of parallel data processing and multi-dimensional coding space, and the hyperchaotic characteristics of 4D MHS make up for the nonlinear deficiency of DRPE. Secondly, a construction method of the deterministic chaotic measurement matrix(DCMM) is proposed. Using DCMM can not only save a lot of transmission bandwidth and storage space, but also ensure good quality of reconstructed images. Thirdly, the confusion method and diffusion method proposed are related to plaintext images, which require both four hyperchaotic sequences of 4D MHS and row and column keys based on plaintext images. The generation process of hyperchaotic sequences is closely related to the hash value of plaintext images. Therefore, this algorithm has high sensitivity to plaintext images. The experimental testing and comparative analysis results show that proposed algorithm has good security and effectiveness.

Cryptanalysis of 2D-SCMCI Hyperchaotic Map Based Image Encryption Algorithm

Chaos-based cryptosystems are considered a secure mode of communication due to their reliability.Chaotic maps are associated with the other domains to construct robust encryption algorithms.There exist numerous encryption schemes in the literature based on chaotic maps.This work aims to propose an attack on a recently proposed hyper-chaotic map-based cryptosystem.The core notion of the original algorithm was based on permutation and diffusion.A bitlevel permutation approach was used to do the permutation row-and column-wise.The diffusion was executed in the forward and backward directions.The statistical strength of the cryptosystem has been demonstrated by extensive testing conducted by the author of the cryptosystem.This cryptanalysis article investigates the robustness of this cryptosystem against a chosen-plaintext attack.The secret keys of the cryptosystem were retrieved by the proposed attack with 258 chosen-plain images.The results in this manuscript suggest that,in addition to standard statistical evaluations,thorough cryptanalysis of each newly suggested cryptosystem is necessary before it can be used in practical application.Moreover,the data retrieved is also passed through some statistical analysis to compare the quality of the original and retrieved data.The results of the performance analysis indicate the exact recovery of the original data.To make the cryptosystem useful for applications requiring secure data exchange,a few further improvement recommendations are also suggested.

Local Dynamics Analysis of a Four-Dimensional Hyperchaotic Lorenz System

The local dynamical behaviors of a four-dimensional hyperchaotic Lorenz system, including stability and bifurcations, are investigated in this paper by analytical and numerical methods. The equilibriums and their stability under different parameter conditions are analyzed by applying Routh-Hurwitz criterion. The results indicate that the system may exist one, three and five equilibrium points for different system parameters. Based on the central manifold theorem and normal form theorem, the pitchfork bifurcation and Hopf bifurcation are studied respectively. By using the Hopf bifurcation theorem and calculating the first Lyapunov coefficient, the Hopf bifurcation of this system is obtained as supercritical for certain parameters. Finally, the results of theoretical parts are verified by some numerical simulations.

Event-Triggered Finite-Time H∞ Filtering for Discrete-Time Nonlinear Stochastic Systems

This paper addresses the problem of event-triggered finite-time H∞ filter design for a class of discrete-time nonlinear stochastic systems with exogenous disturbances. The stochastic Lyapunov-Krasoviskii functional method is adopted to design a filter such that the filtering error system is stochastic finite-time stable (SFTS) and preserves a prescribed performance level according to the pre-defined event-triggered criteria. Based on stochastic differential equations theory, some sufficient conditions for the existence of H∞ filter are obtained for the suggested system by employing linear matrix inequality technique. Finally, the desired H∞ filter gain matrices can be expressed in an explicit form.

The generation of a hyperchaotic system based on a three-dimensional autonomous chaotic system

This paper reports a new four-dimensional hyperchaotic system obtained by adding a controller to a threedimensional autonomous chaotic system. The new system has two parameters, and each equation of the system has one quadratic cross-product term. Some basic properties of the new system are analysed. The different dynamic behaviours of the new system are studied when the system parameter a or b is varied. The system is hyperchaotic in several different regions of the parameter b. Especially, the two positive Lyapunov exponents are both larger, and the hyperchaotic region is also larger when this system is hyperchaotic in the case of varying a. The hyperchaotic system is analysed by Lyapunov-exponents spectrum, bifurcation diagrams and Poincaré sections.

A new image encryption algorithm based on the fractional-order hyperchaotic Lorenz system

We propose a new image encryption algorithm on the basis of the fractional-order hyperchaotic Lorenz system. While in the process of generating a key stream, the system parameters and the derivative order are embedded in the proposed algorithm to enhance the security. Such an algorithm is detailed in terms of security analyses, including correlation analysis, information entropy analysis, run statistic analysis, mean-variance gray value analysis, and key sensitivity analysis. The experimental results demonstrate that the proposed image encryption scheme has the advantages of large key space and high security for practical image encryption.

Generalized projective synchronization of a class of chaotic （hyperchaotic） systems with uncertain parameters

In this paper is investigated the generalized projective synchronization of a class of chaotic （or hyperchaotic） systems, in which certain parameters can be separated from uncertain parameters. Based on the adaptive technique, the globally generalized projective synchronization of two identical chaotic （hyperchaotic） systems is achieved by designing a novel nonlinear controller. Furthermore, the parameter identification is realized simultaneously. A sufficient condition for the globally projective synchronization is obtained. Finally, by taking the hyperchaotic L system as example, some numerical simulations are provided to demonstrate the effectiveness and feasibility of the proposed technique.

Adaptive synchronization of hyperchaotic Lü system with uncertainty

This paper presents a novel adaptive control scheme for synchronization of the latest hyperchaotic Lü system. Based on the Lyapunov stability theory, a feedback controller and a parameter update law are designed for the synchronization of hyperchaotic Lfi systems with uncertainty. Numerical simulations are given to demonstrate the validity of the synchronization technique.

A new four-dimensional hyperchaotic Chen system and its generalized synchronization

Based on the Chen chaotic system, a new four-dimensional hyperchaotic Chen system is constructed, and the basic dynamic behaviours of the system were studied, and the generalized synchronization has been observed in the coupled four-dimensional hyperchaotic Chen system with unknown parameters. The Routh Hurwitz theorem is used to derive the conditions of stability of this system. Furthermore based on Lyapunov stability theory, the control laws and adaptive laws of parameters are obtained to make generalized synchronization of the coupled new four-dimensional hyperchaotic Chen systems. Numerical simulation results are presented to illustrate the effectiveness of this method.

A novel fractional-order hyperchaotic system and its synchronization

In this paper, a novel hyperchaotic system with one nonlinear term and its fractional order system are proposed. Furthermore,synchronization between two fractional-order systems with different fractional-order values is achieved. The proposed synchronization scheme is simple and theoretically rigorous.Numerical simulations are in agreement with the theoretical analysis.

A unified approach to fuzzy modelling and robust synchronization of different hyperchaotic systems

In this paper, a Takagi Sugeno （T-S） fuzzy model-based method is proposed to deal with the problem of synchronization of two identical or different hyperchaotic systems. The T S fuzzy models with a small number of fuzzy IF-THEN rules are employed to represent many typical hyperchaotic systems exactly. The benefit of employing the T-S fuzzy models lies in mathematical simplicity of analysis. Based on the T-S fuzzy hyperchaotic models, two fuzzy controllers arc designed via parallel distributed compensation （PDC） and exact linearization （EL） techniques to synchronize two identical hyperchaotic systems with uncertain parameters and two different hyperchaotic systems, respectively. The sufficient conditions for the robust synchronization of two identical hyperchaotic systems with uncertain parameters and the asymptotic synchronization of two different hyperchaotic systems are derived by applying the Lyapunov stability theory. This method is a universal one of synchronizing two identical or different hyperchaotic systems. Numerical examples are given to demonstrate the validity of the proposed fuzzy model and hyperchaotic synchronization scheme.

A novel color image encryption algorithm based on genetic recombination and the four-dimensional memristive hyperchaotic system

Recently, many image encryption algorithms based on chaos have been proposed. Most of the previous algorithms encrypt components R, G, and B of color images independently and neglect the high correlation between them. In the paper, a novel color image encryption algorithm is introduced. The 24 bit planes of components R, G, and B of the color plain image are obtained and recombined into 4 compound bit planes, and this can make the three components affect each other. A four-dimensional(4D) memristive hyperchaotic system generates the pseudorandom key streams and its initial values come from the SHA 256 hash value of the color plain image. The compound bit planes and key streams are confused according to the principles of genetic recombination, then confusion and diffusion as a union are applied to the bit planes,and the color cipher image is obtained. Experimental results and security analyses demonstrate that the proposed algorithm is secure and effective so that it may be adopted for secure communication.

Modified projective synchronization of a fractional-order hyperchaotic system with a single driving variable

In this paper, the modified projective synchronization （MPS） of a fractional-order hyperchaotic system is inves- tigated. We design the response system corresponding to the drive system on the basis of projective synchronization theory, and determine the sufficient condition for the synchronization of the drive system and the response system based on fractional-order stability theory. The MPS of a fractional-order hyperchaotic system is achieved by transmitting a single variable. This scheme reduces the information transmission in order to achieve the synchronization, and extends the applicable scope of MPS. Numerical simulations further demonstrate the feasibility and the effectiveness of the proposed scheme.