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At present, design and applications of discrete chaotic systems aroused much interests of scholars. A practical way to design new chaotic systems is based on existing classical systems. In this paper, based on the existing Hénon map, a class of new Hénon maps are proposed and investigated. Furthermore, the dynamic characteristics of a chosen map are analyzed by numerical simulations including phase diagrams, bifurcation diagrams and Lyapunov exponents. It shows that the system has rich dynamical behaviors. An analogue circuit of the new system is designed and its real hardware circuit is realized. The experiment demonstrates the succuss of the design. Finally, an application in the design of welded beams is carried out using the golden Jackal optimization algorithm which is improved based on the randomness and ergodicity of the generated chaos. The potential practical application based on the proposed system in the engineering field is demonstrated. [ABSTRACT FROM AUTHOR]

Robust block substitution techniques are essential to protect information against sophisticated cryptographic attacks. Existing techniques use static methods, mainly Sbox substitution, to increase nonlinearity. These static substitutions have significant weaknesses, however. Bit alterations impacts only the specific block where the bit alterations occur without extending their effects to subsequent blocks. Second, current block substitution techniques only leverage alterations in input symbols to boost confusion without considering other significant changes such as the positions of the altered symbols within the input block. To address the limitations, a more dynamic block substitution is presented. This new block substitution technique effectively monitors changes in input symbols and extends the impact of these changes to cover the entire ciphertext. The proposed method utilizes operations based on chaos, deep symbol mutation, and memory loss models to process the input blocks. We analyzed the performance of the technique using robust testing tools. Our experiments demonstrated that the proposed technique surpasses existing methods in nonlinearity and randomness metrics, achieving SAC (0.50190), BIC (111.67), NL (112.0), and entropy (7.99997). Furthermore, we integrated the proposed substitution technique into both text and image encryption techniques. The experimental study showed a substantial improvement in the performance of these techniques when using the proposed substitution technique. [ABSTRACT FROM AUTHOR]

Providing robust security within an image cryptosystem during network communication is more essential than ever, highlighting the fundamental aspects of confusion and diffusion. This study presents a novel approach for securely transmitting images through insecure channels, using array index manipulation for scrambling and simple pixel-level confusion and diffusion through XOR operations, effectively involving row and column permutation to achieve robust scrambling. This method operates two layers of confusion by rearranging array indexes using a Tent map and constructing an S-box algorithm derived from the Henon map, while the diffusion process uses a pseudo-random sequence generator based on the Henon map, with their chaotic dynamics analyzed through investigations of Lyapunov exponents and bifurcation diagrams. The expanded chaotic range and improved effects of chaotic maps produce new S-boxes with satisfactory cryptographic performance, including balancedness and non-linearity, with S-box3 achieving the highest non-linearity at 108. Following a performance and security assessment of the proposed technique, it has been found that the entropy value for the cipher image is nearly 7.998; also, the NPCR and UACI values for the cipher images are 99.6 and 33.4, respectively, and it demonstrates closeness to the ideal value. Finally, the proposed method demonstrates high randomness, undergoes evaluation using the NIST test suite, has good operation efficiency, and exhibits resilience against various attack forms such as statistical, differential, data-loss, and noise attacks, affirming its security and relevance for real-time cryptosystems. [ABSTRACT FROM AUTHOR]

The growing use of the internet results in significant difficulties for the copyright protection of images. To resolve the copyright problem, a scheme is proposed that utilizes the advantages of Zigzag scanning (ZS), Schur decomposition (SD), Discrete cosine transform (DCT) and Redundant discrete wavelet transform (RDWT). In the proposed scheme, Henon map is applied on the watermarked image for its secure transfer and further enhancement of the security. Medical image security assurance is the need of the hour. Therefore, ultrasound images of the liver of six different patients are taken for experimental result analysis. The use of Singular value decomposition (SVD) is avoided here due to its false positive problem. RDWT is used to split the input ultrasound liver image into different sub-bands. Then DCT is applied on the suitable sub-band to improve the robustness of the presented work. The ZS is used to avoid artifacts which may occur at higher value of the gain factor. SD and RDWT are applied to the watermark logo (MNNIT logo) and selective pixel adding is used to embed the watermark logo into the host image. The authentication process is done by using binary robust invariant scalable keypoints (BRISK) features. The presented scheme perform satisfactorily against various attacks such as Histogram equalization, Region of interest (ROI) filtering, Cropping, Scaling, Rotation, Average and Median filtering attacks, Salt and Pepper, Gaussian noise attack, Flipping, etc. The proposed scheme shows better performance as compared to existing schemes in terms of various performance parameters such as Peak signal-to-noise ratio (PSNR), Structural similarity index measurement (SSIM) and Normalized correlation coefficient (NCC). [ABSTRACT FROM AUTHOR]

This research paper focuses on investigating the dynamics and chaos control of two discrete systems corresponding to the singularly perturbed equation associated to the Hénon difference equation with continuous arguments. We investigate the local stability of the two systems. To achieve chaos control, we employ the state feedback control method for both systems. Numerical simulations are conducted to gain insights into the qualitative behavior of these systems. By conducting numerical simulations, we gain valuable insights into the role played by the perturbation parameter and the discrete delays in the bifurcation behavior of each system. Furthermore, we demonstrate the practical application of the chaos control method through the implementation of an on-off strategy.

In the last few years the literature has witnessed a remarkable surge of interest for maps implemented by discrete-time (DT) memristor circuits. One main reason is that from numerical simulations it appears that even for simple memristor circuits the maps can easily display complex dynamics, including chaos and hyperchaos, which are of relevant interest for engineering applications. Goal of this manuscript is to investigate on the reasons underlying this type of complex behavior. To this end, the manuscript considers the map implemented by the simplest memristor circuit given by a capacitor and an ideal flux-controlled memristor or an inductor and an ideal charge-controlled memristor. In particular, the manuscript uses the DT flux-charge analysis method (FCAM) introduced in a recent paper to ensure that the first integrals and foliation in invariant manifolds of continuous-time (CT) memristor circuits are preserved exactly in the discretization for any step size. DT-FCAM yields a two-dimensional map in the voltage-current domain (VCD) and a manifold-dependent one-dimensional map in the flux-charge domain (FCD), i.e., a one-dimensional map on each invariant manifold. One main result is that, for suitable choices of the circuit parameters and memristor nonlinearities, both DT circuits can exactly embed two classic chaotic maps, i.e., the logistic map and the tent map. Moreover, due to the property of extreme multistability, the DT circuits can simultaneously embed in the manifolds all the dynamics displayed by varying one parameter in the logistic and tent map. The paper then considers a DT memristor Murali-Lakshmanan-Chua circuit and its dual. Via DT-FCAM these circuits implement a three-dimensional map in the VCD and a two-dimensional map on each invariant manifold in the FCD. It is shown that both circuits can simultaneously embed in the manifolds all the dynamics displayed by two other classic chaotic maps, i.e., the Hénon map and the Lozi map, when varying one parameter in such maps. In essence, these results provide an explanation of why it is not surprising to observe complex dynamics even in simple DT memristor circuits.

With the rise of emerging technologies, the information security becomes particularly important. Digital images play an important role in network transmission. To prevent the image information about being stolen during network transmission, this paper established a stereo Zigzag transformation and proposed a new multiple-image encryption algorithm. Firstly, the original images make up a cube; secondly, Hénon map is used to randomly select the starting point of the two-dimensional Zigzag transformation; thirdly, the scrambled images can be obtained by the stereo Zigzag transformation; finally, a chaotic sequence is used to randomly diffuse the scrambled images to get the encrypted images. Experimental analyses indicate that the proposed algorithm has the characteristic of high key sensitivity, large key space, high efficiency and resistance to plaintext attacks, statistical attacks, etc. [ABSTRACT FROM AUTHOR]

This research paper focuses on investigating the dynamics and chaos control of two discrete systems corresponding to the singularly perturbed equation associated to the Hénon difference equation with continuous arguments. We investigate the local stability of the two systems. To achieve chaos control, we employ the state feedback control method for both systems. Numerical simulations are conducted to gain insights into the qualitative behavior of these systems. By conducting numerical simulations, we gain valuable insights into the role played by the perturbation parameter and the discrete delays in the bifurcation behavior of each system. Furthermore, we demonstrate the practical application of the chaos control method through the implementation of an on-off strategy. [ABSTRACT FROM AUTHOR]

The renowned 2D invertible Hénon map turns into 1D noninvertible quadratic map when its leading parameter b becomes zero. This well-known link was studied by Mira who demonstrated that the bifurcation set of Hénon diffeomorphism is similar to his "box-within-a-box" bifurcation structure of 1D endomorphism. In general, such similarity has not been strictly established, especially in multidimensional cases. In this paper, we proved that the Mira bifurcation structure of a quadratic noninvertible map persists when the parameter increases from zero and the map turns into an invertible multidimensional generalized Hénon map. The changes of periodic and homoclinic orbits and chaotic attractors at this transition are described. We proved the existence of Newhouse regions is different from those Mira boxes that accumulate to the homoclinic bifurcations. [ABSTRACT FROM AUTHOR]

The rapid evolution of network technologies, where computer vision and the Internet seamlessly merge, poses significant challenges in safeguarding end-user data. To address the growing need for secure data in smart environments, lightweight and efficient security solutions are in high demand. Notably, the Trivium, a well-known encrypt stream cipher (eSTREAM) project, has been the target of several attacks. This paper proposes an innovative approach for fast and highly secured image encryption using byte scrambling and a modified version of the Trivium algorithm. The Henon map is utilized to generate random numbers for permutation, reducing time and increasing security. Performance is measured on 100 colour images with different several tests. The results present a fast and robust security system for the transmission process. It is more efficient than the traditional Trivium method. The encrypted-decryption time is reduced by 1:24 times, making it a quick security system for surgical telepresence and multimedia visuals. [ABSTRACT FROM AUTHOR]

This article is devoted to the first steps of nine mathematicians from five countries on their path to mathematics, chaos and discrete dynamical systems, some from early childhood. In these life stories, the names of outstanding mathematicians arise, crisscrossing the nine stories in unexpected ways. These mathematicians also interacted with each other, forming an intriguing social network worldwide, across all borders of nationality and languages. [ABSTRACT FROM AUTHOR]

Most of the dynamics displayed by highly complicated nonlinear systems also appear in simple nonlinear systems. The purpose of the two-dimensional map with a strange attractor was for it to be a simple mapping that possesses similar properties to the Lorenz system and its Poincare map. Henon map is investigated, periodic points are found, and chaotic attractors are produced. In this paper, we will demonstrate an orbit of the Henon map with 10000 points, which vary on the initial conditions of the orbit and the values of the two parameters of the system. Known that the chaotic attractors in the Henon map are neither area filling (dimension 2) nor a simple curve (of dimension 1), the dimensions of these complicated geometries must be non-integer values between 1 and 2, and the chaotic attractors are then called fractals or strange attractors. Thee capacity or box-counting dimension dbox is the simplest possible way to measure such pathologies. We use The OGY (Ott, Grebogi, and Yorke) Method with the idea to make small time-dependent linear perturbations to the control parameter p in order to nudge the state towards the stable manifold of the desired fixed point. Acting on 500 equally spaced initial points (x0,y0) on a circle or a square, represent an numerical experiment which may give us some hints about why Jupiter's red spot and Saturn's hexagon-shaped hurricane seem to exist forever without contracting. [ABSTRACT FROM AUTHOR]

Effectively identifying and characterizing the various dynamics present in complex and chaotic systems is fundamental for chaos control, chaos classification, and behavior-transition forecasting, among others. It is a complicated task that becomes increasingly difficult as systems involve more dimensions and parameters. Here, we extend methods inspired in ordinal patterns to analyze 2D iterative maps to unveil underlying approximate symmetries of their dynamics. We distinguish different families of chaos within the systems, find similarities among chaotic maps, identify approximate temporal and dynamical symmetries, and anticipate sharp transitions in dynamics. We show how this methodology displays the evolution of the spatial correlations in a dynamical system as the control parameter varies. We prove the power of these techniques, which involve simple quantifiers as well as combinations of them, in extracting relevant information from the complex dynamics of 2D systems, where other techniques are less informative or more computationally demanding.

Effectively identifying and characterizing the various dynamics present in complex and chaotic systems is fundamental for chaos control, chaos classification, and behavior-transition forecasting, among others. It is a complicated task that becomes increasingly difficult as systems involve more dimensions and parameters. Here, we extend methods inspired in ordinal patterns to analyze 2D iterative maps to unveil underlying approximate symmetries of their dynamics. We distinguish different families of chaos within the systems, find similarities among chaotic maps, identify approximate temporal and dynamical symmetries, and anticipate sharp transitions in dynamics. We show how this methodology displays the evolution of the spatial correlations in a dynamical system as the control parameter varies. We prove the power of these techniques, which involve simple quantifiers as well as combinations of them, in extracting relevant information from the complex dynamics of 2D systems, where other techniques are less informative or more computationally demanding. [ABSTRACT FROM AUTHOR]

The conventional PV array configurations fail to eliminate the ill effects of partial shading (PS) thereby resulting in mismatch losses. PV array reconfiguration is considered as the most competent solution for shading-related issues. However, many reconfiguration techniques reported in the literature fail to disperse the shade effectively, not compatible with large-scale arrays, and are not generalized for all array sizes. To overcome these limitations, a highly efficient generalized Henon Map technique which is extensively used in the image encryption process is proposed in this paper to reconfigure the PV array. The proposed technique is extensively analyzed for 9 × 9, 8 × 8, 4 × 4, 3 × 5 and 4 × 3 PV arrays under distinct shading patterns with seven parameters. The obtained results are compared with conventional Series-Parallel, Total-Cross-Tied, and very recently reported Odd-Even, Odd-Even-Prime, Chaotic Baker's Map, Magic square, and Diagonal-TCT reconfiguration techniques. Besides, an experimental prototype model is developed and tested under artificially shading conditions to validate the competence of the proposed configuration. From the comprehensive analysis, it is noted that the proposed technique offers consistently superior performance with the least percentage mismatch and maximum power enhancement of 24.17%, 25.65%, and 47.81% for various shading cases of 9 × 9, 8 × 8, and 4 × 4 PV arrays, respectively. Further, a qualitative comparative assessment of the proposed technique with other state-of-art techniques is presented. [ABSTRACT FROM AUTHOR]

Amidst the growth of the internet and communication technologies, the requirements for the security of data transmitted via these technologies are increasing. Visual data, like images and videos, are commonly utilized for representing the majority of data due to its having more detailed information. Until now, the physical implementation of quantum computers does not have enough capability for hacking any traditional image cryptosystem, but amidst the growth of quantum resources, enough capability may be available in the near future. Most data represented by images has a long lifetime, like personal, medical, military, etc. Therefore, new quantum-inspired-based designs for image cryptosystems are required to be performed on digital resources and have the capability of defying the potential attacks from digital and quantum resources. In this study, a new substitution box (S-box) mechanism is proposed, which is based on quantum-inspired quantum walks, Hénon map, and a customized particle swarm optimization algorithm. Performance analysis of the suggested S-box proves its effectiveness and its reliability in designing various cryptosystems. Based on the effectiveness of the presented S-box, a new image cryptosystem is proposed, in which its experiential outcomes prove its efficacy and security against various attacks. The average outcome of entropy is 7.99977, UACI is 33.484%, NPCR is 99.618%, and Chi-square is 249.481 for the constructed cipher images. [ABSTRACT FROM AUTHOR]

The Hénon map was introduced by Hénon. It is a very rich dynamical model as Hénon himself proved and compared the dynamical properties of his map with those of other dynamical systems such as the Ro¨ssler model. Here, we study the dynamics of the difference equation with continuous arguments corresponding to the Hénon map and its singularly perturbed counterpart. The local stability of fixed points is studied. The system exhibits various types of bifurcation, such as saddle-node and Hopf bifurcations. By letting the perturbation parameter ∊⟶0, we show that the singularly perturbed equation exhibits the same qualitative behavior as its corresponding difference equation. The singularly perturbed equation exhibits the same qualitative behavior as its corresponding delay differential equation when ∊⟶1. The method of steps is used to discretize the original system to simulate the behavior of the original system. Numerical simulations are performed to confirm the theoretical analysis obtained and to illustrate the complex dynamics of the system. Moreover, the numerical simulations illustrate the effect of the perturbation parameter. In particular, the occurrence of halving-period bifurcation due to small change in the perturbation parameter.

In this paper, a new chaotic image encryption and authentication model based on Elliptic Curves is proposed. The Elliptic Curve Diffie-Hellman (ECDH) is used to generate a reliable session key prior to encryption. The scheme uses the Henon map for the initial shuffling of the pixels and the Hopfield chaotic neural network to compute random chaotic values. The chaotic matrix is XORed with henon shuffled image. The model encrypts the scrambled image using improved ElGamal encoding to obtain the cipher image. The model employs a low computational cost digital signature to check for the authenticity of the received encrypted images before decryption. The proposed model makes use of a large key space to resist brute force attacks, and produces randomized cipher images with high average Shannon's entropies, 7.9994 for grayscale and 7.9993 for color images, and lower adjacent pixel correlation to resist statistical attacks and ensure a good quality encryption. The model can withstand the chosen-plaintext and known-plaintext attacks, and attains average Number of Pixel Change Rate (NPCR) and Unified Average Change Intensity (UACI) values of (99.63%, 33.345%) for grayscale and (99.625%, 33.34%) for color images signifying its effectiveness against differential attacks. The ability to recover decipherable images after masking up to 75% of cipher image indicates the robustness of the proposed scheme against the occlusion attacks. [ABSTRACT FROM AUTHOR]

This foreword presents the 32 articles of the special issue devoted to the Lozi, Hénon and other chaotic attractors proposed by 78 authors from 20 countries (Algeria, Brazil, Canada, China, Czech Republic, France, Germany, India, Italy, Jamaica, Japan, Lebanon, New Zealand, Portugal, Russia, Saudi Arabia, Spain, Ukraine, Tunisia, USA). These original research papers consider chaotic attractors, both in their theoretical and applied aspects. [ABSTRACT FROM AUTHOR]

We study numerically the spatio-temporal dynamics of ring networks of coupled discrete-time systems in the presence of additive noise. The robustness of chimera states with respect to noise perturbations is explored for two ensembles in which the individual elements are described by either logistic maps or Henon maps in the chaotic regime. The influence of noise on the behavior of solitary states is investigated for a ring of nonlocally coupled Lozi maps. Numerical simulations are performed for a set of different noise realizations and random initial conditions to provide reliable statistical data. The type of dynamics of the considered networks is quantified by using a cross-correlation coefficient. We find that there is a finite and sufficiently wide region with respect to the coupling strength and the noise intensity where the probability of observing the chimeras and solitary states is high. [ABSTRACT FROM AUTHOR]

We explore stabilization for nonlinear systems of difference equations with modified Target-Oriented Control and a chosen equilibrium as a target, both in deterministic and stochastic settings. The influence of stochastic components in the control parameters is explored. The results are tested on the Hénon and the Lozi maps. [ABSTRACT FROM AUTHOR]

Lightweight cryptographic algorithms must develop to ensure the confidentiality and integrity of the data in resource-constrained devices. Keys are vital to every cryptography algorithm because they provide randomness, complexity, unexpected nature, and robustness. A light encryption device (LED) is considered a lighter version of advanced encryption standard (AES), but it is vulnerable to related key attacks due to using the same key during the whole encryption process. This paper presents a hybrid chaotic key generator (HCKG) based on 3D Lorenz, and 2D Henon maps to generate a highly randomized key that combines with the LED to provide a high level of secure encryption on resource-constrained devices. We modified the HCKG every four rounds via simple operations to get the subkeys and XORed it with the state to increase the complexity of the ciphertext. Moreover, the HCKG with subkeys allows us to decrease the total number of LED rounds from 32 to 24 to minimize the calculation cost while maintaining a high level of security. National Institute of Science and Technology (NIST) test suite proves that the proposed LED-HCKG demonstrates a high-performance increase by nearly 0.3283 higher than LED concerning data integrity and secrecy. [ABSTRACT FROM AUTHOR]

The Henon map forms one of the most-studied two-dimensional discrete-time dynamical systems that exhibits chaotic behavior. The Henon map takes a point (Xn,Yn) in the plane andmaps it to a new point (Xn+1,Yn+1). In this paper, a chaotic pulse generator based on the chaotic Henon map is proposed. It consists of aHenon map function subcircuit to realize the Henon map and another subcircuit to perform the iterative operation. The Henon map subcircuit comprises operational amplifiers, multipliers, delay elements and resistors, whereas, the iterative subcircuit is implemented with a simple design that comprises of an edge forming circuit followed by a monostable multivibrator and a voltage controlled switch without the use of any clock control. The proposed design can be used to realize the Henon map and also to generate a chaotic pulse train, with a controllable time interval and pulse position. The proposed circuit is implemented and simulated using Multisim 13.0 and MATLAB R2019b. The chaotic nature of the generated pulse train and also the time interval between the consecutive pulses is verified by the calculation of its Lyapunov exponents. [ABSTRACT FROM AUTHOR]

Abstract Amidst the growth of the internet and communication technologies, the requirements for the security of data transmitted via these technologies are increasing. Visual data, like images and videos, are commonly utilized for representing the majority of data due to its having more detailed information. Until now, the physical implementation of quantum computers does not have enough capability for hacking any traditional image cryptosystem, but amidst the growth of quantum resources, enough capability may be available in the near future. Most data represented by images has a long lifetime, like personal, medical, military, etc. Therefore, new quantum-inspired-based designs for image cryptosystems are required to be performed on digital resources and have the capability of defying the potential attacks from digital and quantum resources. In this study, a new substitution box (S-box) mechanism is proposed, which is based on quantum-inspired quantum walks, Hénon map, and a customized particle swarm optimization algorithm. Performance analysis of the suggested S-box proves its effectiveness and its reliability in designing various cryptosystems. Based on the effectiveness of the presented S-box, a new image cryptosystem is proposed, in which its experiential outcomes prove its efficacy and security against various attacks. The average outcome of entropy is 7.99977, UACI is 33.484%, NPCR is 99.618%, and Chi-square is 249.481 for the constructed cipher images.

The complexity of chaotic systems, if used in information encryption, can determine the status of security. The paper proposes a novel image encryption scheme that uses chaotic maps and fuzzy numbers for the secure transmission of information. The encryption method combines logistic and sine maps to form the logistic sine map, as well as the fuzzy concept and the Hénon map to form the fuzzy Hénon map, in which these maps are used to generate secure secret keys, respectively. Additionally, a fuzzy triangular membership function is used to modify the initial conditions of the maps during the diffusion process. The encryption process involves scrambling the image pixels, summing adjacent row values, and XORing the result with randomly generated numbers from the chaotic maps. The proposed method is tested against various attacks, including statistical attack analysis, local entropy analysis, differential attack analysis, signal-to-noise ratio, signal-to-noise distortion ratio, mean error square, brute force attack analysis, and information entropy analysis, while the randomness number has been evaluated using the NIST test. This scheme also has a high key sensitivity, which means that a small change in the secret keys can result in a significant change in the encrypted image The results demonstrate the effectiveness of the proposed scheme in ensuring the secure transmission of information. [ABSTRACT FROM AUTHOR]

In this paper, we propose a hybrid method called sliding-window amplitude-based dispersion entropy, which combines dispersion entropy with sliding-window amplitude, to characterize nonlinear time series. This hybrid method not only inherits the fast calculation speed and the ability to characterize nonlinear time series of dispersion entropy, but also has higher noise resistance than dispersion entropy. We firstly utilize three artificial data (logistic map, Hénon map, ARFIMA model) to qualify the effectiveness of the proposed method, results show that our method can correctly characterize the nonlinear time series, and has stronger robustness to noise. Next, the method is applied to analyze stock market system, the data of stock market are composed of six main indices from different countries, the result shows that the proposed method can easily distinguish the emerging markets and developed markets, and can reveal some features under the financial time series. [ABSTRACT FROM AUTHOR]

Reservoir Computing (RC) is a type of machine learning inspired by neural processes, which excels at handling complex and time-dependent data while maintaining low training costs. RC systems generate diverse reservoir states by extracting features from raw input and projecting them into a high-dimensional space. One key advantage of RC networks is that only the readout layer needs training, reducing overall training expenses. Memristors have gained popularity due to their similarities to biological synapses and compatibility with hardware implementation using various devices and systems. Chaotic events, which are highly sensitive to initial conditions, undergo drastic changes with minor adjustments. Cascade chaotic maps, in particular, possess greater chaotic properties, making them difficult to predict with memoryless devices. This study aims to predict 1D and 2D cascade chaotic time series using a memristor-based hierarchical RC system. [ABSTRACT FROM AUTHOR]

The Hénon map was introduced by Hénon. It is a very rich dynamical model as Hénon himself proved and compared the dynamical properties of his map with those of other dynamical systems such as the R o ¨ ssler model. Here, we study the dynamics of the difference equation with continuous arguments corresponding to the Hénon map and its singularly perturbed counterpart. The local stability of fixed points is studied. The system exhibits various types of bifurcation, such as saddle-node and Hopf bifurcations. By letting the perturbation parameter ∊ ⟶ 0 , we show that the singularly perturbed equation exhibits the same qualitative behavior as its corresponding difference equation. The singularly perturbed equation exhibits the same qualitative behavior as its corresponding delay differential equation when ∊ ⟶ 1. The method of steps is used to discretize the original system to simulate the behavior of the original system. Numerical simulations are performed to confirm the theoretical analysis obtained and to illustrate the complex dynamics of the system. Moreover, the numerical simulations illustrate the effect of the perturbation parameter. In particular, the occurrence of halving-period bifurcation due to small change in the perturbation parameter. [ABSTRACT FROM AUTHOR]

Methods of image encryption based on chaos maps are very useful and effective to provide information security. Based on mixed chaos maps, a novel image encoding method has been introduced. A novel procedure for PRN (known as Pseudo Random Number) generation taking the mixed chaotic Henon-Logistic-Tent map has been introduced, where the two dimensional henon chaos map, logistic chaos map, and tent chaos maps, are taken together to create unique integer sequences. These are xored to create the secret key. A grayscale image of size 256 × 256 is converted to encoded image using the operation of xor with the value of the secret_key. In decryption phase, the encoded picture is also xored with the same key to get the original picture. Our encryption method is applied to encrypt handwritten signature images to produce authentic government documents. Detailed analysis according to security parameters shows that our method is robust against different security attacks. It is also efficient in terms of encryption time than other existing methods. [ABSTRACT FROM AUTHOR]

Abstract Cloud computing plays a vital task in our daily lives, in which an enormous amount of data is stored daily on cloud storage. The highest priority for cloud data storage is guaranteeing the security of confidential data. The security of confidential data can be realised through utilising one of the cryptographic mechanisms including encryption and data hiding. With the rapid development for the realization of quantum computers, modern cryptosystems may be cracked including cloud systems. Accordingly, it is a crucial task for achieving confidentiality of data stored on cloud storage before the availability of quantum computers. Therefore, this study aims to utilise one of the quantum computational models, as a quantum-inspired system, to layout a new data confidentiality technique that can be applied in digital devices to have the capability for resisting the potential attacks from quantum and digital computers. In this paper, a new image security algorithm for real-time cloud applications using Gray code, quantum walks (QW), and Henon map is proposed. In the proposed image cryptosystem, the generated key streams from QW and Henon map is related to the plain image with high sensitivity of slight bit changes on the plain image. The outcomes based on deep analysis proves that the presented algorithm is efficient with high security for real-time application.

Cloud computing plays a vital task in our daily lives, in which an enormous amount of data is stored daily on cloud storage. The highest priority for cloud data storage is guaranteeing the security of confidential data. The security of confidential data can be realised through utilising one of the cryptographic mechanisms including encryption and data hiding. With the rapid development for the realization of quantum computers, modern cryptosystems may be cracked including cloud systems. Accordingly, it is a crucial task for achieving confidentiality of data stored on cloud storage before the availability of quantum computers. Therefore, this study aims to utilise one of the quantum computational models, as a quantum-inspired system, to layout a new data confidentiality technique that can be applied in digital devices to have the capability for resisting the potential attacks from quantum and digital computers. In this paper, a new image security algorithm for real-time cloud applications using Gray code, quantum walks (QW), and Henon map is proposed. In the proposed image cryptosystem, the generated key streams from QW and Henon map is related to the plain image with high sensitivity of slight bit changes on the plain image. The outcomes based on deep analysis proves that the presented algorithm is efficient with high security for real-time application. [ABSTRACT FROM AUTHOR]

Most existent DeoxyriboNucleic Acid (DNA) based cryptosystems suffer from low processing capability and need submitting a long key. This article provides a robust cryptosystem which depends on both DNA and Hyperchaotic encryption techniques to solve the drawbacks of using DNA technique. The proposed algorithm uses a hyperchaotic system which hypers between 3D Hénon map and 1D Logistic to generate a greater key space than that generated by using either one. This system has high security and efficiency to produce secret keys for diffusion in the DNA encryption. After that data is encoded exploiting the hypothesis of traditional cryptography, DNA Digital coding, and the advancements of DNA synthesis. Test results prove that the proposed algorithm has large key space (up to 1084) which provides a high resistance against attack of exhaustion, strong sensitivity of encryption key, and good statistical characteristics. Compared to other existing techniques, the proposed algorithm is extremely secure against differential attack and Chosen/Known plaintext attack. Also, the execution time of the proposed cryptosystem is convenient for different applications. [ABSTRACT FROM AUTHOR]

Nonorthogonal multiple access (NOMA) is a relevant technology for realizing the primary goals of next-generation wireless networks, such as high connectivity and stability. Because a rising number of users are becoming connected, user data security has become a critical issue. Many chaotic communication systems have been established to address this important issue via exhibition of affordable physical-layer-security solutions. In this study, we propose a chaotic downlink NOMA (C-DL-NOMA) system over the additive white Gaussian noise and Rayleigh-fading channels to enhance the security of the DL-NOMA system. The proposed algorithm is based on a coherent analog modulation technique that combines various chaotic maps for chaotic masking of encrypted data. On the transmitter, chaotic encryption was used for transmitted data with fixed power-allocation-level control, whereas on the receiver, successive interference-cancellation demodulation was utilized to detect multiple users, after which chaotic decryption was performed. Simulation results were evaluated based on security analyses, such as statistical analysis (histogram and correlation analyses and information entropy), bit-error-rate performance, and achievable-data-rate performance. According to these security analyses and numerical results, the proposed C-DL-NOMA system outperformed traditional unencrypted NOMA systems. [ABSTRACT FROM AUTHOR]

Image encryption is a technique used to protect the confidentiality of digital images. Existing encryption algorithms use chaotic maps for generating random sequences. These sequences purely depend on the fixed initial parameters, which are not adaptive to different input images, thereby limiting their ability to produce the best-possible encrypted image. This paper introduces a new Harris Hawk optimization (HHO) based adaptive image encryption algorithm that integrates Hilbert vibrational decomposition (HVD), and multiple chaotic maps. The new fitness function is specifically designed to optimize the security and robustness of the encrypted image, considering statistical and differential attack parameters (NPCR, UACI, Entropy, and correlation coefficient values). HHO is used to optimize the initial parameters of chaotic maps (Henon map, Duffing map, Lorenz equation) subject to variations in the input image. In the confusion stage, the input image is decomposed into four components of same dimension but decreasing energy using HVD. Subsequently, the block based pixel scrambling is performed on alternate components using the Henon map and Duffing map. In the diffusion stage, the image from the confusion stage undergoes a first level diffusion by bitwise XOR operation with a random image generated from the Lorenz equation. Further, second level of diffusion is achieved with a complex operation generated image from the input image, enhancing its adaptability. Finally, Arnold Cat map having number of iterations determined by the input image is used for final pixel permutation to produces the encrypted image. The optimized initial parameters and input image dependent confusion and diffusion operations enable the encryption system to adapt to changes in the input image, thereby enhancing its resistance against known plain text and known cipher text attacks. Moreover, the integration of HVD with multiple chaotic maps makes the scheme more complex to decrypt with plain text attacks. The performance of proposed algorithm is evaluated on five standard test images in terms of security and quality measures. Experimental results exhibit statistical and differential attack parameters close to ideal values (with Max. Entropy≈7.9996, Min. CC≈ 10−5, Max. NPCR≈99.64, Max. UACI≈33.62). The use of multiple chaotic maps at different stages makes the key space larger. Robustness analysis demonstrates the algorithm's capability in withstanding noise and image enhancement attacks. The comparison with state of the arts metaheuristic based and classical encryption algorithms highlights the superior performance of proposed HHO driven algorithm. The significant improvement in security parameters and overall fitness compared to fixed initial parameters is achieved by optimizing the encryption process. • A new Harris Hawk optimization driven adaptive image encryption algorithm is proposed. • New Mult objective fitness function is designed considering statistical and differential attack parameters. • Hilbert vibrational decomposition is introduced to enhance the complexity and provide flexibility in number of components. • Initial parameters of the chaotic maps are adaptively tuned to generate the optimal encrypted image. • Input image adaptive confusion and diffusion operations to improve the security and robustness. [ABSTRACT FROM AUTHOR]

Chaos maps are widely used in information security due to some attractive properties, but many low-dimensional chaotic maps have several disadvantages such as small parameter space, small Lyapunov exponent, and long calculation time. To better remedy these problems, the paper presents a new spatiotemporal chaotic system with high complex dynamical behavior and good randomness, which is called the general two-dimensional Hénon map (GTDHM). We mathematically investigate the Lyapunov exponent and spatial dynamical behavior of the new system and give the corresponding theorems and corollaries which reflects the good chaotic performance of the proposed system. Moreover, a novel pseudo-random sequence generator based on GTDHM is investigated, and the statistical testes are performed with NIST SP 800-22 statistical test suite. We explicitly show that pseudo-random sequences generated by GTDHM are random and have a weak correlation and possess a larger parameter space, which are good theoretical guarantees for the security of its application in encryption. [ABSTRACT FROM AUTHOR]

The complexity of chaotic systems, if used in information encryption, can determine the status of security. The paper proposes a novel image encryption scheme that uses chaotic maps and fuzzy numbers for the secure transmission of information. The encryption method combines logistic and sine maps to form the logistic sine map, as well as the fuzzy concept and the Hénon map to form the fuzzy Hénon map, in which these maps are used to generate secure secret keys, respectively. Additionally, a fuzzy triangular membership function is used to modify the initial conditions of the maps during the diffusion process. The encryption process involves scrambling the image pixels, summing adjacent row values, and XORing the result with randomly generated numbers from the chaotic maps. The proposed method is tested against various attacks, including statistical attack analysis, local entropy analysis, differential attack analysis, signal-to-noise ratio, signal-to-noise distortion ratio, mean error square, brute force attack analysis, and information entropy analysis, while the randomness number has been evaluated using the NIST test. This scheme also has a high key sensitivity, which means that a small change in the secret keys can result in a significant change in the encrypted image The results demonstrate the effectiveness of the proposed scheme in ensuring the secure transmission of information.

In the last decade, the communication of images through the internet has increased. Due to the growing demands for data transfer through images, protection of data and safe communication is very important. For this purpose, many encryption techniques have been designed and developed. New and secured encryption schemes based on chaos theory have introduced methods for secure as well as fast communication. A modified image encryption process is proposed in this work with chaotic maps and orthogonal matrix in Hill cipher. Image encryption involves three phases. In the first phase, a chaotic Henon map is used for permuting the digital image. In the second phase, a Hill cipher is used whose encryption key is generated by an orthogonal matrix which further is produced from the equation of the plane. In the third phase, a sequence is generated by a chaotic tent map which is later XORed. Chaotic maps play an important role in the encryption process. To deal with the issues of fast and highly secured image processing, the prominent properties of non-periodical movement and non-convergence of chaotic theory play an important role. The proposed scheme is resistant to different attacks on the cipher image. Different tests have been applied to evaluate the proposed technique. The results of the tests such as key space analysis, key sensitivity analysis, and information entropy, histogram correlation of the adjacent pixels, number of pixel change rate (NPCR), peak signal to noise ratio (PSNR), and unified average changing intensity (UCAI) showed that our proposed scheme is an efficient encryption technique. The proposed approach is also compared with some state-of-the-art image encryption techniques. In the view of statistical analysis, we claim that our proposed encryption algorithm is secured. [ABSTRACT FROM AUTHOR]

In this paper, we propose an adaptive encryption scheme for color images using Multiple Distinct Chaotic Maps (MDCM) and DNA computing. We have chosen three distinct chaotic maps, including a 2D-Henon map, a Tent map, and a Logistic map, to separately encrypt the red, green, and blue channels of the original image. The proposed scheme adaptively modifies the parameters of the maps, utilizing various statistical characteristics such as mean, variance, and median of the image to be encrypted. Thus, whenever there is a change in the plain image, the secret keys also change. This makes the proposed scheme robust against the chosen and known plaintext attacks. DNA encoding has also been used to add another layer of security. The experimental analysis of the proposed scheme shows that the average value of entropy is approximately eight, the Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) are 99.61% and 33%, respectively, and correlation coefficients close to zero, making the scheme not only reliable but also resilient against many attacks. Moreover, the use of low-dimensional maps reduces the computational costs of the scheme to a large extent. [ABSTRACT FROM AUTHOR]

In quantum wavelet transform, the pixel values of an image may appear signed decimal instead of integer. Consequently, a generalized representation to signed decimal is presented to carry the data type of pixels in a quantum image. Subsequently, common quantum modules including addition, subtraction, multiplication and division arithmetic units based on the generalized representation to signed decimal are designed to implement quantum Haar wavelet transform. The proposed quantum Haar wavelet transform strategy is more efficient than the classical Haar wavelet transform method. To avoid the periodicity of the quantum Arnold scrambling algorithm, a new quantum scrambling mechanism based on the Henon map is presented. Experimental results indicate that the proposed quantum watermarking algorithm has good performance in security and invisibility. [ABSTRACT FROM AUTHOR]

Domains that are increasing union of balls (up to biholomorphism) and on which the Kobayashi metric vanishes identically arise inexorably in complex analysis. In this article, we show that in higher dimensions these domains have infinite volume and the Bergman spaces of these domains are trivial. As a consequence they fail to be strictly pseudo-convex at each of their boundary points although these domains are pseudo-convex by definition. These domains can be of different types and one of them is Short C k ’s. In pursuit of identifying the Runge Short C k ’s (up to biholomorphism), we introduce a special class of Short C k ’s, called Loewner Short C k ’s. These are those Short C k ’s which can be exhausted in a continuous manner by a strictly increasing parametrized family of open sets, each of which is biholomorphically equivalent to the unit ball and therefore, they are Runge up to biholomorphism. Although, the question of whether all Short C k ’s are Runge (up to biholomorphism), or whether all Short C k ’s are Loewner remains unsettled, we show that the typical Short C k ’s are Loewner. In the final section, we construct a bunch of non-autonomous basins of attraction, which serve as interesting examples of Short C 2 ’s. [ABSTRACT FROM AUTHOR]

The topic of the paper is strange attractors of multidimensional maps and flows. Strange attractors can be divided into two groups: genuine attractors, that keep their chaoticity under small perturbations, and quasi-attractors (according to Afraimovich–Shilnikov), inside which stable periodic orbits can arise under small perturbations. Main goal of this work is to construct effective criteria that make it possible to distinguish such attractors, as well as to verify these criteria by means of numerical experiments. Under «genuine» attractors, we mean the so-called pseudohyperbolic attractors. We give their definition and describe characteristic properties, on the basis of which two numerical methods are constructed, which allow to check the principally important property of pseudohyperbolic attractors: the continuity of strong contracting subspaces and subspaces where volumes are expanded. As examples on which numerical methods for checking pseudohyperbolicity have been tested, we consider the classical Henon map, the singularly hyperbolic Lozi map, the Anosov diffeomorphism of two-dimensional torus, the classical Lorenz and Shimizu–Morioka systems, as well as a three-dimensional Henon-like maps.

The information stored or shared via the internet is growing massively and includes images primarily. The images are vulnerable to attacks when transferred over the internet as they contain confidential information about a person. We propose an adaptive image encryption scheme for the security of images based on twin chaotic maps; Quadratic map and 2-dimensional chaotic Henon map. In the proposed encryption scheme, the Pseudo-Random Number (PRN) sequence for shuffling the pixels has been generated from the Henon map. The PRN sequence for the diffusion process has been produced from the classical Quadratic map. The significant contribution of the presented work is that both the chaotic sequences (for confusion and diffusion) are plain image dependent, making the encryption scheme adaptive and hence highly resilient to brute force attack. Image quality analysis, histogram analysis, correlation coefficient analysis, entropy analysis, key sensitivity analysis, and differential attack analysis have been carried out on eight natural images of size 512 × 512 to validate the performance of the proposed encryption scheme. The proposed encryption algorithm has correlation coefficient values that are very close to zero, entropy value of 7.9993 (average), Net Pixels Change Rate (NPCR), and Unified Average Changing Intensity (UACI) values of 99.62% (average) and 33.3% (average) respectively. The histogram of the encrypted images is flat, meaning no information can be deduced about the plain image. The comparison of the proposed encryption technique with other recent encryption schemes validates the performance of the proposed cryptosystem. The computational time taken to encrypt Lena image of size 512 × 512 using the proposed encryption scheme is 0.25 s. [ABSTRACT FROM AUTHOR]

A novel image encryption scheme generating visually meaningful encrypted images (VMEIs) is presented based on the gyrator transform (GT) and the Hénon chaotic map. The optical encryption system exploits the polarization degree of freedom to reduce the system complexity by allowing the use of a single GT system to concurrently apply to both the secret image and the VMEI. The cryptosystem yields a VMEI with the same size as the secret plain image, which is more secure in either storage or transmission. The results demonstrate that even if the identity of the VMEI is unveiled, any attempt to recover the original image from that VMEI −without knowing the whole secret key− will fail. Unlike previous meaningful cryptosystems that are mainly based on steganographic methods like image hiding and watermarking, the proposed optical cryptosystem adopts a non-steganographic approach by assigning an independent VMEI carrying no information related to the original secret image. The cryptosystem is verified robust against occlusion, noise, statistical attacks, as well as the classical attacks including the chosen-plaintext and chosen-ciphertext attacks. [ABSTRACT FROM AUTHOR]

The partial shading phenomenon significantly limits the PV array output. To maximize the output power during shading, various reconfiguration techniques have been reported in the literature. However, many of these techniques are not scalable, ineffective, and inconsistent in uniform shade dispersion. Therefore, two highly efficient Arnold's Cat Map and Henon Map-based chaotic approaches which are widely used in image encryption are employed to effectively disperse the shade by reconfiguring the modules without disturbing the electrical circuitry. The proposed approaches are evaluated in a MATLAB environment for symmetrical 8 × 8 PV array and unsymmetrical 5 × 7 PV array under different groups of progressive shading like left-to-right, triangular, top-to-bottom, and diagonal shading conditions. The efficacy of the proposed approaches is compared with conventional Series-Parallel, Total-Cross-Tied, existing Chaotic Baker-Map, Odd-Even, and Odd-Even-Prime-based reconfiguration techniques and extensively analyzed using different performance indices. A laboratory experimental setup of a 4 × 4 PV array reconfiguration system is developed and examined under distinct progressive shading cases. From the quantitative results obtained, it is noted that the proposed approaches offer consistently superior and reliable performance compared to existing state-of-art configurations under shading reinforcing their effectiveness. [ABSTRACT FROM AUTHOR]

In tele-medicine, images based on patient diagnostic tests and reports are usually required to broadcast securely so that recipient can receive them without any error. For the secure communication of sensitive medical data, there is a need of authenticated and unforgeable cryptosystem. In this paper, we propose a novel signcryption technique for medical images that fulfills the necessary security requirements of sensitive medical data during its communication. The design of novel medical image signcryption scheme is empowered by the combination strategy of hybrid cryptography. It employs a technique of public key cryptography furnished with elliptic curve cryptography for the generation of secret encryption key. The proposed scheme uses elliptic curve for signcryption purpose and chaotic maps for performing encryption of the medical images. It gives a combined mechanism of key exchange, digital signature and chaotic image encryption. Image encryption phase contains permutation and diffusion for pixel scrambling. The proposed image signcryption scheme provides confidentiality, authentication, integrity, unforgability, forward secrecy and non-repudiation. The use of chaos-based symmetric image encryption scheme shows good results for key space analysis, key sensitivity, correlation analysis, number of pixel change rate, unified average changing intensity, entropy analysis and histogram analysis. On the basis of the results obtained from all these analyses, we believe that the proposed medical image signcryption scheme is efficient and robust, providing good security for sensitive medical images.

The evolution of wireless and mobile communication from 0G to the upcoming 5G gives rise to data sharing through the Internet. This data transfer via open public networks are susceptible to several types of attacks. Encryption is a method that can protect information from hackers and hence confidential data can be secured through a cryptosystem. Due to the increased number of cyber attacks, encryption has become an important component of modern-day communication. In this article, a new image encryption algorithm is presented using chaos theory and dynamic substitution. The proposed scheme is based on two-dimensional Henon, Ikeda chaotic maps, and substitution box (S-box) transformation. Through Henon, a random S-Box is selected and the image pixel is substituted randomly. To analyze security and robustness of the proposed algorithm, several security tests such as information entropy, histogram investigation, correlation analysis, energy, homogeneity, and mean square error are performed. The entropy values of the test images are greater than 7.99 and the key space of the proposed algorithm is 2798. Furthermore, the correlation values of the encrypted images using the proposed scheme are close to zero when compared with other conventional schemes. The number of pixel change rate (NPCR) and unified average change intensity (UACI) for the proposed scheme are higher than 99.50% and 33, respectively. The simulation results and comparison with the state-of-the-art algorithms prove the efficiency and security of the proposed scheme.

We propose a fast nonlinear method for assessing quantitatively both the existence and directionality of linear and nonlinear couplings between a pair of time series. We test this method, called Boolean Slope Coherence (BSC), on bivariate time series generated by various models, and compare our results with those obtained from different well-known methods. A similar approach is employed to test the BSC's capability to determine the prevalent coupling directionality. Our results show that the BSC method is successful for both quantifying the coupling level between a pair of signals and determining their directionality. Moreover, the BSC method also works for noisy as well as chaotic signals and, as an example of its application to real data, we tested it by analyzing neurophysiological recordings from visual cortices. [ABSTRACT FROM AUTHOR]

For every m ∈ N, we establish the equidistribution of the sequence of the averaged pullbacks of a Dirac measure at any given value in C\{0} under the mth order derivatives of the iterates of a polynomials ƒ ∈ C[z] of degree d > 1 towards the harmonic measure of the filled-in Julia set of ƒ with pole at ∞. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field k for a sequence of effective divisors on P1(k) having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Hénon-type polynomial automorphism of C2 has a given eigenvalue. [ABSTRACT FROM AUTHOR]