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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.

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.

In the past decade, a large amount of important digital data has been created and stored in the form of color images; the protection of such data from undesirable accesses has become an important problem in information security. In this paper, a new approach based on an evolutionary framework is proposed for the secure encryption of color images. The image contents in a color image are first fully scrambled with a sequence of bit-level operations determined by a number of integer keys. A scrambled image is then encrypted with keys generated from an evolutionary process controlled by a set of chaotic systems. Analysis and experiments show that the proposed approach can generate encrypted color images with high security. In addition, the performance of the proposed approach is compared with that of a few state-of-the-art approaches for color image encryption. The results of the comparison suggest that the proposed approach outperforms the other approaches in the overall security of encrypted images. The proposed approach is thus potentially useful for applications that require color image encryption.

The database of faces containing sensitive information is at risk of being targeted by unauthorized automatic recognition systems, which is a significant concern for privacy. Although there are existing methods that aim to conceal identifiable information by adding adversarial perturbations to faces, they suffer from noticeable distortions that significantly compromise visual perception, and therefore, offer limited protection to privacy. Furthermore, the increasing prevalence of appearance anxiety on social media has led to users preferring to beautify their faces before uploading images. In this paper, we design a novel face database protection scheme via beautification with chaotic systems. Specifically, we construct the adversarial face with better visual perception via beautification for each face in the database. In the training, the face matcher and the beautification discriminator are federated against the generator, prompting it to generate beauty-like perturbations on the face to confuse the face matcher. Namely, the pixel changes produced by face beautification mask the adversarial perturbations. Moreover, we use chaotic systems to disrupt the order of adversarial faces in the database, further mitigating the risk of privacy leakage. Our scheme has been extensively evaluated through experiments, which show that it effectively defends against unauthorized attacks while also yielding good visual results.

Despite tangible advances in machine learning (ML) over the last few decades, many of the ML techniques still suffer from fundamental issues like overfitting and lack of explainability. These issues mandate requirements for mathematical rigor to ensure robust learning from observed data. In this context, topological invariants in data manifolds provide a rich representation of the underlying dynamical system, which can be utilized for developing a mathematically rigorous ML tool to characterize the dynamical behaviour and operational phases of the underlying process. This paper aims to investigate spectral invariants of symbolic systems for detecting changes in topological characteristics of data manifolds. A novel ML approach is proposed, where commutator norms are used on sequences of endomorphisms to symbolically describe dynamical systems on probability spaces with ergodic measures. The objective here is to detect topological invariants of data manifolds that can be used for signal processing, pattern recognition, and anomaly detection. The proposed ML approach is validated on models of selected chaotic dynamical systems for prompt detection of phase transitions. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Multimedia data play an important role in our daily lives. The evolution of internet technologies means that multimedia data can easily participate amongst various users for specific purposes, in which multimedia data confidentiality and integrity have serious security issues. Chaos models play an important role in designing robust multimedia data cryptosystems. In this paper, a novel chaotic oscillator is presented. The oscillator has a particular property in which the chaotic dynamics are around pre-located manifolds. Various dynamics of the oscillator are studied. After analyzing the complex dynamics of the oscillator, it is applied to designing a new image cryptosystem, in which the results of the presented cryptosystem are tested from various viewpoints such as randomness, time encryption, correlation, plain image sensitivity, key-space, key sensitivity, histogram, entropy, resistance to classical types of attacks, and data loss analyses. The goal of the paper is proposing an applicable encryption method based on a novel chaotic oscillator with an attractor around a pre-located manifold. All the investigations confirm the reliability of using the presented cryptosystem for various IoT applications from image capture to use it.

In chaotic entanglement, pairs of interacting classically-chaotic systems are induced into a state of mutual stabilization that can be maintained without external controls and that exhibits several properties consistent with quantum entanglement. In such a state, the chaotic behavior of each system is stabilized onto one of the system's many unstable periodic orbits (generally located densely on the associated attractor), and the ensuing periodicity of each system is sustained by the symbolic dynamics of its partner system, and vice versa. Notably, chaotic entanglement is an entropy-reversing event: the entropy of each member of an entangled pair decreases to zero when each system collapses onto a given period orbit. In this paper, we discuss the role that entropy plays in chaotic entanglement. We also describe the geometry that arises when pairs of entangled chaotic systems organize into coherent structures that range in complexity from simple tripartite lattices to more involved patterns. We conclude with a discussion of future research directions.

To address the problems of the high complexity and low security of the existing image encryption algorithms, this paper proposes a dynamic key chaotic image encryption algorithm with low complexity and high security associated with plaintext. Firstly, the RGB components of the color image are read, and the RGB components are normalized to obtain the key that is closely related to the plaintext, and then the Arnold transform is used to stretch and fold the RGB components of the color image to change the position of the pixel points in space, so as to destroy the correlation between the adjacent pixel points of the image. Next, the generated sequences are independently encrypted with the Arnold-transformed RGB matrix. Finally, the three encrypted images are combined to obtain the final encrypted image. Since the key acquisition of this encryption algorithm is related to the plaintext, it is possible to achieve one key per image, so the key acquisition is dynamic. This encryption algorithm introduces chaotic mapping, so that the key space size is 10180. The key acquisition is closely related to the plaintext, which makes the ciphertext more random and resistant to differential attacks, and ensures that the ciphertext is more secure after encryption. The experiments show that the algorithm can encrypt the image effectively and can resist attack on the encrypted image.

In this paper, the stabilization and synchronization of a complex hidden chaotic attractor is shown. This article begins with the dynamic analysis of a complex Lorenz chaotic system considering the vector field properties of the analyzed system in the Cn domain. Then, considering first the original domain of attraction of the complex Lorenz chaotic system in the equilibrium point, by using the required set topology of this domain of attraction, one hidden chaotic attractor is found by finding the intersection of two sets in which two of the parameters, r and b , can be varied in order to find hidden chaotic attractors. Then, a backstepping controller is derived by selecting extra state variables and establishing the required Lyapunov functionals in a recursive methodology. For the control synchronization law, a similar procedure is implemented, but this time, taking into consideration the error variable which comprise the difference of the response system and drive system, to synchronize the response system with the original drive system which is the original complex Lorenz system.

The COVID-19 epidemic challenges humanity in 2020. It has already taken an enormous number of human lives and had a substantial negative economic impact. Traditional compartmental epidemiological models demonstrated limited ability to predict the scale and dynamics of COVID-19 epidemic in different countries. In order to gain a deeper understanding of its behavior, we turn to chaotic dynamics, which proved fruitful in analyzing previous diseases such as measles. We hypothesize that the unpredictability of the pandemic could be a fundamental property if the disease spread is a chaotic dynamical system. Our mathematical examination of COVID-19 epidemic data in different countries reveals similarity of this dynamic to the chaotic behavior of many dynamics systems, such as logistic maps. We conclude that the data does suggest that the COVID-19 epidemic demonstrates chaotic behavior, which should be taken into account by public policy makers. Furthermore, the scale and behavior of the epidemic may be essentially unpredictable due to the properties of chaotic systems, rather than due to the limited data available for model parameterization., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2020 Elsevier Ltd. All rights reserved.)

Electroencephalogram (EEG) signals accumulate the brain's spiking activities using standardized electrodes placed at the scalp. These cumulative brain signals are chaotic in nature and vary depending upon current physical and/or mental activities. The anatomy of the brain is altered when dopamine releasing neurons die because of Parkinson Disease (PD), a neurodegenerative disorder. The resulting alterations force synchronized neuronal activity in β frequency components deep within motor region of the brain. This synchronization in the motor region affects the dynamical behavior of the brain activities, which induce motor related impairments in patient's limbs. Identification of reliable bio-markers for PD is active research area since there are no tests or scans to diagnose PD. We use embedding reconstruction, a tool from chaos theory, to highlight PD-related alterations in dynamical properties of EEG and present it as a potentially reliable bio-marker for PD related classification. We use Individual Component Analysis (ICA) to demonstrate that the strengthened synchronizations can be cumulatively collected from EEG channels over the motor region of the brain. We use this information to select the 12 EEG channels for classification of On and Off medication PD patients. Additionally, there is the strong synchronization between amplitude of higher frequency components and phase of β components for PD patients. This information is used to improve the performance of this classification. We apply embedding reconstruction to design a new architecture of a deep neural network called Dynamical system Generated Hybrid Network. We report that this network outperforms the state of the art in terms of classification accuracy of 99.2%(+0.52%) with approximately 24% of the computational resources. Apart from classification accuracy, we use well known statistical measures like specificity, sensitivity, Matthews Correlation Coefficient (MCC), F1 score, and Cohen Kappa score for the analysis and comparison of classification performances., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2020 Elsevier Ltd. All rights reserved.)

Introduction: The Internet of Things (IoT) comprises of various smart devices for the sharing of sensed data through online services. People will be directly contacted to check their health parameters and the reports will be collected centrally through smart devices. The requirement is protection of messages during the exchange of data between sender and receiver in order to tackle human malicious attacks. Various signature-based schemes are discussed in the literature to provide secure communication. Smart devices however require lightweight tasks by ensuring critical safety strengths. An important problem in the signature based method is that it incurs more computational expenses for signing and verification process in large numbers., Objectives: In this study, we introduced an efficient Short Signature Scheme (SSS) that uses Fractional Chaotic Map (FCM) for secure communication in IoT based smart devices, the security of which is closely related to a random oracle based on FCM assumption., Methods: In this study, we have designed new short signature scheme using FCM. The presented scheme consist of four sub-algorithm as follows: setup, key generation, signing and verification. We have used less rigorous operations based on the FCM to carry out signing and verification procedures, similar to human signing on valid documents and then verifying them as per witness., Results: The proposed SSS offers a better security assurance than currently established signature schemes. The key advantage of the SSS over the DSA schemes is that at the verification stage and signing period it takes less computation; it retains the degree of protection. The presented SSS takes less bandwidth for storage, communication, and computing resources; particularly applicable to wireless devices and smart cards., Conclusion: We concluded that the uses of fractional chaotic maps is more effective for secure communication in human-centered IoT to present a provably secure short signature technique., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2021 The Authors. Published by Elsevier B.V. on behalf of Cairo University.)

This paper introduces an intensive discussion for the dynamical model of the love triangle in both integer and fractional-order domains. Three different types of nonlinearities soft, hard, and mixed between soft and hard, are used in this study. MATLAB numerical simulations for the different three categories are presented. Also, a discussion for how the kind of personalities affects the behavior of chaotic attractors is introduced. This paper suggests some explanations for the complex love relationships depending on the impact of memory (IoM) principle. Lyapunov exponents, Kaplan-Yorke dimension, and bifurcation diagrams for three different integer-order cases show a significant dependency on system parameters. Hardware digital realization of the system is done using the Xilinx Artix-7 XC7A100T FPGA kit. Version 14.7 from the Xilinx ISE platform is used in both Verilog simulation and hardware implementation stages. The digital approach of such a system opens the door to predict the love relation after sensing the human personality. Also, this study will help in justifying more human emotions like happiness, panic, and fear accurately. Perhaps shortly, this study may combine with artificial intelligence to demonstrate Human-Computer interaction products., (© 2020 The Authors. Published by Elsevier B.V. on behalf of Cairo University.)

In order to obtain chaos with a wider chaotic scope and better chaotic behavior, this paper combines the several existing one-dimensional chaos and forms a new one-dimensional chaotic map by using a modular operation which is named by LLS system and abbreviated as LLSS. To get a better encryption effect, a new image encryption method based on double chaos and DNA coding technology is proposed in this paper. A new one-dimensional chaotic map is combined with a hyperchaotic Qi system to encrypt by using DNA coding. The first stage involves three rounds of scrambling; a diffusion algorithm is applied to the plaintext image, and then the intermediate ciphertext image is partitioned. The final encrypted image is formed by using DNA operation. Experimental simulation and security analysis show that this algorithm increases the key space, has high sensitivity, and can resist several common attacks. At the same time, the algorithm in this paper can reduce the correlation between adjacent pixels, making it close to 0, and increase the information entropy, making it close to the ideal value and achieving a good encryption effect.

We examine the quantum-classical correspondence from a classical perspective by discussing the potential for chaotic systems to support behaviors normally associated with quantum mechanical systems. Our main analytical tool is a chaotic system's set of cupolets, which are highly-accurate stabilizations of its unstable periodic orbits. Our discussion is motivated by the bound or entangled states that we have recently detected between interacting chaotic systems, wherein pairs of cupolets are induced into a state of mutually-sustaining stabilization that can be maintained without external controls. This state is known as chaotic entanglement as it has been shown to exhibit several properties consistent with quantum entanglement. For instance, should the interaction be disturbed, the chaotic entanglement would then be broken. In this paper, we further describe chaotic entanglement and go on to address the capacity for chaotic systems to exhibit other characteristics that are conventionally associated with quantum mechanics, namely analogs to wave function collapse, various entropy definitions, the superposition of states, and the measurement problem. In doing so, we argue that these characteristics need not be regarded exclusively as quantum mechanical. We also discuss several characteristics of quantum systems that are not fully compatible with chaotic entanglement and that make quantum entanglement unique.

In this research article, we propose a new structure namely inverse left almost semigroup (LA-semigroup) by adding confusion in our proposed image encryption scheme along with discrete and continuous chaotic systems in order to complement the diffusion characteristics. The performance analysis was conducted in terms of the correlation analysis, the pixels uniformity analysis, the sensitivity analysis, the information entropy analysis, and pixels difference based measurements. The results show that the proposed algorithm has better information security properties, which can provide strong security and a high performance.

Robust synchronization of master slave chaotic systems are considered in this work. First an approximate model of the error system is obtained using the ultra-local model concept. Then a Continuous Singular Terminal Sliding-Mode (CSTSM) Controller is designed for the purpose of synchronization. The proposed approach is output feedback-based and uses fixed-time higher order sliding-mode (HOSM) differentiator for state estimation. Numerical simulation and experimental results are given to show the effectiveness of the proposed technique., (Copyright © 2018 ISA. Published by Elsevier Ltd. All rights reserved.)

In this paper, we introduce a new chaotic system that is used for an engineering application of the signal encryption. It has some interesting features, and its successful implementation and manufacturing were performed via a real circuit as a random number generator. In addition, we provide a parameter estimation method to extract chaotic model parameters from the real data of the chaotic circuit. The parameter estimation method is based on the attractor distribution modeling in the state space, which is compatible with the chaotic system characteristics. Here, a Gaussian mixture model (GMM) is used as a main part of cost function computations in the parameter estimation method. To optimize the cost function, we also apply two recent efficient optimization methods: WOA (Whale Optimization Algorithm), and MVO (Multi-Verse Optimizer) algorithms. The results show the success of the parameter estimation procedure., Competing Interests: The authors declare no conflict of interest.

In this paper, the stabilization problem of actuators saturation in uncertain chaotic systems is investigated via an adaptive PID control method. The PID control parameters are auto-tuned adaptively via adaptive control laws. A multi-level augmented error is designed to account for the extra terms appearing due to the use of PID and saturation. The proposed control technique uses both the state-feedback and the output-feedback methodologies. Based on Lyapunov׳s stability theory, new anti-windup adaptive controllers are proposed. Demonstrative examples with MATLAB simulations are studied. The simulation results show the efficiency of the proposed adaptive PID controllers., (Copyright © 2016 ISA. Published by Elsevier Ltd. All rights reserved.)

We aimed to determine how age-associated changes in mechanisms extrinsic and intrinsic to pacemaker cells relate to basal beating interval variability (BIV) reduction in vivo. Beating intervals (BIs) were measured in aged (23-25 months) and adult (3-4 months) C57BL/6 male mice (i) via ECG in vivo during light anesthesia in the basal state, or in the presence of 0.5 mg mL(-1) atropine + 1 mg mL(-1) propranolol (in vivo intrinsic conditions), and (ii) via a surface electrogram, in intact isolated pacemaker tissue. BIV was quantified in both time and frequency domains using linear and nonlinear indices. Although the average basal BI did not significantly change with age under intrinsic conditions in vivo and in the intact isolated pacemaker tissue, the average BI was prolonged in advanced age. In vivo basal BIV indices were found to be reduced with age, but this reduction diminished in the intrinsic state. However, in pacemaker tissue BIV indices increased in advanced age vs. adults. In the isolated pacemaker tissue, the sensitivity of the average BI and BIV in response to autonomic receptor stimulation or activation of mechanisms intrinsic to pacemaker cells by broad-spectrum phosphodiesterase inhibition declined in advanced age. Thus, changes in mechanisms intrinsic to pacemaker cells increase the average BIs and BIV in the mice of advanced age. Autonomic neural input to pacemaker tissue compensates for failure of molecular intrinsic mechanisms to preserve average BI. But this compensation reduces the BIV due to both the imbalance of autonomic neural input to the pacemaker cells and altered pacemaker cell responses to neural input., (© 2016 The Authors. Aging Cell published by the Anatomical Society and John Wiley & Sons Ltd.)

Many physical systems are described by nonlinear differential equations that are too complicated to solve in full. A natural way to proceed is to divide the variables into those that are of direct interest and those that are not, formulate solvable approximate equations for the variables of greater interest, and use data and statistical methods to account for the impact of the other variables. In the present paper we consider time-dependent problems and introduce a fully discrete solution method, which simplifies both the analysis of the data and the numerical algorithms. The resulting time series are identified by a NARMAX (nonlinear autoregression moving average with exogenous input) representation familiar from engineering practice. The connections with the Mori-Zwanzig formalism of statistical physics are discussed, as well as an application to the Lorenz 96 system.

This paper discusses the continuous effect of the fractional order parameter of the Lü system where the system response starts stable, passing by chaotic behavior then reaching periodic response as the fractional-order increases. In addition, this paper presents the concept of synchronization of different fractional order chaotic systems using active control technique. Four different synchronization cases are introduced based on the switching parameters. Also, the static and dynamic synchronizations can be obtained when the switching parameters are functions of time. The nonstandard finite difference method is used for the numerical solution of the fractional order master and slave systems. Many numeric simulations are presented to validate the concept for different fractional order parameters.

In this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values. We show that the extremes of so-called physical observables are distributed according to the classical generalised Pareto distribution and derive explicit expressions for the scaling and the shape parameter. In particular, we derive that the shape parameter does not depend on the chosen observables, but only on the partial dimensions of the invariant measure on the stable, unstable, and neutral manifolds. The shape parameter is negative and is close to zero when high-dimensional systems are considered. This result agrees with what was derived recently using the generalized extreme value approach. Combining the results obtained using such physical observables and the properties of the extremes of distance observables, it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow. Moreover, by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory, we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments, of the partial dimensions, and of the Kaplan-Yorke dimension of the attractor. Preliminary numerical investigations provide encouraging results on the applicability of the theory presented here. The results presented here do not apply for all combinations of Axiom A systems and observables, but the breakdown seems to be related to very special geometrical configurations.