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At present, there is a growing emphasis on safeguarding image data, yet conventional encryption methods are full of numerous limitations. In order to tackle the limitations of conventional color image encryption methodologies, such as inefficiency and insufficient security, this paper designs an expedited encryption method for color images that uses DNA coding in conjunction with multiple chaotic systems. The encryption algorithm proposed in this paper is based on three-dimensional permutation, global scrambling, one-dimensional diffusion and DNA coding. First of all, the encryption algorithm uses three-dimensional permutation algorithms to scramble the image, which disrupts the high correlation among the image pixels. Second, the RSA algorithm and the SHA-256 hashing algorithm are utilized to derive the starting value necessary for the chaotic system to produce the key. Third, the image is encrypted by using global scrambling and one-dimensional diffusion. Finally, DNA coding rules are used to perform DNA computing. The experimental results indicate that the encryption scheme exhibits a relatively weak inter-pixel correlation, uniform histogram distribution, and an information entropy value approaching eight. This shows that the proposed algorithm is able to protect the image safely and efficiently. [ABSTRACT FROM AUTHOR]

Discrete-time chaotic systems exhibit nonlinear and unpredictable dynamic behavior, making them very difficult to classify. They have dynamic properties such as the stability of equilibrium points, symmetric behaviors, and a transition to chaos. This study aims to classify the time series images of discrete-time chaotic systems by integrating deep learning methods and classification algorithms. The most important innovation of this study is the use of a unique dataset created using the time series of discrete-time chaotic systems. In this context, a large and unique dataset representing various dynamic behaviors was created for nine discrete-time chaotic systems using different initial conditions, control parameters, and iteration numbers. The dataset was based on existing chaotic system solutions in the literature, but the classification of the images representing the different dynamic structures of these systems was much more complex than ordinary image datasets due to their nonlinear and unpredictable nature. Although there are studies in the literature on the classification of continuous-time chaotic systems, no studies have been found on the classification of discrete-time chaotic systems. The obtained time series images were classified with deep learning models such as DenseNet121, VGG16, VGG19, InceptionV3, MobileNetV2, and Xception. In addition, these models were integrated with classification algorithms such as XGBOOST, k-NN, SVM, and RF, providing a methodological innovation. As the best result, a 95.76% accuracy rate was obtained with the DenseNet121 model and XGBOOST algorithm. This study takes the use of deep learning methods with the graphical representations of chaotic time series to an advanced level and provides a powerful tool for the classification of these systems. In this respect, classifying the dynamic structures of chaotic systems offers an important innovation in adapting deep learning models to complex datasets. The findings are thought to provide new perspectives for future research and further advance deep learning and chaotic system studies. [ABSTRACT FROM AUTHOR]

Mathematical models such as Fitzhugh–Nagoma and Hodgkin–Huxley models have been used to understand complex nervous systems. Still, due to their complexity, these models have made it challenging to analyze neural function. The discrete Rulkov model allows the analysis of neural function to facilitate the investigation of neuronal dynamics or others. This paper introduces a fractional memristor Rulkov neuron model and analyzes its dynamic effects, investigating how to improve neuron models by combining discrete memristors and fractional derivatives. These improvements include the more accurate generation of heritable properties compared to full-order models, the treatment of dynamic firing activity at multiple time scales for a single neuron, and the better performance of firing frequency responses in fractional designs compared to integer models. Initially, we combined a Rulkov neuron model with a memristor and evaluated all system parameters using bifurcation diagrams and the 0–1 chaos test. Subsequently, we applied a discrete fractional-order approach to the Rulkov memristor map. We investigated the impact of all parameters and the fractional order on the model and observed that the system exhibited various behaviors, including tonic firing, periodic firing, and chaotic firing. We also found that the more I tend towards the correct order, the more chaotic modes in the range of parameters. Following this, we coupled the proposed model with a similar one and assessed how the fractional order influences synchronization. Our results demonstrated that the fractional order significantly improves synchronization. The results of this research emphasize that the combination of memristor and discrete neurons provides an effective tool for modeling and estimating biophysical effects in neurons and artificial neural networks. [ABSTRACT FROM AUTHOR]

In this research work, we investigate a new three-dimensional jerk system with three parameters in which one of the nonlinear terms is a sinusoidal nonlinearity. We show that the new jerk system has two unstable equilibrium points on the ��-axis. Numerical integrations show the existence of periodic and chaotic states, as well as unbounded solutions. Consideration of the Poincaré sphere at infinity found no periodic states. We show that the new jerk system exhibits multistability with coexisting attractors. We also present results for the offset boosting of the proposed chaotic jerk system. Using MultiSim version 14.1, we design an electronic circuit for the new jerk system with a sinusoidal nonlinearity. As a control application, we design complete synchronization for the master-slave jerk systems using backstepping control technique. Simulations are presented to illustrate the main results of this research work.

In this paper, we present the design, development, and pilot implementation of a Teaching–Learning Sequence (TLS) about the physics of deterministic chaos. The main aim of the activities is to let students become aware of two key ideas about deterministic chaos: (1) the role of initial conditions and (2) the graphical representation in a momentum–position graph. To do so, the TLS is based on the observation and analysis of the trajectory of the free end of a double pendulum through the modeling software Tracker. In particular, the Tracker-based activities help students understand that, by modifying the well-known simple pendulum dynamic system into a double pendulum, long-time-scale predictability is lost, and a completely new behavior appears. The TLS was pilot tested in a remote teaching setting with about 70 Italian high school students (16–17 years old). The pretest analysis shows that before participating in the activities, students held typical misconceptions about chaotic behavior. Analysis of the written responses collected during and after implementation shows that the proposed activities allowed students to grasp the two key ideas about nondeterministic chaos. A possible integration of the TLS with an online simulation is finally discussed. [ABSTRACT FROM AUTHOR]

In recent years, memristors have received widespread attention due to their inherent nonlinear characteristics. To this end, a three-dimensional memristor chaotic system based on secondary nonlinear magnetron memristors has been constructed. Through the analysis of phase portraits, 0–1 tests, Poincare sections, bifurcation diagrams, and Lyapunov exponent spectra, it was found that the system exhibits rich dynamical behaviors, including immovable points, periodic orbits, and chaos. The coexistence attractors of the system were studied, and their basins were also plotted. Comparative complexity analysis indicates that the chaotic system possesses high complexity, thus providing higher security for image encryption. The practicality of the chaotic system was verified through circuit simulation and FPGA. Finally, the chaotic system was applied to DNA image encryption. The focus was on analyzing the histogram, adjacent pixel correlation, information entropy, security, and runtime. The analysis results demonstrate that the algorithm has high security and practicality. [ABSTRACT FROM AUTHOR]

The present work presents the investigation of the dynamics and influence of chaotic behavior on energy capture for a U-shaped structure (portal frame) that contains shape memory alloy (SMA), piezoelectric material (PZT), a nonlinear energy sink (NES) and a non-ideal excitation source represented by an unbalanced electric motor coupled to the U-structure. The mathematical model presents nonlinearities arising from the nonlinear stiffness of the U-structure, the NES system, the SMA, and the PZT material. Chaotic behavior is assessed through time history, bifurcation diagrams, phase diagrams, and the 0–1 test. Energy capture is carried out through a piezoelectric material (PZT), represented by a non-linear electromechanical coupling model, and electromagnetic induction generated by the non-linear electromagnetic energy sink coupled to the structure (NES). Dynamic analysis is performed through parametric analysis of parameters related to piezoelectric coupling and NES parameters. Numerical simulations demonstrate that the system has chaotic behavior for specific parameters and that its energy capture is influenced by parametric variation. It is shown numerically that the parameters of the SMA material, the PZT material, and the NES significantly influence the chaotic behavior and energy capture of the investigated electromechanical system. [ABSTRACT FROM AUTHOR]

This manuscript introduces a piecewise linear decomposition method devoted to a class of fractional-order dynamical systems composed of piecewise linear (PWL) functions. Inspired by the Adomian decomposition method, the proposed technique computes an approximated solution of fractional-order PWL systems using only linear operators and specific constants vectors for each sub-domain of the PWL functions, with no need for the Adomian polynomials. The proposed decomposition method can be applied to fractional-order PWL systems composed of nth PWL functions, where each PWL function may have any number of affine segments. In particular, we demonstrate various examples of how to solve fractional-order systems with 1D 2-scroll, 4-scroll, and 4 × 4 -grid scroll chaotic attractors by applying the proposed approach. From the theoretical and implementation results, we found the proposed approach eliminates the unneeded terms, has a low computational cost, and permits a straightforward physical implementation of multi-scroll chaotic attractors on ARMs and FPGAs digital platforms. [ABSTRACT FROM AUTHOR]

Encryption of images is an important method that can effectively improve the security and privacy of crucial image data. Existing methods generally encrypt an image with a combination of scrambling and encoding operations. Currently, many applications require highly secure results for image encryption. New methods that can achieve improved randomness for both the scrambling and encoding processes in encryption are thus needed to further enhance the security of a cipher image. This paper proposes a new method that can securely encrypt color images. As the first step of the proposed method, a complete bit-level operation is utilized to scramble the binary bits in a color image to a full extent. For the second step, the bits in the scrambled image are processed with a sweeping operation to improve the encryption security. In the final step of encryption, a codebook that varies with evolutionary operations based on several chaotic systems is utilized to encrypt the partially encrypted image obtained in the second step. Experimental results on benchmark color images suggest that this new approach can securely encrypt color images and generate cipher images that remain secure under different types of attacks. The proposed approach is compared with several other state-of-the-art encryption approaches and the results show that it can achieve improved encryption security for cipher images. Experimental results thus suggest that this new approach can possibly be utilized practically in applications where color images need to be encrypted for content protection. [ABSTRACT FROM AUTHOR]

Chaotic systems are highly sensitive to initial conditions, meaning small changes can lead to significant outcomes. This sensitivity can be leveraged to enhance the flexibility and adaptability of supply chains, allowing them to respond quickly to changing market demands and conditions. In this research work, we obtain a New Chaotic Supply Chain Model (NCSCM) by introducing a sinusoidal model uncertainty in the third differential equation of the Hamidzadeh chaotic supply chain model (2023). The sinusoidal uncertainty in the model accounts for the volatility and fluctuations of the supply chain model. We investigate the dynamic behaviors of NCSCM by performing a detailed Bifurcation Diagram (BD) analysis and Lyapunov Exponents (LEs). Next, we carry out a 0−1 test for the NCSCM to confirm the chaotic attractor of the system. We also study the multi-stability properties of the NCSCM. Finally, with the help of backstepping control, we derive new results for the complete synchronization of the novel chaotic supply chain models.

Objective: In this article, time-varying chaotic systems with uncertainties, including external disturbances, are considered, and sliding mode control (SMC) is used to control such systems. To control these systems, an autonomous differential equation is first introduced. Then, based on this differential equation, a sliding surface is defined to control this chaotic system. This kind of controller is remarkable in that it removes the effects of disturbances, whether bounded or unbounded. Therefore, the system is known to be fixed-time stable. where the trajectories of this chaotic system are not placed on the sliding surface, we have created creative controllers to place the trajectories on the sliding surface in finite time. Theoretical investigations show that such chaotic systems can be made fixed-time stable by applying the controls proposed in this study. Based on the findings of this study, the controllers are designed to eliminate all disturbances, whether bounded or unbounded. the results be said to apply chaotic, time-dependent, and time-independent systems. To further consolidate the results obtained in this article, two examples, namely the time-dependent system of the Gyro and the time-independent system of the Liu, are investigated, and the results were compared with previous works by other researchers.

Smartphones, digital cameras, and other imaging devices generate vast amounts of high-resolution colored images daily, stored on devices equipped with multi-core central processing units or on the cloud. Safeguarding these images from potential attackers has become a pressing concern. This paper introduces a set of six innovative image ciphers designed to be stronger, faster, and more efficient. Three of these algorithms incorporate the State-Based Tent Map (SBTM) Chaotic Pseudo Random Number Generator (CPRNG), while the remaining three employ a proposed modified variant, SBTMPi. The Grayscale Image Cipher (GIC), Colored Image Cipher Single-Thread RGB (CIC1), and Colored Image Cipher Three-Thread RGB (CIC3) showcase the application of the proposed algorithms. By incorporating novel techniques in the confusion and diffusion phases, these ciphers demonstrate remarkable performance, particularly with large colored images. The study underscores the potential of SBTM-based image ciphers, contributing to the advancement of secure image encryption techniques with robust random number generation capabilities. [ABSTRACT FROM AUTHOR]

In this research work, we investigate a new three-dimensional jerk system with three parameters in which one of the nonlinear terms is a sinusoidal nonlinearity. We show that the new jerk system has two unstable equilibrium points on the x-axis. Numerical integrations show the existence of periodic and chaotic states, as well as unbounded solutions. Consideration of the Poincaré sphere at infinity found no periodic states. We show that the new jerk system exhibits multistability with coexisting attractors. We also present results for the offset boosting of the proposed chaotic jerk system. Using MultiSim version 14.1, we design an electronic circuit for the new jerk system with a sinusoidal nonlinearity. As a control application, we design complete synchronization for the master-slave jerk systems using backstepping control technique. Simulations are presented to illustrate the main results of this research work. [ABSTRACT FROM AUTHOR]

This paper studies the synchronization and control of chaotic systems while proposing a novel chaotic-based path-tracking application for mobile robots (MRs) to ensure their safety and security. In security-based applications that use MRs, such as patrol MRs, the path of the MRs must be complex enough to prevent easy prediction. Multiple chaotic systems with a chaotic switching mechanism are introduced for secure path planning. The main challenges are that the dynamics of MRs are entirely unknown. The modeled dynamics of the MRs are unreliable in practice due to a broad range of uncertainties related to the parameters, operating conditions, environmental impacts, time delays, unmodeled frictions, noisy sensors, and faulty actuators. Also, the chaotic switching of reference signals between chaotic signals imposes a high dynamic perturbation. The main novelties are as follows: (1) a strong secure path is introduced for MRs. (2) A powerful fractional-order predictive controller using type-3 (T3) fuzzy-logic systems (FLSs) is developed. (3) The estimation and prediction errors of T3-FLSs are compensated by a designed parallel compensator. (4) T3-FLSs are tuned online, such that stability is ensured, and prediction accuracy is guaranteed. (5) The suggested scheme is implemented on a real-world MR, and the results demonstrate the feasibility and accuracy of the proposed method. Also, in several simulations, the efficacy of the introduced controller is examined. [ABSTRACT FROM AUTHOR]

Multistep prediction of high-dimensional time series is an essential and challenging task. In this study, we propose an integrated reservoir predictor for making accurate and robust multistep-ahead forecasts based on short-term high-dimensional time series. Initially, a conjugated pair of Spatiotemporal Information (STI) equations is derived using Takens' embedding theory to transform the spatial information of high-dimensional variables into one-dimensional temporal information of the target variable and vice versa. Next, by exploiting reservoir networks, reservoir-based STI equations are established to efficiently capture nonlinear dynamics of the target system with only linear optimization. Then, through an integration phase, the integrated reservoir predictor can output precise and robust predictions of the multistep-ahead states of any target variable. The integrated reservoir predictor outperforms some other prediction methods (including reservoir computing, long-short-term-memory network, convolutional neural network and support vector regression), when applied to classical dynamic systems (e.g. 60D double scroll model, 40D Lorenz 96 model, and 60D Rössler model) and real-world datasets (solar generation data and PM2.5 concentration records), as indicated by evaluation metrics such as Pearson correlation coefficients exceeding 0.9 and root-mean-square errors below 0.3, even in the presence of noise in training data. [ABSTRACT FROM AUTHOR]

Existing biohashing authentication schemes usually store bioinformation in the cloud, whose storage method can face the problems of bioinformation leakage and tampering, and single point of failure. To address the above problems, a ciphertext speech biohashing authentication scheme based on chaotic system and improved public chain is proposed. First, the fusion features are extracted and generated with the inner product of 3D chaotic system to generate the biohashing sequence. Second, the original speech is encrypted and uploaded to the InterPlanetary file system and then, uploaded to the blockchain after the data are encrypted. Finally, the server matches the hash sequence queried on the chain with the to-be-authenticated hash at the user's end using the Hamming distance. The experimental results show that the biohashing algorithm in this paper has better distinguishability, robustness, security and time efficiency. The key space of the speech encryption algorithm is 2 256 , which can well resist the exhaustive attack and ensure the security during speech authentication. In this paper, the combination of on-chain and off-chain storage of speech information can prevent the problem of bioinformation leakage and tampering, as well as avoiding the problem of single point of failure. [ABSTRACT FROM AUTHOR]

The present study investigates unsteady wake dynamics in flow past an oscillating cylinder. A two-dimensional computational analysis is undertaken for flow past a sinusoidally oscillating cylinder. Operating parameters like Reynolds number, forcing frequency, amplitude, and wavelength are systematically varied, and their effect on wake dynamics is investigated. Lift and drag coefficients are plotted with cylinder position to characterize the flow. The flow field is analyzed using Proper Orthogonal Decomposition (POD). POD mode shapes and Power Spectral Density (PSD) plots characterize the flow response to cylinder oscillations. The effect of frequency ratio is examined in achieving synchronized vortex shedding. Previous studies focus on two extremes- the cylinder is either stationary or oscillating with perfect phase-locking. Flow past an oscillating cylinder where Karman and forcing frequencies compete for dominance is rarely reported. This paper considers such a scenario and demonstrates how POD can identify coherent structures, their significance, and associated frequencies. Further, non-symmetric vortex shedding, which is not a common observation, is captured and discussed. Large amplitude and large wavelength regimes are investigated, which remained unexplored in previous research. Coherent structures associated with those regimes are identified. [ABSTRACT FROM AUTHOR]

In the recent years, significant research interest has been devoted in the modelling and applications of chaotic systems with stable equilibria. In this research study, we propose a new 3-D chaotic system with two stable node-foci equilibria and an unstable saddle-focus equilibrium. We conduct a detailed bifurcation analysis for the new chaotic system with the aid of bifurcation diagrams and Lyapunov exponents. We also show that the new chaotic system has multistability with coexisting attractors. Using MultiSim version 14.1, we design an electronic circuit for the proposed 3-D chaotic system. Finally, as an application, we introduce a voice cryptosystem using the new chaotic system with two stable node-foci equilibrium points and an unstable saddle-focus equilibrium point and its hardware realization on the FPGA platform.

The swift development of the Internet of Things (IoT) has accelerated digitalization across several industries, offering networked applications in fields such as security, home automation, logistics, and quality control. The growth of connected devices, on the other hand, raises worries about data breaches and security hazards. Because of IoT devices’ computational and energy limits, traditional cryptographic methods face issues. In this context, we emphasize the importance of our contribution to image encryption in IoT environments through the proposal of Multiple Map Chaos Based Image Encryption (MMCBIE), a novel method that leverages the power of multiple chaotic maps. MMCBIE uses multiple chaotic maps to construct a strong encryption framework that considers the inherent features of digital images. Our proposed method, MMCBIE, distinguishes itself by integrating multiple chaotic maps like Henon Chaotic Transform and 2D-Logistic Chaotic Transform in a novel combination, a unique approach that sets it apart from existing schemes. Compared to other chaotic-based encryption systems, this feature renders them practically indistinguishable from pure visual noise. Security evaluations and cryptanalysis confirm MMCBIE’s high-level security properties, indicating its superiority over existing image encryption techniques. MMCBIE demonstrated superior performance with NPCR (Number of Pixel Changing Rate) score of 99.603, UACI (Unified Average Changing Intensity) score of 32.8828, MSE (Mean Square Error) score of 6625.4198, RMSE (Root Mean Square Error) score of 80.0063, PSNR (Peak Signal to Noise Ratio) score of 10.2114, and other security analyses.

We propose a double image encryption scheme combining DNA, parallel compressed sensing, enhanced Zigzag confusion, chaotic sparse basis matrix, and chaotic permutation. Firstly, use the logistic map to scramble a Discrete Wavelet Transformation (DWT) matrix to obtain a chaotic DWT matrix. Next, 2D-LICM generates the measurement matrix. Parallel compressed sensing is then performed separately on the two images. Then, enhanced Zigzag confusion and DNA encryption are performed on the new matrix using chaotic sequences to obtain an encrypted image. The chaotic system’s initial value is generated using SHA-256, which has a strong sensitivity to the plain image. We improve on the third-order Colpitts chaotic circuit to construct a more general higher-order Colpitts chaotic oscillator and propose a new fourth-order Colpitts chaotic system. The performance analysis indicates that our encryption algorithm provides excellent security and performance while saving storage space.

A hyperjerk system pertains to a dynamical system regulated by an ordinary differential equation of $n$ th order, where $n \geq 4$ . The main contribution of this work is the finding of a new autonomous hyperjerk system with a half line equilibrium. The mathematical framework of the proposed hyperjerk system contains eight terms with an absolute function nonlinearity. The essential dynamic characteristics of the model are explored, encompassing analysis of equilibrium points and their stability, depiction of the phase trajectories, illustration of bifurcation patterns, and visualization of Lyapunov exponent graphs. Our finding shows that the new 4D hyperjerk system exhibits special behavior like multistability, period doubling reversals and antimonotonocity. The proposed hyperjerk system has been implemented with an electronic circuit using MultiSim 14.0. Moreover, the FPGA implementation of the proposed hyperjerk system is performed by applying two numerical methods: Forward Euler and Trapezoidal. Experimental attractors are given from an oscilloscope by using the Zybo Z7-20 FPGA development board, which are in good agreement with the MATLAB and MultiSim 14.0 simulations. Finally, based on the chaotic dynamical behavior of the proposed chaotic hyperjerk system, a new image encryption approach is proposed. The experimental outcomes of the presented encryption algorithm prove its efficiency and security.

The use of wheeled mobile robots (MRs) with symmetrical structure in engineering is rapidly increasing, with applications in various fields, such as industry, agriculture, forestry, healthcare, mining, rehabilitation, search and rescue, household tasks, remote locations, and entertainment. As MRs become more common, researchers are focusing on developing better ways to model and control these robots to improve their performance and adaptability. The main challenges in this area include uncertain dynamics, non-holonomic constraints, and various perturbations, which complicate the design of the control system. This paper presents a new predictive control scheme for MRs that is independent of the dynamics and the robot’s working environment. A Type-3 fuzzy logic system is developed to identify the MR dynamics online. The designed predictive scheme improves accuracy and speeds up convergence, while also addressing uncertainties and considering constraints on control input. Additionally, a chaotic-based system is proposed for secure path planning, generating a complex and unpredictable reference trajectory that is useful for patrol MR applications. The effectiveness of the suggested controller is demonstrated through simulations and experiments.

In this work, a new 3-D modified WINDMI chaotic jerk system with exponential and sinusoidal nonlinearities is presented and its dynamical behaviours and properties are investigated. Firstly, some properties of the system are studied such as equilibrium points and their stability, Lyapunov exponents and Kaplan-Yorke dimension. Also, we study the new jerk system dynamics using numerical simulations and analyses, including phase portraits, Lyapunouv exponent spectrum, bifurcation diagram and Poincaré map, 0-1 test. Next, we exhibit that the new 3-D chaotic modified WINDMI jerk system has multistability with coexisting chaotic attractors. Moreover, we design an electronic circuit using MultiSim 14.1 for real implementation of the modified WINDMI chaotic jerk system. Finally, we design an active synchronization scheme for the complete synchronization of the modified WINDMI chaotic jerk systems via backstepping control.

At present, there is a growing emphasis on safeguarding image data, yet conventional encryption methods are full of numerous limitations. In order to tackle the limitations of conventional color image encryption methodologies, such as inefficiency and insufficient security, this paper designs an expedited encryption method for color images that uses DNA coding in conjunction with multiple chaotic systems. The encryption algorithm proposed in this paper is based on three-dimensional permutation, global scrambling, one-dimensional diffusion and DNA coding. First of all, the encryption algorithm uses three-dimensional permutation algorithms to scramble the image, which disrupts the high correlation among the image pixels. Second, the RSA algorithm and the SHA-256 hashing algorithm are utilized to derive the starting value necessary for the chaotic system to produce the key. Third, the image is encrypted by using global scrambling and one-dimensional diffusion. Finally, DNA coding rules are used to perform DNA computing. The experimental results indicate that the encryption scheme exhibits a relatively weak inter-pixel correlation, uniform histogram distribution, and an information entropy value approaching eight. This shows that the proposed algorithm is able to protect the image safely and efficiently.

Discrete-time chaotic systems exhibit nonlinear and unpredictable dynamic behavior, making them very difficult to classify. They have dynamic properties such as the stability of equilibrium points, symmetric behaviors, and a transition to chaos. This study aims to classify the time series images of discrete-time chaotic systems by integrating deep learning methods and classification algorithms. The most important innovation of this study is the use of a unique dataset created using the time series of discrete-time chaotic systems. In this context, a large and unique dataset representing various dynamic behaviors was created for nine discrete-time chaotic systems using different initial conditions, control parameters, and iteration numbers. The dataset was based on existing chaotic system solutions in the literature, but the classification of the images representing the different dynamic structures of these systems was much more complex than ordinary image datasets due to their nonlinear and unpredictable nature. Although there are studies in the literature on the classification of continuous-time chaotic systems, no studies have been found on the classification of discrete-time chaotic systems. The obtained time series images were classified with deep learning models such as DenseNet121, VGG16, VGG19, InceptionV3, MobileNetV2, and Xception. In addition, these models were integrated with classification algorithms such as XGBOOST, k-NN, SVM, and RF, providing a methodological innovation. As the best result, a 95.76% accuracy rate was obtained with the DenseNet121 model and XGBOOST algorithm. This study takes the use of deep learning methods with the graphical representations of chaotic time series to an advanced level and provides a powerful tool for the classification of these systems. In this respect, classifying the dynamic structures of chaotic systems offers an important innovation in adapting deep learning models to complex datasets. The findings are thought to provide new perspectives for future research and further advance deep learning and chaotic system studies.

Mathematical models such as Fitzhugh–Nagoma and Hodgkin–Huxley models have been used to understand complex nervous systems. Still, due to their complexity, these models have made it challenging to analyze neural function. The discrete Rulkov model allows the analysis of neural function to facilitate the investigation of neuronal dynamics or others. This paper introduces a fractional memristor Rulkov neuron model and analyzes its dynamic effects, investigating how to improve neuron models by combining discrete memristors and fractional derivatives. These improvements include the more accurate generation of heritable properties compared to full-order models, the treatment of dynamic firing activity at multiple time scales for a single neuron, and the better performance of firing frequency responses in fractional designs compared to integer models. Initially, we combined a Rulkov neuron model with a memristor and evaluated all system parameters using bifurcation diagrams and the 0–1 chaos test. Subsequently, we applied a discrete fractional-order approach to the Rulkov memristor map. We investigated the impact of all parameters and the fractional order on the model and observed that the system exhibited various behaviors, including tonic firing, periodic firing, and chaotic firing. We also found that the more I tend towards the correct order, the more chaotic modes in the range of parameters. Following this, we coupled the proposed model with a similar one and assessed how the fractional order influences synchronization. Our results demonstrated that the fractional order significantly improves synchronization. The results of this research emphasize that the combination of memristor and discrete neurons provides an effective tool for modeling and estimating biophysical effects in neurons and artificial neural networks.

In this paper, we use two Fractional-Order Chaotic Systems (FOCS)—one for the sender and one for the receiver—to determine the optimal synchronisation for a secure communication technique. With the help of the Step-By-Step Sliding-Mode Observer (SBS-SMO), this synchronisation is accomplished. An innovative optimisation method, known as the artificial Harris hawks optimisation (HHO), was employed to enhance the observer's performance. This method eliminates the random parameter selection process and instead selects the optimal values for the observer. In a short amount of time, the secret message that could have been in the receiver portion (signal, voice, etc.) was successfully recovered using the proposed scheme. The experimental validation of the numerical results was carried out with the assistance of an Arduino microcontroller and several electronic components. In addition, the findings of the experiments were compared with the theoretical calculations, revealing a satisfactory level of agreement. [ABSTRACT FROM AUTHOR]

Predicting and understanding the behavior of dynamic systems have driven advancements in various approaches, including physics-based models and data-driven techniques like deep neural networks. Chaotic systems, with their stochastic nature and unpredictable behavior, pose challenges for accurate modeling and forecasting, especially during extreme events. In this paper, we propose a novel deep learning framework called Attractor-Inspired Deep Learning (AiDL), which seamlessly integrates actual statistics and mathematical models of system kinetics. AiDL combines the strengths of physics-informed machine learning and data-driven methods, offering a promising solution for modeling nonlinear systems. By leveraging the intricate dynamics of attractors, AiDL bridges the gap between physics-based models and deep neural networks. We demonstrate the effectiveness of AiDL using real-world data from various domains, including catastrophic weather mechanics, El Niño cycles, and disease transmission. Our empirical results showcase AiDL's ability to substantially enhance the modeling of extreme events. The proposed AiDL paradigm holds promise for advancing research in Time Series Prediction of Extreme Events and has applications in real-world chaotic system transformations. [ABSTRACT FROM AUTHOR]

In the field of Electronic Health (e-Health), Electronic Health Records (EHR) are transmitted between health professionals using e-Health systems for cooperative medical practice, medical monitoring, telemedical expertise, and telemedical imaging. Medical images are a crucial component of EHR and are used in various aspects of telemedicine systems such as expertise, consultation, teaching, and research. However, protecting the authenticity and copyrights of medical images is essential to prevent duplication, modification, or unauthorized distribution. This paper proposes a robust medical image copyright protection method that uses patient palm-print template as watermark and Lorenz chaotic map for template concealing and selecting the appropriate embedding positions in medical images. The novelty of the method lies in optimizing the expected number of modifications per pixel of the medical images after being watermarked. Experimental results indicate that this approach has a high performance with a genuine accept rate of 99.86% and can withstand various image processing attacks, including Gaussian noise, compression, and image rotations, while ensuring personal data security during telemedicine data exchange.

Abstract Predicting and understanding the behavior of dynamic systems have driven advancements in various approaches, including physics-based models and data-driven techniques like deep neural networks. Chaotic systems, with their stochastic nature and unpredictable behavior, pose challenges for accurate modeling and forecasting, especially during extreme events. In this paper, we propose a novel deep learning framework called Attractor-Inspired Deep Learning (AiDL), which seamlessly integrates actual statistics and mathematical models of system kinetics. AiDL combines the strengths of physics-informed machine learning and data-driven methods, offering a promising solution for modeling nonlinear systems. By leveraging the intricate dynamics of attractors, AiDL bridges the gap between physics-based models and deep neural networks. We demonstrate the effectiveness of AiDL using real-world data from various domains, including catastrophic weather mechanics, El Niño cycles, and disease transmission. Our empirical results showcase AiDL’s ability to substantially enhance the modeling of extreme events. The proposed AiDL paradigm holds promise for advancing research in Time Series Prediction of Extreme Events and has applications in real-world chaotic system transformations.

Digital data is increasing rapidly in the world day by day. Information security is important during data exchange over the Internet. The way to securely transmit images over the network is through the image encryption technique. In the proposed cryptography system, the hybridization of Lorenz-Rossler chaotic systems is used, and a random number sequence is generated. The security analyses such as histogram, correlation, differential attack, information entropy, and duration analysis of the study are performed. It is seen that the proposed system performs well, especially in terms of correlation. Additionally, the performance of the developed embedded system platforms is compared after testing on Nvidia Jetson Nano and Xilinx PYNQ Z1 boards. The Nvidia Jetson Nano board is more performant than the Xilinx PYNQ Z1 board. The safety and feasibility of the proposed system have been demonstrated.

Abstract This paper studies the synchronization and control of chaotic systems while proposing a novel chaotic-based path-tracking application for mobile robots (MRs) to ensure their safety and security. In security-based applications that use MRs, such as patrol MRs, the path of the MRs must be complex enough to prevent easy prediction. Multiple chaotic systems with a chaotic switching mechanism are introduced for secure path planning. The main challenges are that the dynamics of MRs are entirely unknown. The modeled dynamics of the MRs are unreliable in practice due to a broad range of uncertainties related to the parameters, operating conditions, environmental impacts, time delays, unmodeled frictions, noisy sensors, and faulty actuators. Also, the chaotic switching of reference signals between chaotic signals imposes a high dynamic perturbation. The main novelties are as follows: (1) a strong secure path is introduced for MRs. (2) A powerful fractional-order predictive controller using type-3 (T3) fuzzy-logic systems (FLSs) is developed. (3) The estimation and prediction errors of T3-FLSs are compensated by a designed parallel compensator. (4) T3-FLSs are tuned online, such that stability is ensured, and prediction accuracy is guaranteed. (5) The suggested scheme is implemented on a real-world MR, and the results demonstrate the feasibility and accuracy of the proposed method. Also, in several simulations, the efficacy of the introduced controller is examined.

In this paper, we present the design, development, and pilot implementation of a Teaching–Learning Sequence (TLS) about the physics of deterministic chaos. The main aim of the activities is to let students become aware of two key ideas about deterministic chaos: (1) the role of initial conditions and (2) the graphical representation in a momentum–position graph. To do so, the TLS is based on the observation and analysis of the trajectory of the free end of a double pendulum through the modeling software Tracker. In particular, the Tracker-based activities help students understand that, by modifying the well-known simple pendulum dynamic system into a double pendulum, long-time-scale predictability is lost, and a completely new behavior appears. The TLS was pilot tested in a remote teaching setting with about 70 Italian high school students (16–17 years old). The pretest analysis shows that before participating in the activities, students held typical misconceptions about chaotic behavior. Analysis of the written responses collected during and after implementation shows that the proposed activities allowed students to grasp the two key ideas about nondeterministic chaos. A possible integration of the TLS with an online simulation is finally discussed.

Encryption of images is an important method that can effectively improve the security and privacy of crucial image data. Existing methods generally encrypt an image with a combination of scrambling and encoding operations. Currently, many applications require highly secure results for image encryption. New methods that can achieve improved randomness for both the scrambling and encoding processes in encryption are thus needed to further enhance the security of a cipher image. This paper proposes a new method that can securely encrypt color images. As the first step of the proposed method, a complete bit-level operation is utilized to scramble the binary bits in a color image to a full extent. For the second step, the bits in the scrambled image are processed with a sweeping operation to improve the encryption security. In the final step of encryption, a codebook that varies with evolutionary operations based on several chaotic systems is utilized to encrypt the partially encrypted image obtained in the second step. Experimental results on benchmark color images suggest that this new approach can securely encrypt color images and generate cipher images that remain secure under different types of attacks. The proposed approach is compared with several other state-of-the-art encryption approaches and the results show that it can achieve improved encryption security for cipher images. Experimental results thus suggest that this new approach can possibly be utilized practically in applications where color images need to be encrypted for content protection.

This manuscript introduces a piecewise linear decomposition method devoted to a class of fractional-order dynamical systems composed of piecewise linear (PWL) functions. Inspired by the Adomian decomposition method, the proposed technique computes an approximated solution of fractional-order PWL systems using only linear operators and specific constants vectors for each sub-domain of the PWL functions, with no need for the Adomian polynomials. The proposed decomposition method can be applied to fractional-order PWL systems composed of nth PWL functions, where each PWL function may have any number of affine segments. In particular, we demonstrate various examples of how to solve fractional-order systems with 1D 2-scroll, 4-scroll, and 4×4-grid scroll chaotic attractors by applying the proposed approach. From the theoretical and implementation results, we found the proposed approach eliminates the unneeded terms, has a low computational cost, and permits a straightforward physical implementation of multi-scroll chaotic attractors on ARMs and FPGAs digital platforms.

Secure data transmission is critical in public video surveillance applications such as tracking missing persons in large crowd gatherings. It may help the security personnel to monitor & record the live feeds from multiple cameras. However, this live streaming could be falsified. Various cryptographic techniques have been proposed to ensure the integrity of transmitted images, but continuous attacks still pose serious challenges to ensuring the integrity and authenticity of transmitted images. This work proposes a new chaotic-based watermarking technique to ensure the integrity of the transmitted video frames, and since the frames come from multiple cameras, the scheme should be as simple as possible and as fast as possible, the simplicity manifests in designing a fragile watermarking scheme therefore, the watermark is to be embedded in the spatial domain by direct manipulation of the values of the cover’s pixels. To ensure the security of the proposed scheme, chaotic maps have been deployed to reach an acceptable level of security with minimum computational overhead. The proposed scheme is novel in design in terms of the chaotic maps used and it is tested in challenging large gathering scenario. Experimental results demonstrate that the proposed method successfully locates any falsification in the transmitted images, and the performance of the scheme evaluates different types of attacks. Our testing results in good rate tamper of detection with TPR=99.2005, TNR=99.0777 and with a good quality of imperceptibility for the watermarked image with PSNR=43.82dB, SSIM=00.9955.

Information security has emerged as a key problem in encryption because of the rapid evolution of the internet and networks. Thus, the progress of image encryption techniques is becoming an increasingly serious issue and considerable problem. Small space of the key, encryption-based low confidentiality, low key sensitivity, and easily exploitable existing image encryption techniques integrating chaotic system and DNA computing are purposing the main problems to propose a new encryption technique in this study. In our proposed scheme, a three-dimensional Chen's map and a one-dimensional Logistic map are employed to construct a double-layer image encryption scheme. In the confusion stage, different scrambling operations related to the original plain image pixels are designed using Chen's map. A stream pixel scrambling operation related to the plain image is constructed. Then, a block scrambling-based image encryption-related stream pixel scrambled image is designed. In the diffusion stage, two rounds of pixel diffusion are generated related to the confusing image for intra-image diffusion. Chen's map, logistic map, and DNA computing are employed to construct diffusion operations. A reverse complementary rule is applied to obtain a new form of DNA. A Chen's map is used to produce a pseudorandom DNA sequence, and then another DNA form is constructed from a reverse pseudorandom DNA sequence. Finally, the XOR operation is performed multiple times to obtain the encrypted image. According to the simulation of experiments and security analysis, this approach extends the key space, has great sensitivity, and is able to withstand various typical attacks. An adequate encryption effect is achieved by the proposed algorithm, which can simultaneously decrease the correlation between adjacent pixels bymaking it near zero, also the information entropy increased. The number of pixels changing rate (NPCR) and the unified average change intensity (UACI) both are very near to optimal values. [ABSTRACT FROM AUTHOR]

This manuscript includes the design and evaluation of the new four 16×16 S-boxes for subbyte operation in image encryption applications and estimates their strength using the following parameters: Dynamic Distance, BIC non-linearity, Bijective, Non-linearity, Strict Avalanche Criterion (SAC), and Balanced criterion. The S-box matrix is designed by a new reconfigurable 3D-Chaotic PRNG. This PRNG is designed using four different 3D chaotic systems i.e. Lorenz, Chen, Lu, and Pehlivan's chaotic systems. This reconfigurable architecture of PRNG exploits the ODEs of these four attractors that fit all four chaotic systems in a single circuit. The first part of this manuscript is focused to develop hardware-efficient VLSI architecture. To demonstrate the hardware performance, the PRNG circuit is implemented in Virtex-5 (XC5VLX50T) FPGA. A performance comparison of proposed and existing PRNGs (in terms of timing performance, area constraint, power dissipation and statistical testing) has been presented in this work. The PRNG generates the 24-bit random number at 96.438-MHz. The area of FPGA is occupied by only 16.66 %, 1.08%, 0.33 %, and 1.15% of the available DSP blocks, slice LUTs, slice registers and slices respectively. The designed S-boxes using reconfigurable PRNG fulfill the following criteria: Dynamic Distance, BIC non-linearity, Bijective, Non-linearity, Strict Avalanche Criterion (SAC), and Balanced criterion. [ABSTRACT FROM AUTHOR]

Digital data is increasing rapidly in the world day by day. Information security is important during data exchange over the Internet. The way to securely transmit images over the network is through the image encryption technique. In the proposed cryptography system, the hybridization of Lorenz-Rossler chaotic systems is used, and a random number sequence is generated. The security analyses such as histogram, correlation, differential attack, information entropy, and duration analysis of the study are performed. It is seen that the proposed system performs well, especially in terms of correlation. Additionally, the performance of the developed embedded system platforms is compared after testing on Nvidia Jetson Nano and Xilinx PYNQ Z1 boards. The Nvidia Jetson Nano board is more performant than the Xilinx PYNQ Z1 board. The safety and feasibility of the proposed system have been demonstrated. [ABSTRACT FROM AUTHOR]

In this work, a new 3-D modified WINDMI chaotic jerk system with exponential and sinusoidal nonlinearities is presented and its dynamical behaviours and properties are investigated. Firstly, some properties of the system are studied such as equilibrium points and their stability, Lyapunov exponents and Kaplan-Yorke dimension. Also, we study the new jerk system dynamics using numerical simulations and analyses, including phase portraits, Lyapunouv exponent spectrum, bifurcation diagram and Poincaré map, 0-1 test. Next, we exhibit that the new 3-D chaotic modified WINDMI jerk system has multistability with coexisting chaotic attractors. Moreover, we design an electronic circuit using MultiSim 14.1 for real implementation of the modified WINDMI chaotic jerk system. Finally, we design an active synchronization scheme for the complete synchronization of the modified WINDMI chaotic jerk systems via backstepping control. [ABSTRACT FROM AUTHOR]

Schmidt, Jones, and Kennedy's (SJK) (2023, https://doi.org/10.1029/2022GL102530) critique of Scafetta (2022, https://doi.org/10.1029/2022GL097716) is flawed. Their assessment of the error of the ERA‐T2m 2011–2021 mean (≈0.10°C) is 5–10 times overestimated and contradicts published literature. SJK confused natural variability with random noise and mistook the error of the mean of a temperature chronology for the stochastic error of the regression parameter M of a nonphysical isothermal climate model (T(t) = M). SJK's allegations regarding the internal variability of the models, the role of the global climate model ensemble members, and other issues were partially addressed in Scafetta (2022, https://doi.org/10.1029/2022GL097716) and, later, more extensively in Scafetta (2023a, https://doi.org/10.1007/s00382-022-06493-w) where Scafetta's (2022, https://doi.org/10.1029/2022GL097716) conclusions were confirmed. Plain Language Summary: Schmidt, Jones, and Kennedy's (SJK) (2023, https://doi.org/10.1029/2022GL102530) assessment of the error of the ERA‐T2m 2011–2021 mean (σμ,95% = 0.10°C) incorrectly assumes that, during such a period, the global surface temperature was constant (T(t) = M) and that its interannual variability (ΔTi = Ti − T(ti) = Ti − M) was random noise. This is a nonphysical interpretation of the climate system that inflates the real error of the temperature mean by 5–10 times. In fact, the analysis of the ensemble of the global surface temperature members yields a decadal‐scale error of about 0.01–0.02°C, as reported in published records and deduced from the Gaussian error propagation formula (GEPF) of a function of several variables (such as the mean of a temperature sequence of 11 different years). Instead, SJK assessed such error using the standard deviation of the mean (SDOM), which is an equation that can only be used when there exists a distribution of repeated measurements of the same variable, which is not the present case. Furthermore, SJK misinterpreted Scafetta (2022, https://doi.org/10.1029/2022GL097716) and ignored published literature such as Scafetta (2023a, https://doi.org/10.1007/s00382-022-06493-w) that already contradicted their main claim about the role of the internal variability of the models and confirmed the results of Scafetta (2022, https://doi.org/10.1029/2022GL097716). Key Points: Schmidt, Jones, and Kennedy (2023, https://doi.org/10.1029/2022GL102530) assessed the error of the temperature mean by erroneously using the standard deviation of the mean instead of the error propagation formulaThe interannual temperature variability is physical signal, not noise; the 2011–2021 ERA5‐T2m mean error is not 0.10°C, but 0.01–0.02°CThe role of the internal variability of the models was extensively addressed in Scafetta (2023a, https://doi.org/10.1007/s00382-022-06493-w), who confirmed Scafetta (2022, https://doi.org/10.1029/2022GL097716) [ABSTRACT FROM AUTHOR]

In this work, we report a new chaotic population biology system with one prey and two predators. Our new chaotic population model is derived by introducing two nonlinear interaction terms between the prey and predator-2 to the Samardzija-Greller population biology system (1988).We show that the new chaotic population biology system has a greater value of Maximal Lyapunov Exponent (MLE) than the Maximal Lyapunov Exponent (MLE) of the Samardzija- Greller population biology system (1988).We carry out a detailed bifurcation analysis of the new chaotic population biology system with one prey and two predators. We also show that the new chaotic population biology model exhibits multistability with coexisting chaotic attractors. Next, we use the integral sliding mode control (ISMC) for the complete synchronization of the new chaotic population biology system with itself, taken as the master and slave chaotic population biology systems. Finally, for practical use of the new chaotic population biology system, we design an electronic circuit design using Multisim (Version 14.0).

Pandemics can have a significant impact on international health systems. Researchers have found that there is a correlation between weather conditions and respiratory diseases. This paper focuses on the non-linear analysis of respiratory diseases and their relationship to weather conditions. Chaos events may appear random, but they may actually have underlying patterns. Edward Lorenz referred to this phenomenon in the context of weather conditions as the butterfly effect. This inspired us to define a chaotic system that could capture the properties of respiratory diseases. The chaotic analysis was performed and was related to the difference in the daily number of cases received from real data. Stability analysis was conducted to determine the stability of the system and it was found that the new chaotic system was unstable. Lyapunov exponent analysis was performed and found that the new chaotic system had Lyapunov exponents of (+, 0, -, -). A dynamic neural architecture for input-output modeling of nonlinear dynamic systems was developed to analyze the findings from the chaotic system and real data. A NARX network with inputs (maximum temperature, pressure, and humidity) and one output was used to to overcome any delay effects and analyze derived variables and real data (patients number). Upon solving the system equations, it was found that the correlation between the daily predicted number of patients and the solution of the new chaotic equation was 90.16%. In the future, this equation could be implemented in a real-time warning system for use by national health services.

The damped van der Pol oscillator is a chaotic non-linear system. Small perturbations in initial conditions may result in wildly different trajectories. Controlling, or forcing, the behavior of a van der Pol oscillator is difficult to achieve through traditional adaptive control methods. Connecting two van der Pol oscillators together where the output of one oscillator, the driver, drives the behavior of its partner, the responder, is a proven technique for controlling the van der Pol oscillator. Deterministic artificial intelligence is a feedforward and feedback control method that leverages the known physics of the van der Pol system to learn optimal system parameters for the forcing function. We assessed the performance of deterministic artificial intelligence employing three different online parameter estimation algorithms. Our evaluation criteria include mean absolute error between the target trajectory and the response oscillator trajectory over time. Two algorithms performed better than the benchmark with necessary discussion of the conditions under which they perform best. Recursive least squares with exponential forgetting had the lowest mean absolute error overall, with a 2.46% reduction in error compared to the baseline, feedforward without deterministic artificial intelligence. While least mean squares with normalized gradient adaptation had worse initial error in the first 10% of the simulation, after that point it exhibited consistently lower error. Over the last 90% of the simulation, deterministic artificial intelligence with least mean squares with normalized gradient adaptation achieved a 48.7% reduction in mean absolute error compared to baseline.

Smartphones, digital cameras, and other imaging devices generate vast amounts of high-resolution colored images daily, stored on devices equipped with multi-core central processing units or on the cloud. Safeguarding these images from potential attackers has become a pressing concern. This paper introduces a set of six innovative image ciphers designed to be stronger, faster, and more efficient. Three of these algorithms incorporate the State-Based Tent Map (SBTM) Chaotic Pseudo Random Number Generator (CPRNG), while the remaining three employ a proposed modified variant, SBTMPi. The Grayscale Image Cipher (GIC), Colored Image Cipher Single-Thread RGB (CIC1), and Colored Image Cipher Three-Thread RGB (CIC3) showcase the application of the proposed algorithms. By incorporating novel techniques in the confusion and diffusion phases, these ciphers demonstrate remarkable performance, particularly with large colored images. The study underscores the potential of SBTM-based image ciphers, contributing to the advancement of secure image encryption techniques with robust random number generation capabilities.

Abstract Schmidt, Jones, and Kennedy’s (SJK) (2023, https://doi.org/10.1029/2022GL102530) critique of Scafetta (2022, https://doi.org/10.1029/2022GL097716) is flawed. Their assessment of the error of the ERA‐T2m 2011–2021 mean (≈0.10°C) is 5–10 times overestimated and contradicts published literature. SJK confused natural variability with random noise and mistook the error of the mean of a temperature chronology for the stochastic error of the regression parameter M of a nonphysical isothermal climate model (T(t) = M). SJK's allegations regarding the internal variability of the models, the role of the global climate model ensemble members, and other issues were partially addressed in Scafetta (2022, https://doi.org/10.1029/2022GL097716) and, later, more extensively in Scafetta (2023a, https://doi.org/10.1007/s00382-022-06493-w) where Scafetta's (2022, https://doi.org/10.1029/2022GL097716) conclusions were confirmed.

In this research, we examine the solution of ordinary fractional differential equations using Atangana’s beta derivative. Our approach is divided into two parts. First, we establish conditions under which the fractional differential equation has a unique solution. Next, we develop a numerical technique based on the Adams–Bashforth method and demonstrate its convergence. We then apply the numerical method to a test example to evaluate its efficacy. Finally, we analyze three nonlinear chaotic and hyperchaotic fractional dynamical systems, calculating the strange attractors for various fractional orders. We analyzed the fractional chaotic systems and showed that by changing the order of the fractional derivative, some properties of the system change, such as the area of the chaos attracting region.

This paper addresses the design of an observer-based adaptive neural network chaos synchronization scheme for a general class of uncertain chaotic systems. The controller consists of an adaptive neural network control law and an extended state observer. The parameterization of the designed extended observer and the sufficient stability conditions are derived in the light of the singular perturbation theory. The extended observer is incorporated into the controller to reconstruct the synchronization error vector as well as to estimate the error between an ideal control law and the actual control. These estimated error signals are utilized in the adaptation mechanism of the neural network weight vector. In the presented chaos synchronization method, the knowledge of the models of the master–slave systems is not required and the controller only needs the projective synchronization error for its implementation. Numerical simulations are performed along with a comparative study to demonstrate the efficiency and effectiveness of the suggested chaos synchronization approach. [ABSTRACT FROM AUTHOR]