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Recently, Yin et al. published the article titled "Identifying critical nodes in complex networks based on distance Laplacian energy" in which they proposed a new vertex centrality called distance Laplacian centrality (D L C) to identify critical nodes. However, we find that D L C can only be used in specific networks, which are still connected after removing any individual node. Meanwhile, some incorrect properties were used. Therefore, in this paper, we explain the reason that the property is incorrect and make corrections. In addition, we provide two centrality methods based on two different distance energies to solve the problem caused by disconnected networks. • We find some errors in the main results of the paper (ID: 114487). • We make correction on the main theorem. • We provide two centrality methods based on two different distance energies. [ABSTRACT FROM AUTHOR]

A serious error exists in the above paper. [ABSTRACT FROM AUTHOR]

In the paper by Jalal et al. [1], the authors present phase portraits of a differential system exhibiting chaotic behavior with line equilibria. This commentary identifies inaccuracies in the provided figures and offers corrected versions. Specifically, discrepancies were found in Fig. 1 (phase portraits with parameters a = 15 and b = 1) and Fig. 2 (phase portraits with parameters a = 0 and b = 5). Corrected figures, along with MATLAB codes for verification, are provided. These corrections are discussed in the context of chaotic systems with line equilibria, referencing Jafari and Sprott's work [2], which explores similar systems. [ABSTRACT FROM AUTHOR]

Two very serious errors exist in the above paper. [ABSTRACT FROM AUTHOR]

This comment pertains to the novel modified sub-ODE method introduced in the aforementioned paper. We have established, through a straightforward calculation, that the method employed in the paper is flawed. Moreover, certain exact solutions, as presented in the paper, are also erroneous in the context of both the auxiliary equation method associated with the aforementioned method and the model under investigation, namely the nonlinear low-pass electrical transmission lines model. It is noteworthy that the authors of the paper have asserted on two occasions that they meticulously validated all derived solutions by re-substituting them into the relevant equations, utilizing the Mathematica software. [ABSTRACT FROM AUTHOR]

Some serious errors exist in the above paper. [ABSTRACT FROM AUTHOR]

In ecosystems, cooperative behavior is universal and can dramatically improve a species' chances of survival. Nevertheless, the situations that can occur when different species with cooperative tendencies interact are veiled. To explore such a situation, in this paper, we investigate how cooperative behavior can affect biodiversity in the population system. Based on the spatial rock–paper–scissors (RPS) game, which incorporates the relative power between predator and prey species, we redefine the competition rate to facilitate cooperative behavior. Competition rates are modulated by the sensitivity parameter, which regulates alterations in competition rates stemming from variations in predator–prey population disparities. Through comprehensive numerical analysis, we have demonstrated compelling evidence confirming the nature of cooperative behavior in maintaining biodiversity. The sensitivity parameter acts as a double-edged sword; it hampers biodiversity when it falls below a certain level. Conversely, when it exceeds the threshold, it supports the maintenance of biodiversity. From snapshots and the coefficient analysis based on spatial autocorrelation, we found that empty sites are essential to promote coexistence as resource nodes. Compared with previous studies in spatial RPS games, our findings suggest that simple modification of a competition rate rather than exploiting cooperative games can realize the cooperative behavior of cyclically competing populations, and biodiversity is sensitively affected by cooperation. • We define cooperative behavior between species by redefining competition rates. • Cooperation affects competition and exchange sensitively, affecting biodiversity. • Empty spaces play an important role and are crucial in promoting coexistence. [ABSTRACT FROM AUTHOR]

At the forefront of brain-computer interface (BCI) technology exploration, the synchronization phenomenon in neural networks demonstrates significant potential for application. This paper focuses on the memory characteristics and nonlinear dynamic behavior of memristors, an emerging electronic component. By innovatively integrating memristors, phototubes, and FitzHugh-Nagumo (FHN) circuits, we construct a memristive photosensitive neuron (MPN) model, aiming to explore its unique role in neural network synchronization. Utilizing the memristors' inherent memory capabilities and nonlinear properties, the MPN model mimics the dynamic behavior of biological neurons. Experiments show that synchronization is highly sensitive to the memristor's initial values, revealing intricate synchronization regulation mechanisms. Furthermore, we find that chaotic electrical currents, as environmental disturbances, have dual effects—either promoting or inhibiting MPN network synchronization—depending on specific parametric conditions. To simulate a realistic biological neural network and achieve efficient coupling among multiple MPN units, this paper adopts a Hopfield network structure. The results indicate that this structure significantly enhances the synchronization stability of the system, reduces sensitivity to initial conditions, and mitigates the adverse effects of chaotic currents. This research not only enhances our understanding of neural network synchronization but also provides novel theoretical support and technical pathways for the development of BCI technology, indicating broad application prospects in neuroscience, rehabilitative medicine, and other fields. • A model was established to explore how initial values of memristors affect firing patterns and synchronization behaviors. • Chaotic currents were introduced to investigate the impact of external environments on neuronal synchronization. • A realistic Hopfield neural network was built for a further and deeper exploration of the network synchronization. [ABSTRACT FROM AUTHOR]

The level of privacy may vary across different parts of an image. This paper proposes a semi-visual obfuscation algorithm for images that takes into account the varying levels of privacy in different areas of the image. Firstly, we present a novel One-dimensional Uniform Chaotic Amplifier (1_DUCA) aimed at expanding the parameter range and enhancing the uniformity of the standard one-dimensional chaotic map. Next, we employ a detection algorithm or autonomous frame selection to identify the precise location of the area with strong privacy. Finally, we apply noise to blur the selected area and conceal vital bit information within the image. At this point, the image has certain visual effects, and only people with prior knowledge can recognize the image. Furthermore, in the last stage of image encryption, we employ a weight scrambling and high-low bit coupled diffusion technology to completely obscure the visual effects of the image. It is noteworthy that the experimental results and performance analysis have verified the practicality and security of the encryption algorithm. Moreover, they have also demonstrated the robust amplification effect of the employed amplifier. • A semi-visual fuzzy encryption scheme for images with varying privacy levels is proposed. • This paper presents a self-hiding fuzzy algorithm using noise blur and information hiding techniques. • Proposes a scrambling method that perturbs pixel positions based on their own weight, enhancing security and robustness. • This encryption scheme introduces a novel high-low coupling diffusion method to enhance pixel coupling. [ABSTRACT FROM AUTHOR]

Variational inequalities are a universal optimization paradigm that incorporate classical minimization and saddle point problems. Nowadays more and more tasks require to consider stochastic formulations of optimization problems. In this paper, we present an analysis of a method that gives optimal convergence estimates for monotone stochastic finite-sum variational inequalities. In contrast to the previous works, our method supports batching, does not lose the oracle complexity optimality and uses an arbitrary Bregman distance to take into account geometry of the problem. Paper provides experimental confirmation to algorithm's effectiveness. • We consider stochastic variational inequalities and present a new method for them. • An analysis of the method gives optimal convergence estimates. • Our method supports batching, does not lose the oracle complexity optimality. • Our method uses an arbitrary Bregman distance to take into account geometry of the problem. [ABSTRACT FROM AUTHOR]

Aiming at the security problems of medical images at present, the paper proposed a retinal image compression and encryption scheme based on chaotic mapping. The retina image is compressed and the compressed information is embedded into multiple color images. This paper adopts different compression methods for different areas to ensure the compression rate. The sensitive area of retina image is identified by Sobel operator edge detection method. Huffman lossless compression is used for sensitive area, and JPEG lossless compression is used for non-sensitive area, which meets the accuracy requirement of medical image and attains good compression efficiency. The Least Significant Bit(LSB) algorithm is combined with chaotic mapping, and the CLSB algorithm is proposed to embed the compressed information into the color image. The embedding position of the algorithm is determined by the random chaotic sequence. Compared with recent research schemes, the CLSB algorithm improves the PSNR of decrypted images by 5.64 %. Correspondingly, to further ensure the security of retinal images, multiple carrier images are jointly encrypted. The proposed Sine-Singer chaotic system(SSCS) has better chaotic characteristics and more complex dynamical behaviors. Experimental results have shown that when applied in encryption algorithms, it exhibits improved robustness and is capable of resisting common attacks such as cropping and salt-and-pepper noise. • A new chaotic system SSCS is proposed, and proved that the chaos effect is better than that of Sine and Singer maps. • This paper proposes the chaotic LSB algorithm, combining the LSB method with the SSCS , improving the embedding rate. • This paper proposes a novel hidden-embed encryption model designed for encrypting medical or other sensitive images. [ABSTRACT FROM AUTHOR]

The term resilience has become a relevant research topic in last years due to the interest of being applied to different fields and to many types of complex networks. A recently published paper proposed a theoretical and mathematical framework for a concept of local and discrete resilience, from a system performance recovery point of view. This paper deepens on this definition and provides bounds and exact values for the local resilience of some classes of networks. In addition, a new measure is defined, the composed resilience, which allows to study and compare networks from a general point of view. Several computational experiments are presented for this new parameter in different types of synthetic and real networks, providing a general measure for the resilience in complex networks. • New analytical results on local resilience concept are developed. • General bounds are obtained for any performance function and any recovery process. • Exact expressions are provided when the performance function is the global efficiency. • A new general tool to measure the resilience is introduced, the composed resilience. • Composed resilience has been computed on graphs, on synthetic and real networks. • The key point is the combination of advances in basic science with the applications. [ABSTRACT FROM AUTHOR]

In the framework of evidence theory, this paper proposes a new measure of uncertainty and introduces an entirely new unified framework for analyzing the complexity of multivariate time series. The theoretical foundation of this paper is random sets, viewing multivariate sequences as a new type of set variable, and using the belief theory framework to describe their behavior. Set variables are a natural extension of point variables, thus providing stronger universality in the analysis of multivariate time series. We tested our proposed new analytical framework on the logistic map and used the Tsallis-Deng Structure Entropy introduced in this paper to analyze the generated multivariate time series. Compared to the traditional Deng entropy, Tsallis-Deng Structure Entropy can adjust the sensitivity to time series complexity by tuning its structural parameter p. As the parameters of the logistic map change, the changes in Tsallis-Deng Structure Entropy show a high correlation with the Lyapunov exponent. In addition, we applied the proposed framework to analyze a multivariate epileptic EEG dataset. The results showed that using the novel evidence theory-based analysis framework for the 23-channel multivariate time series, the average classification accuracies of Deng entropy, Tsallis-Deng entropy, and Tsallis-Deng structural entropy for 24 epilepsy patients were 79 %, 74 %, and 82 %, respectively. In contrast, the average recognition accuracy using the traditional probabilistic framework based on multivariate embedding theory was 64 %. The evidence theory-based framework generally outperformed the multivariate embedding theory, with Tsallis-Deng structural entropy achieving the best performance. These results indicate that the proposed evidence theory-based framework not only avoids the dimensionality curse of the multivariate embedding theory but also better quantifies the complexity of multivariate time series from the same system and distinguishes different types of multivariate time series. • Using DS evidence theory and ordinal methods to transform multivariate time series data into mass functions. • A new measure of uncertainty: Belief Tsallis-Deng Structure Entropy. • In the classification of multichannel electric time series, the proposed method achieves better classification accuracy. [ABSTRACT FROM AUTHOR]

Propagation is crucial for acquiring information, and learning involves deep information processing. Imitation dynamics, commonly used in spatial public goods games, represents strategy propagation in society, while learning dynamics enables agents to self-learn through environmental interactions. In this paper, Q-learning and the Fermi update rule are used to compare differences between learning dynamics and imitation dynamics in simulation experiments. Our study finds that Q-learning is continuously updating the Q-table during its evolution, forming a heterogeneous gain matrix. However, Q-learning's interaction with the environment is indirect, with almost no network reciprocity, and cannot form clusters in contrast to the Fermi update rule. The Fermi update rule focus only on immediate payoffs. Furthermore, we combine imitation dynamics and learning dynamics, integrating the advantages of both. Defectors and cooperators form a special semi-stable checkerboard-like state. The update rules, the number of states, and the configuration of conversion actions as strategies have led to this special structure. Moreover, we found that parameter choices in the new evolutionary dynamics affect the percentage of cooperation and evolutionary convergence. Overall, this paper offers new insights into learning dynamics and imitation dynamics and provides a foundation for integrating complex mechanisms of evolution. [ABSTRACT FROM AUTHOR]

Knock-in external barrier options as a special kind of barrier options are popular in the option exchange markets, which contain two assets, one decides the payoffs, and the other decides whether the options can be exercised. In this paper, the normal pricing formulas of knock-in external barrier options based on uncertain stock model including down-in call options and up-in put options are derived by means of the fairness principle. To determine how parameters affect the normal price, sensitivity analysis for the normal pricing formulas of down-in call options and up-in put options are presented. Assume that the price of assets are modeled by Liu's stock model, the explicit pricing formulas for down-in call options and up-in put options are obtained successfully. Meanwhile, the sensitivity analysis for the explicit price of down-in call options and up-in put options are investigated. Moreover, the numerical algorithms and numerical examples for the above explicit pricing formulas of down-in call options and up-in put options are discussed in this paper. • The normal pricing formulas of knock-in external barrier options based on uncertain stock model are derived by the fairness principle. • The explicit pricing formulas of knock-in external barrier options based Liu's stock model are obtained. • Sensitivity analysis for the normal and explicit pricing formulas of knock-in external barrier options are discussed. • Numerical algorithms for the explicit pricing formulas of knock-in external barrier options are designed. [ABSTRACT FROM AUTHOR]

In previous works (Inaba and Kousaka, 2020; Inaba and Tsubone, 2020; Kato et al. 2022; Inaba et al., 2023), significant bifurcation structures referred to as nested mixed-mode oscillations (MMOs) were discovered in the forced Bonhoeffer-van der Pol (BVP) oscillator. Since the experimental discovery of MMOs, unnested MMO-incrementing bifurcations (MMOIBs) have been observed in numerous systems, including the Belousov–Zhabotinsky reaction, chlorite–thiosulfate reaction, and hydrodynamic and electrochemical systems; in such systems, daughter oscillations [ L 1 s 1 , L 2 s 2 × m ] can be observed for successive values of m. In this paper, we investigate high-order singly nested MMOs of the form [ C 0 , m , D 0 , m × p ] for increasing values of m , i.e., C 0 , m is followed by D 0 , m repeated p times, where C 0 , m ≡ [ C 0 , D 0 × m ] , D 0 , m ≡ [ C 0 , D 0 × (m + 1) ] , and C 0 and D 0 are adjacent fundamental MMOs, e.g., C 0 = 1 s and D 0 = 1 s + 1 . This work extends our previous analysis that was conducted on the lowest-order case (m = 1), and we demonstrate that the high-order (m ≥ 2) singly nested MMOs can indeed occur. We investigate the cases of m = 2 , 3 , 4 , and 10, although we note that the regions in which nested MMOs can be observed become increasingly narrow for larger values of m. These bifurcations are confirmed via one-parameter bifurcation diagrams, time-series waveforms, and first return plots. Furthermore, we present notable similarities between the Nagumo–Sato piecewise-linear discontinuous one-dimensional map and the forced BVP oscillator. • Nested mixed-mode oscillations (NMMOs) represent a significant bifurcation phenomenon. • It is known that parent–daughter processes generate [ C 0 , D 0 × m ] for successive m. • NMMOs generate [ C 0 , m , D 0 , m × p ] for successive p. • In the above notation, C 0 , m = [ C 0 , D 0 × m ] and D 0 , m = [ C 0 , D 0 × (m + 1) ]. • This paper shows that NMMOs occur for successive p and increasing values of m. [ABSTRACT FROM AUTHOR]

The existence, uniqueness, exponential stability with the trajectory (T-)controllability and optimal control results for mild solutions to the fractional neutral stochastic differential system (FNSDSs) are presented in this article. To demonstrate the results, the concept of bounded integral contractors combined with the regularity, a weaker notion of Lipschitz continuity, with stochastic approach and sequencing techniques are used. In contrast to previous publications, we do not need to specify the induced inverse of the controllability operator to prove the stability results, and the relevant nonlinear function does not have to meet the Lipschitz condition. Furthermore, exponential stability result for FNSDSs with Poisson jump via impulsive integral inequality is established following the trajectory (T-) controllability for higher-order FNSDSs via integral contractors with the help of Gronwall's inequality and the optimal control problem for higher-order FNSDSs via Balder's theorem. A numerical example is discussed to justify the theory. Finally, a filtration model and the real life stochastic Kelvin–Voigt and Maxwell models with the numerical simulation are demonstrated to satisfy the acquired results. This paper extends all previous works having the nonlinear Lipschitz continuous operators. • The wellposedness for the fractional neutral stochastic differential system are presented. • The concept of bounded integral contractors and sequencing technique are used. • Based on the editor request we study a new article called Trajectory (T-) Controllability of the proposed system. • T-controllability via Integral contractor is not available in the literature for the stochastic fractional system. • Then we study the optimal controllability. This paper is the unique combinations of all three concepts. • We do not need to match with the induced inverse of the controllability operator to prove the stability results. • The relevant nonlinear function does not have to meet the Lipschitz condition. • Integral contractor is a weaker notion of the Lipschitz continuity. • A real-life stochastic Kelvin–Voigt and Mexwell applications are demonstrated to satisfy the acquired results. • This paper extends all previous works in this direction. [ABSTRACT FROM AUTHOR]

In the last decade, hundreds of papers have been published on fractional modeling. The approach is often similar, with the motivation of incorporating a "memory effect": take the integer-order differential-equation model, and replace the ordinary derivative to the left-hand side by a fractional operator/derivative. Does this make sense? Which is the memory there? What is the physical and biological interpretation of fractional models? This paper aims at investigating these important issues. With fluxes, transitions, and concepts from survival analysis (hazard function), we give a meaning for Caputo compartmental models and their non-Markovian property. Essentially, a fractional index is connected with the instantaneous risk of leaving a compartment, conditioned on the past stay in there. We distinguish between Caputo models that are purely fractional (i.e., the usual ones) and partially fractional (i.e., a mix of ordinary and Riemann–Liouville components), and we explore initialization as well. We build new Euler-type numerical schemes, rooted in probability, and assess them for different models: a simple death process, logistic growth, SIR equations, the dynamics of an HIV infection, and the evolution of the smoking habit. A detailed discussion on the pros and cons of the fractional methodology is made along the article. • Which is the role of a fractional operator in an equation? What is memory? • The paper investigates these issues, for Caputo fractional compartmental models. • We analyze the hazard function (risk of transition), initialization, non-Markovianity. • Some Caputo models in the literature are unphysical. Critical evaluation is included. • Numerical experiments are conducted with probabilistic Euler-type discrete schemes. [ABSTRACT FROM AUTHOR]

The exponential stabilization of stochastic quantum systems (SQS) under time-delay noise-assisted feedback is addressed based on the exponential stability of the stochastic differential delay equation (SDDE) in this paper. We show that if the stochastic differential equation (SDE), which covers the dynamical models of SQS under noise-assisted feedback, is exponentially stable, then the corresponding SDDE is also exponentially stable when the time-varying delay is bounded by a positive number, which is given in an implicit expression and can be computed numerically. On this basis, the results for the general SDDE are applied to the SQS under time-delay noise-assisted feedback, and the corresponding time-delay boundary is obtained. Furthermore, we show the time-delay boundary in numerical simulations. • A upper bound on delay time that ensures exponential stabilization of SDDE is given. • The results from general SDE are applied to the stochastic quantum systems (SQS). • The results of this paper are suitable for a wide range of feedback strategies. • We present a way of studying exponential stabilization of SQS under delay feedback. [ABSTRACT FROM AUTHOR]

The tracking problem of Julia sets of the SIS (Susceptible–Infectious–Susceptible) model with saturated healing function under noise perturbation is investigated. Firstly, a discrete version of the SIS model with saturated healing function and its Julia set are introduced. Secondly, the structure of the Julia set are discussed, and the result shows that the filled-in Julia set of this model can be presented as a bounded set with positive measure and an unbounded set. The numerical result shows that the measure of the latter is almost zero. Then, the tracking problem of the Julia sets for the SIS model with saturated healing function is proposed. To address this problem, differential dynamic programming (DDP) and model predictive control (MPC) are used to design controllers. Controllers with different objective functions are compared across their performance. At last, a metric for evaluating the tracking performance is suggested, and a more effective objective function is proposed based on this metric. • The structure of the Julia set is given. Many papers only discuss the shape of the Julia set by showing the figure of the Julia set. Few of them give a certain expression of the Julia set. However, in this paper, the structure of the Julia set is provided. • The tracking problem of the Julia set under noisy disturbance is proposed. The Julia set is important in distinguishing whether the value makes the system keep bounded. However, a small disturbance may cause the value in the filled-in Julia set to go out of the Julia set and finally tend to be infinite. The tracking problem of the Julia sets under noisy disturbance is introduced to keep the value in the filled-in Julia set. • The metric for tracking performance is proposed. To evaluate the tracking effect, the average tracking rate for single point and filled-in Julia set is proposed. It can numerically represent the difference between the Julia set under noisy disturbance and the Julia set without noisy disturbance. In addition, It can measure the difference between the value of iteration under noise disturbance and the value of iteration without noise disturbance. [ABSTRACT FROM AUTHOR]

The slowly forced Duffing system has been found to exhibit unique fast–slow behaviors in descending frequency switching scheme, the typical of which is the sliding bursting, which is instructive for understanding the dynamics of ubiquitous frequency switching systems in scientific research and practical engineering. This paper is devoted to further refining the fast–slow dynamics of the slowly forced Duffing system with another commonly encountered frequency switching scheme, i.e., the ascending frequency switching, characterized by switching the frequency synchronously according to the increase or decrease of state variable values. Taking the forcing amplitude as an example, this paper provides a theoretical way to fully summarize the fast–slow behaviors in frequency switching systems with the variation of parameter values based on the proposed superposition analysis of one- and two-parameter bifurcations of subsystems. As a result, two typical bifurcation structures and eight threshold windows with distinct switching vector fields therein are identified, inducing up to ten different bursting patterns. Among them, several novel fast–slow dynamics, such as multiple jumps hysteresis loop formed by boundary equilibrium bifurcations and the switching failure phenomena, are presented and investigated. In particular, the underlying mechanism of threshold modulation, i.e., the evolution of unconventional bifurcations, is also found. These findings contribute to complementing and contrasting the existing studies on the fast–slow dynamics of frequency switching systems. • Theoretical way for identifying threshold windows under different parameter values. • Novel bursting patterns induced by several novel multi-jump hysteresis loops. • Underlying mechanisms of the threshold modulating the dynamical behaviors. • Effect of frequency switching scheme and system parameters on the fast–slow dynamics. [ABSTRACT FROM AUTHOR]

In recent years, the connection between emotions and cooperative behavior has received widespread attention. Aiming at the limitations of the emotional state hypothesis in the traditional spatial evolutionary game model, this study introduces a novel memory-based emotion mechanism. By treating emotion as a variable that can be measured and accumulated, and incorporating it into a network reciprocity model, this paper explores how emotion affects cooperative behavior in social dilemmas with different dilemma strengths. It is worth noting that the accumulated length of memory for emotions is variable. Using Monte Carlo simulation, the study analyzed the roles of positive and negative emotions in three social dilemmas and their specific effects on cooperative tendencies. The findings of this paper emphasize the importance of emotional dynamics and suggest that emotions can address some difficult problems. We propose potential applications of this mechanism in social governance and policymaking. • Introduce a memory-based emotion mechanism into network reciprocity, which quantifies and accumulates emotions as factors influencing cooperative behavior. • Emotions emerge along with the process of comparison with local neighbors, with the length of memory being variable. • Emotions are categorized as positive and negative. Positive emotion can facilitate emergence of cooperative behavior, while negative emotion makes the evolution of cooperation more difficult. • From the analysis of dilemma strength across various models, it is evident that integrating emotion into evolutionary games can solve the dilemma problem. [ABSTRACT FROM AUTHOR]

Noise, as a form of destructive signal interference, brings challenges of varying degrees in the field of wireless communications. In order to reduce the impact of noise on the communication system, this paper combines four kinds of noise reduction schemes to compare and analyze the anti-noise performance in the communication system. In this paper, a chaotic keying system based on carrier modulation is firstly proposed as the basis, which can implicitly propagate an extra part of bits to improve the transmission rate. The system is compared with other systems through formula derivation and simulation analysis to analyze the advantages brought by carrier modulation. Next, four noise reduction schemes, namely, moving average (MA) noise reduction, autocorrelation (AC) noise reduction, variational modal decomposition (VMD) and wavelet packet decomposition (WPD), are combined to compare and analyze the performance of each scheme. By improving or balancing the advantages of some of the schemes, the overall performance is more obviously improved. Finally, LDPC channel coding is introduced, and the anti-noise performance of the communication system after the introduction of LDPC coding is analyzed through Monte Carlo simulation. After all the above simulations, the carrier modulation can efficiently improve the system transmission rate, which is the best choice for today's high-speed communication. The four noise reduction schemes can effectively improve the anti-noise performance of the system, each has its own advantages and disadvantages, and the performance is better after improvement and balancing. After the introduction of the LDPC coding, the performance of the system has a qualitative leap, which provides a reference for the future research. All the above schemes have good theoretical value and can provide good reference for practical engineering applications. • A superior chaotic system is proposed to be used as a test system for detecting the effects of signal processing schemes. • The noise reduction effects of the four schemes when applied to chaotic system are proposed and comparatively analyzed. • The paper introduces LDPC code and explores its combination with the chaotic system. [ABSTRACT FROM AUTHOR]

Hammerstein structure is commonly used for describing nonlinear dynamic characteristics, and its identification is a basic premise of nonlinear system analysis and control. This paper investigates online identification methods for a class of Hammerstein nonlinear systems, which consists of a nonlinear memoryless element followed by a linear output-error subsystem. The unmeasurable noise-free output of the linear subsystem makes the model parameters cannot be directly estimated by traditional identification methods. To address this difficulty, by using a series of weighted particles to adaptively approximate the posterior probability density function of the unmeasurable noise-free output, this paper proposes a particle filter-based stochastic gradient algorithm. Moreover, to enhance the data utilization and estimation accuracy, a particle filter-based multi-innovation stochastic gradient algorithm is developed through the innovation expansion technique. The simulation results demonstrate that compared with the existing benchmark algorithms, the proposed algorithms need a little more computational time due to the introduction of the adaptive particle filter, but they have the improved identification accuracies. [ABSTRACT FROM AUTHOR]

Manakov and Santini had studied the formal solutions and utilized the Riemann–Hilbert (RH) dressing method to investigate longtime behavior of the solutions and asymptotical implicit solutions of Pavlov equation (Manakov and Santini, 2007) and heavenly equation (Manakov and Santini, 2006). In the paper, we introduce a new Lax pair to construct an integrable system, which can be reduced to the standard Pavlov equation. Therefore, we refer to it as a generalized Pavlov equation. By utilizing the inverse scattering transform (IST) technique, we successfully derive the general formal solutions for the general Pavlov equation. Additionally, through the construction of a new RH problem, we investigate the longtime behavior of the solutions to the general Pavlov equation. The results presented in this paper expand upon the results previously presented by Manakov and Santini in their work (Manakov and Santini, 2007). [ABSTRACT FROM AUTHOR]

In this paper, we create the rumor propagation model with diffusion behavior by considering the state of the rumor in both the time dimension and the space dimension comprehensively. Meanwhile, we demonstrate the reaction–diffusion model using Turing patterns after determining the prerequisites for their occurrence. In order to achieve the purpose of predicting and controlling rumors in time, we choose to utilize the parameter identification technique based on the Barzilai–Borwein (BB) algorithm and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. In the numerical simulation section, we first investigate how the rumor avoidance rate and cross-diffusion coefficients affect the propagation of rumors. Then, based on a continuous spatio-temporal system and complex network system, respectively, we perform parameter identification for the propagation model. We thoroughly examine how the type of algorithm, the quantity of unknown parameters, and the network structure affect the identification outcomes in terms of the cost function, error curve, and program function time. When the model constructed in this paper is used for parameter identification on different network structures, the error gap between the final value and the target value is not significant. However, the cost function and time consumption for parameter identification on complex networks are much smaller than on the continuous medium. • This paper is based on a rumor propagation system with diffusion behavior. • We further analyze the parameter identification based on the optimal control theory. • We explore the different factors that influence the results of parameter identification. [ABSTRACT FROM AUTHOR]

Modern power grids, functioning as a cyber-physical system (CPS), accommodate high penetration level of distributed generations (DGs) to ensure sustainability. Achieving sustainability through grid-scale DGs, however, increases the likelihood of islanding occurrences, which jeopardizes a major parameter of CPS implementation: security. This paper proposes an artificial intelligence-based approach for detecting islanding in modern power grids. The method utilizes measured voltage and frequency to develop a fuzzy center of gravity (COG)-based equivalent model of the system. The model is derived by combining data collected from phasor measurement units (PMUs) with fundamental dynamical equations that govern power system dynamics. In this model, the system is represented using a set of fictitious reactances calculated using goal programming, which are then utilized to connect the COG to the local centers of inertia (COIs). By having a set of fictitious reactances for various operating points fed into a fuzzy model, one could develop an online model to calculate the fictitious reactances with high accuracy and speed at specific snapshots. By incorporating the maximum allowable phase differences between areas into the COG model to ensure transient stability, one could enhance the developed model to be robust against cyber-physical contingencies and cyber-attacks, albeit at the expense of a slight reduction in accuracy. The calculated fictitious reactances, represented in terms of local frequencies throughout the system, serve as valuable indicators for detecting islanding. By classifying the calculated fictitious reactances using support vector clustering over a period of time, islanding could be detected with high accuracy. Furthermore, incorporating the COG concept with the clustering based on fictitious reactances makes it possible to detect false data injection in an area of the system. The efficacy of the proposed method is assessed using simulated data from the renewable integrated 73-bus IEEE test system. • This paper deals with the application of computational intelligence for stability assessment of large-scale power systems. • Data-driven based approach is proposed to detect islanding occurrence. • A combination of Fuzzy-modeling and support vector clustering is used to propose a fast and accurate farmwork. • The highly non-linear model of power system is represented using a combination of COIs and center of gravity. [ABSTRACT FROM AUTHOR]

Uncertain differential equation is a type of differential equation involving uncertain processes. This paper analyzes a simple pendulum system with a varying drag coefficient by the tool of uncertain differential equation, and derives the uncertain simple pendulum equation. Afterwards, the numerical method of solving the uncertain simple pendulum equation and its parameter estimation method are given in this paper. Finally, a real-world example is provided to illustrate the uncertain simple pendulum equation. [ABSTRACT FROM AUTHOR]

The multi-layer perceptron architecture in PINNs model severely limits the model's ability to learn the temporal evolution of equation features. Instead, the GRU network is capable of capturing these complex temporal correlation features. This paper replaces the multi-layer perceptron structure of the PINNs model with the GRU network and called the modified model as physics-informed gated recurrent units network method (PGNM model). The PGNM model exhibits enhanced performance in assimilating insights from historical data and providing accurate predictions for future data. This paper compares the predictive performance of the proposed PGNM model and the classical PINNs model using L2 error and mean squared error (MSE) as metrics. Additionally, it evaluates how altering various parameter settings, such as the number of neurons, hidden layers and iterations, affects the predictive capabilities of both models. In conclusion, PGNM model shows significant improvement in prediction accuracy compared to PINNs model. Furthermore, PGNM model achieves better long-range prediction results than PINNs model when the training data does not include samples from the time period to be predicted. [ABSTRACT FROM AUTHOR]

This paper investigates the Sundaresan technique for modeling fractional order systems. Sundaresan, Prasad, and Krishnaswamy published this method in 1978 for modeling oscillatory and non-oscillatory systems based on the second-order integer transfer function. This technique is based on the transient response parameters. A problem of convergence of the derivative of the response in the frequency domain makes it impossible to follow Sundaresan's solution in his original paper with integer order when it is a fractional order case. The paper proposes an equation that outlines this problem. Due to the limited knowledge of the inverse Mittag-Leffler function, a reduced form of this equation is explicit to avoid the inverse problem. Results with simulated and real curve shapes show that the method works well, with a good approximation to the curve, both with simulation and real system curves. • Develop an equation for fine-tuning the parameters of the pseudo-second-order system. • Novel fractional-order identification based on Sundaresan technique. • The method proposed is a generalized method of the classic Sundaresan technique. [ABSTRACT FROM AUTHOR]

Nonlinear dynamical systems excited by combined Gaussian and Poisson white noises are widely found in the scientific field. However, the introduction of Poisson white noise results in the associated forward Kolmogorov equation becoming an integro-differential equation (IDE), which makes it difficult to solve the response of these dynamical systems. At present, traditional numerical IDE solvers are limited due to both the mesh generation and high computational costs. In this context, this paper integrates the multi-element Gauss–Legendre (GL) quadrature into the physics-informed neural networks (PINNs) algorithm to predict the response of systems excited by combined Gaussian and Poisson white noises. A residual-based adaptive distribution (RAD) sampling method, which replaces Latin hypercube sampling (LHS) method, is implemented to adjust the residual points to improve the accuracy and training efficiency of the proposed algorithm. The results demonstrate that the integration of the PINNs algorithm with RAD sampling method dramatically enhances the tolerance of noisy data and expedites the convergence of the PINNs algorithm compared with the LHS sampling method. It has the ability to achieve better performance in a shorter training time and reduces the demand for data during the training process. Additionally, the Monte Carlo (MC) simulation is carried out to demonstrate the effectiveness of the numerical results in this paper. • A deep neural network algorithm is proposed to predict the response of the system. • The Gauss–Legendre quadrature is integrated to solve the forward Kolmogorov equation. • A residual-based sampling approach enhances the convergence of our algorithm. • Two examples are worked out to illustrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]

The object of this study is to define existence, controllability and exponential stability results within a Sobolev-type fractional stochastic neutral equations of order 1 < ω < 2 with sectorial operator and optimal control. To do this, first, this study relies on the confluence of stochastic analysis, fractional calculus, sectorial operator, and the applicability of Sadovskii's fixed point method. First, we highlight the existence of mild solutions to the fractional stochastic control equation and then introduce the concept of approximate controllability. Next, employing an impulsive Poisson system yields sufficient conditions for guaranteeing the exponential stability of the mild solution in the mean square moment. Further, our inquiry expands into the existence of optimal control. Finally, an example is provided to illustrate the obtained theory. • This paper explores the stability of the fractional delay system of order ω ∈ (1 , 2). • This paper is the first study of a fractional jumps system with sectorial operators. • We establish conditions for the fractional system and extend to the controllability. • We determine mean square stability of fractional system ω ∈ (1 , 2) with Poisson jumps. • Finally, an optimal control pair is provided to show the solvability of the problem. [ABSTRACT FROM AUTHOR]

This paper creatively proposes a type of cubic fractal matrix and a new spatiotemporal chaotic system named logistic-logistic cascade coupled map lattice (L-LCCML). Furthermore, based on cubic fractal matrix and L-LCCML, this paper proposes a novel image encryption algorithm. The proposed cubic fractal matrix is three-dimensional, irregular, and self-similar. In particular, the proposed L-LCCML system adopts cascade coupling parameter to ensure the dynamic effect of coupling. Therefore, L-LCCML has excellent chaos and is suitable for information encryption. To provide a more secure approach, the proposed algorithm contains two diffusion operations, one permutation operation, and does not require multiple rounds of encryption. The diffusion operation is based on the cubic fractal matrix, which has good security and high encryption efficiency. In addition, the algorithm adopts sorting permutation based on L-LCCML, which provides good randomness for encryption. Experimental results show that the proposed algorithm has the characteristics of large key space, high sensitivity, fast encryption speed, good statistical properties of cipherd images, and etc. Therefore, the proposed algorithm is a usable alternative for practical secure communication. [ABSTRACT FROM AUTHOR]

This paper focuses on solving a stochastic saddle point problem (SPP) under an overparameterized regime for the case, when the gradient computation is impractical. As an intermediate step, we generalize Same-sample Stochastic Extra-gradient algorithm (Gorbunov et al., 2022) to a biased oracle and estimate novel convergence rates. As the result of the paper we introduce an algorithm, which uses gradient approximation instead of a gradient oracle. We also conduct an analysis to find the maximum admissible level of adversarial noise and the optimal number of iterations at which our algorithm can guarantee achieving the desired accuracy. • Generalized the S-SEG algorithm to a biased oracle. • Specified the result with Uniform Sampling and Importance Sampling based algorithm. • Proposed Zero-Order Same-sample Stochastic Extragradient algorithm for SPP. • Corroborated our theoretical results with experimental testing. • Compared our algorithm with several other algorithms, used for the solution of SPP [ABSTRACT FROM AUTHOR]

Aiming at the problem that speech information has a strong correlation in adjacent times and the data type is floating point, the image encryption algorithm of integer type is not suitable for speech encryption. This paper proposed a speech encryption algorithm based on robust chaotic mapping, which mainly utilizes the nonlinearities and dynamics of robust chaos to adapt to the characteristics of speech signals. Furthermore, a new 3D sine robust chaotic mapping (3D-SRCM) model is proposed in this paper, which effectively solves the problems of discontinuous parameter ranges, prone to chaotic degradation and lack of robustness in existing chaotic systems, and improves the robustness and complexity of chaos. In the speech encryption algorithm, the parameters of the chaotic mapping are adjusted according to the changes in speech signal characteristics to generate unique keys for different speech signals. The encryption algorithm compresses and denoises the signal through the Fast Walsh–Hadamard Transform (FWHT) before using chaotic sequences for initial scrambling encryption. Then, the signal is transformed by Discrete Wavelet Transform (DWT) to realize the second round of scrambling and diffusion encryption. This structure increases the security of the encryption algorithm and ensures the efficiency and reliability of the encryption process. The experimental results show that the algorithm has a large key space, good resistance to exhaustive attack, and statistical attack, which can effectively resist chosen plaintext attack. In the decryption process, the algorithm can quickly and accurately decrypt the encrypted speech with good decryption performance. • A speech encryption algorithm based on robust chaos was proposed. • A new 3D-SRCM model is proposed for existing chaotic systems. • The 3D-SRCM model solves chaotic degradation, improving robustness and complexity. • Control parameters were adjusted to adapt to speech signal, linking key and the signal. • In the encryption algorithm, the parameters of the chaotic map are adjusted to fit the speech signal. [ABSTRACT FROM AUTHOR]

The movements of the plankton in the ocean are driven by random diffusion and cognitive judgement with herd-taxis. In this paper, we formulate a phytoplankton–zooplankton model with time delay in the herd-taxis effect diffusion and homogeneous Neumann boundary conditions. The conditions to guarantee the existence of the coexistence equilibrium of the model are given. By analyzing the distribution of the eigenvalues of the characteristic equation, the local asymptotic stability of the coexistence equilibrium is achieved under certain condition. When there is no time delay in the herd-taxis effect, the model can possess the Turing bifurcation when we consider the nonlinear diffusion term, which leads to instability. When taking the time delay into account, the Hopf bifurcation occurs instead as the time delay varies. Furthermore, we investigate the situation without the fact of time, that is the steady-state bifurcation and the stability of bifurcating solution. Finally, the stability of the coexistence equilibrium, the Turing bifurcation and the Hopf bifurcation of the system are modeled by numerical simulation. The simulations shown are coordinated with the theoretical results which we arrive at in the former part of the paper. The results illustrate that the time delay in the herd-taxis effect of the zooplankton influence the dynamics of the plankton system. [ABSTRACT FROM AUTHOR]

Videos contain temporal information, enabling them to capture the dynamic changes of actions and provide richer visual effects. Traditional video encryption methods involve decomposing videos into frames and encrypting them frame by frame, which results in significant resource consumption. This paper proposes a video encryption method based on temporal action segmentation. This methodology involves the identification and extraction of pivotal frames from a video dataset, followed by the encryption of these significant key frames. This approach serves to enhance the efficacy of the video encryption algorithm. The method consists of three modules. The first module uses temporal action segmentation to classify video frames and extract important frames for the second module's input. The second module encrypts the extracted key frames using a chaos-based encryption algorithm, thereby reducing the time cost of video encryption. The third module outputs the encrypted video. During the encryption process, a large amount of key stream is required. To address this, the paper introduces a new pseudo-random sequence generation method called two-dimensional Gramacy&Lee map (2D-GLM). Comprehensive comparative analysis clearly demonstrates that compared to other systems, 2D-GLM exhibits superior performance and can generate a large number of high-performance pseudo-random sequences. The proposed algorithm is tested on GTEA, and the simulation results demonstrate that it can accomplish video encryption tasks with high security. • Novel 2D-GLM: Outperforms others, ideal for encryption. • Temporal action segmentation boosts video encryption. • Algorithm tested on GTEA dataset, ensuring security. • Efficient video encryption validated with high security. [ABSTRACT FROM AUTHOR]

Ever since the link between nonlinear science and cryptography became apparent, the problem of applying chaotic dynamics to the construction of cryptographic systems has gained a broad audience and has been the subject of thousands of papers. Yet, the field has not found its place in mainstream cryptography, largely due to persistent weaknesses in the presented systems. The goal of this paper is to help remedy this problem in two ways. The first is by providing a new algorithm that can be used to attack – and hence test the security of – stream ciphers based on the iteration of a chaotic map of the interval. The second is to cast discrete dynamical systems problems in a modern cryptographic and complexity theoretic language, so that researchers working in chaos-based cryptography can begin designing cryptographic protocols that have a better chance of meeting the extreme standards of modern cryptography. [ABSTRACT FROM AUTHOR]

The construction of fractional derivatives with the right properties for use in field theory is reputed to be a difficult task, essentially because of the absence of a unique definition and uniform properties. The conformable fractional derivative introduced in 2014 by Khalil et al. in their seminal paper is a novel and well-behaved definition of fractional derivative for a function that is derivable in the usual sense. In this paper, we investigate the consistency of the Euler–Lagrange formalism for a field theory defined on such a fractional space–time. We especially focus on the relation between symmetries and conservation laws (Noether's currents), about the symmetry group introduced to construct the Lagrangian of the field. In particular, we show that the use of the conformable derivative induces additional terms in the calculation of the action variation. We also investigate the conservation of the Noether current and show that this property only takes place on condition that the equations of motion are verified with a new definition of the conserved law. [ABSTRACT FROM AUTHOR]

In this paper, we focus on stochastic delay differential equations in the G-framework (G-SDDEs). We introduce the practical stability to examine whether the performance of G-SDDE near an unstable equilibrium point is acceptable. We establish a new generalized Gronwall inequality based on which we prove the practical mean-square (PMS) exponential stability of G-SDDE. We also establish the stability equivalence between the discrete and the continuous EM approximations for G-SDDE and then show that the continuous EM approximation can preserve the PMS exponential stability of G-SDDE. One numerical experiment is conducted to confirm our theoretical results. • In solving stochastic systems, we usually encounter a probability problem with Knightian uncertainty which can often be characterized by G-Brownian motion. Thus we need to consider the stability and the numerical approximations for the stochastic systems in the G-framework. • In this paper, we have introduced the definitions of practical stability of a dynamical system disturbed by G-Brownian motion (G-SDDE). We have studied the practical mean square (PMS) exponential stability of the G-SDDE under the case that origin is not an equilibrium point by establishing a new generalized Gronwall integral inequality. • We have also introduced the EM method and extended it to a continuous form based on which we have proved that the numerical solution can reproduce the PMS exponential stability of the G-SDDE. [ABSTRACT FROM AUTHOR]

Oil palm cultivation stands as a crucial industry in Indonesia, significantly contributing to the nation's economy by generating employment opportunities and fostering social welfare for communities residing near plantations. Despite its economic importance, oil palm plantations face various challenges, with one prominent issue being the infestation of nettle caterpillar pests. These pests cause leaf skeletonization, resulting in a staggering 36% reduction in oil palm productivity over a two-year period. This paper explores diverse strategies for pest management in oil palm plantations, encompassing biological control through the stimulation of natural predators, mechanical control involving the collection and incineration of cocoons, and chemical control through pesticide application. The research introduces a predator–prey mathematical model for oil palm plantation pests, where the leaf area serves as the primary food source for caterpillars, acting as prey. Through dynamic model analysis, four equilibrium points are identified, with interconnected conditions dictating their existence and stability. These conditions are visually represented in a bifurcation plane, providing concise information. The study further includes bifurcation diagrams of equilibrium points to elucidate the influence of each parameter on pests, predators, and the leaf area of oil palm plants. Additionally, sensitivity analysis of the stable interior equilibrium point is conducted to understand the impact of individual parameters. The paper extends its investigation to optimal control strategies, evaluating six scenarios categorized into two population conditions: with predators and without predators. Within each population condition, three control strategies are considered—chemical control only, mechanical control only, and a combination of chemical and mechanical control. Simulation results from the optimal control study reveal that the presence of natural predators emerges as a pivotal strategy in effectively managing nettle caterpillars. Notably, the control of resistant pests through pupa incineration has a substantial impact on reducing the caterpillar population in subsequent life cycles. • In current study, the MELP-S-B predator-prey model is proposed for managing nettle caterpillar pests in oil palm plantations involving the control measures. • The inclusion of prey populations, specifically oil palm leaves, in this predator-prey model constitutes the most fundamental novelty of this study. • The main objective of this research is to determine the dynamics of pests in plantations and avoid economic losses due to pests in oil palm plantations. • In-depth analysis of system dynamics around four equilibrium points conducted. Sensitivity analysis is carried out to measure the influence of parameters at the interior point. • Optimal control study conducted to manage pest abundance in oil palm plantations under varying predator presence. Numerical simulations of six strategies, blending mechanical and chemical measures, offer insights into effective pest control combinations. [ABSTRACT FROM AUTHOR]

Describing the dynamical properties of explored systems, one finds the need to distinguish between various types of trajectories. The nature of trajectories is often split into regular and irregular, which will be shown in this paper as too crude. Hence, the main aim of this paper is to give a classification of trajectories reflecting persistence, regularity, chaos, intermittency, and transiency. To depict such phenomena, classical examples from discrete (the Rulkov map) and continuous (the Lorenz system) dynamical systems are applied. In these cases, the maximal Lyapunov exponent, the 0-1 test for chaos, the bifurcation diagram, and the Fourier analysis are applied, and these dynamics characteristics are confronted with trajectory types. • Trajectory type classification in terms of persistency, regularity, and chaos. • Dynamics characteristics detection tools: bif. diagrams, MLE, the 0-1test for chaos. • Exploration of proposed trajectory classification on discrete and continuous systems. [ABSTRACT FROM AUTHOR]

This paper focus on studying the two-dimensional tabu learning neurons in conjunction with applied currents using phase, bifurcation, eigenvalues and sequence diagrams by semi-analytical method. The result demonstrates that neurons under the influence of different amplitudes can exhibit multi-periodic coexisting attractors. Two special kinds of bifurcation are investigated in depth through the analysis of orbit motions. The accurate tracking of neural spike events through phase diagrams is feasible. This paper also demonstrates the coexistence of steady and unsteady firing patterns, which cannot be obtained by the conventional numerical method. In addition, the correctness of the obtained results is verified by means of field programmable gate array. Lastly, the sequences generated by unsteady motions in the system are combined with DNA image encryption, enhancing the security for image encryption. • The semi-analytical method is used to study the bifurcation tree of a two-dimensional tabu learning neuron systems. • The evolution of supercritical pitchfork bifurcation to subcritical pitchfork bifurcation is investigated. • The coexisting firing behavior of tabu learning neuron is investigated at particular bifurcations. • The initial values of unsteady periodic motions are used as the keys applied to DNA based image encryption. [ABSTRACT FROM AUTHOR]

A study of nonlinear axisymmetric Rayleigh-Bénard magnetoconvection in a cylindrical enclosure filled with a dilute concentration of carbon-based nanotubes in a weakly electrically conducting Newtonian liquid heated from below for various aspect ratios is carried out. Cylindrical geometry is the prototype for heat storage devices and thermal coolant systems with a controlled environment. There is an analogy between porous media and magnetohydrodynamic problems and hence Rayleigh-Bénard magnetoconvection problem is practically important. The solution of the velocity and the temperature is in terms of the Bessel functions of the first kind and hyperbolic functions that are further used to study the marginal stability curves, heat transport, and the dynamical system. Symmetric and asymmetric boundaries of the realistic-type are considered on the horizontal and vertical bounding surfaces. The results of these boundaries are compared with those of the idealistic-type which are symmetric. A unified analysis approach is adopted for all boundary combinations in deriving the Lorenz model and studying the nonlinear dynamics. The time-dependent Nusselt numbers incorporating the effect of the curvature of the cylinder accurately captures the enhanced heat transport situation in the regular convective regime. Further, the influence of various parameters on the indicators of chaos such as the r H -plots, Lorenz attractor, bifurcation diagram, and the time series plot is investigated. The r H -plots clearly point to the appearance of chaos and also assist in determining its intensity and periodicity. The trapping region of the solution of the Lorenz model having the shape like that of a rugby-ball is highlighted in the paper. The size of the ellipsoid shrinks with increase in the strength of the magnetic field and also depends on the boundary conditions. • Axisymmetric convection in shallow cylindrical enclosures is considered. • Investigation is made for symmetric and asymmetric boundary conditions. • Convective instability, heat transports and chaos are studied. • The rH-plots Lorenz attractor, bifurcation diagrams and times-series plots are used to explore the chaotic regime. • Trapping region in the form of a rugby-ball is highlighted in the paper. [ABSTRACT FROM AUTHOR]

In this paper we consider a simple dissipative oscillator, determined by a two-parameter three-dimensional system of ordinary differential equations, obtained from the Nosé–Hoover oscillator by adding a small anti-damping term in its third equation. Based on numerical evidence, complex dynamics of this system was presented in a recent paper, such as the coexistence of periodic orbits, chaotic attractors and a stable invariant torus. Here we analytically prove the existence of a small periodic orbit from which a stable invariant torus bifurcates near the origin of the dissipative oscillator. We also show that the oscillations near the torus present a kind of relaxation oscillation behavior, like canard-type oscillations, commonly found in singularly perturbed systems. The obtained results extend and provide analytical proofs for some dynamical properties of the considered system, which were numerically described in the literature. [Display omitted] • A dissipative oscillator, based on Nosé–Hoover with a small term, is studied. • Analytical proof of small orbit and stable torus existence near origin. • Near-torus oscillations with canard-like behavior, as in singularly perturbed systems. [ABSTRACT FROM AUTHOR]

High-dimensional chaotic maps offer a larger parameter space, increased complexity, and enhanced resilience against dynamical degradation compared to their one-dimensional counterparts. Therefore, they are gradually replacing one-dimensional chaotic maps in various applications. However, many methods for generating high-dimensional chaotic maps lack mathematical proofs, which cannot theoretically ensure their chaotic nature. Even high-dimensional chaotic maps with theoretical support often lack global transitivity and exhibit local chaos. Applying such chaotic maps in chaos-based stream ciphers or random number generators results in poor randomness of generated chaotic sequences, reduced internal state space, and numerous weak keys, which is not ideal. This paper proposes a systematic method for constructing high-dimensional chaotic maps (called dispersal maps). The paper proves that the maps constructed are topologically mixing across the entire space and are hyper-chaotic on an invariant subset of full measure. These properties make them satisfy almost all definitions of chaos, and their chaotic dynamical behavior is global: exhibiting transitivity across the entire phase space rather than a local subregion, a dense scrambled subset rather than a tiny one, and being hyper-chaotic almost everywhere rather than on a local attractor. Therefore, dispersal maps can improve the existing problems of locally chaotic maps in application. The experiments also indicate that dispersal maps exhibit ergodicity on the phase space, with highly uniform trajectory distributions and sensitivity to initial perturbations. The findings provide researchers with ideal chaotic maps and a feasible method for constructing high-dimensional chaotic maps with global chaos. • Offer a general method for constructing chaotic maps with any dimension • Ensure the chaos of the constructed maps through rigorous theory • The maps are topologically mixing, resulting in global transitivity and chaos in multiple senses. • Introduce the concept of diffusion maps to provide a new idea to prove the chaos of maps [ABSTRACT FROM AUTHOR]

In this paper, we derive new results on the averaging principle for a class of Caputo neutral stochastic system driven by Markovian switching and Lévy noise with variable delays and time-varying fractional order. Under a set of appropriate conditions, we showed that solutions of the averaged stochastic systems approach the solutions of the original stochastic systems in the sense of both convergences in mean square and convergence in probability. Finally, we attach two examples with numerical simulations to justify the validity of our theory. • Our paper presents a general class of variable-order Caputo neutral differential equations. • We investigate the averaging principle under a new set of suitable assumptions. • Averaging result is proved in both senses, convergence in mean square and convergence in probability. • We provide some numerical simulations to illustrate the validity of our results. • Our findings improve and extend some related conclusions on the topic of averaging principle. [ABSTRACT FROM AUTHOR]

The implications of neural synchronization extend beyond brain function, and can impact the development of artificial neural networks. This paper explores the synchronization of multiple fractional-order fuzzy complex-valued spatiotemporal neural networks (MFOFCVSNNs), which is novel and characterized using fuzzy logic and fractional-order partial differential equations, making it more adaptable and versatile. We first establish a new fractional-order complex-valued partial differential inequality, an integer-order complex-valued partial differential inequality, and an equation. Then, by combining the Lyapunov method with fuzzy set theory, employing newly established inequalities and equations, along with a newly designed fuzzy pinning controller, we derive two linear matrix inequality (LMI) formulations of synchronization criteria for MFOFCVSNNs using a direct non-complex decomposition approach. These criteria exhibit different dependencies on the membership function, with one being independent and the other dependent. Importantly, the criterion based on the membership function demonstrates reduced conservatism compared to its independent counterpart. By leveraging M -matrix theory, we present the synchronization criteria in a concise low-dimensional form. Moreover, this paper extends and enhances previous findings, resulting in reduced conservatism. Finally, we validate our theoretical analysis through numerical simulations. • A new model is proposed-multiple fractional-order fuzzy complex-valued spatiotemporal neural networks, which is novel and characterized using fuzzy logic and fractional-order partial differential equations, making it more adaptable and versatile. • Proposing a novel fuzzy pinning controller for large-scale networks, it streamlines implementation complexity and enhances design flexibility by not requiring identical fuzzy parameters as the model. • Exploring synchronization among multiple fractional-order fuzzy complex-valued delayed spatiotemporal neural networks without utilizing the complex-valued decomposition method. • The membership-function-independent synchronization criterion and the membership-function-dependent synchronization criterion are established. [ABSTRACT FROM AUTHOR]

A new fractional-order 5D hyperchaotic system based on memristor is constructed in this paper, with the speciality that the system exists chaotic and hyperchaotic states in the fractional-order case, while in periodic state in the integer-order. In addition, it has a variety of special phenomena at fractional-order such as infinite initial value range, parameter-dependent offset-boosting and amplitude control, attractor coexistence, and fractional order complexity greater than integer order. The correctness and feasibility of the system is verified by analog circuit simulation and hardware circuit implementation. Combining this system with image encryption algorithms, two new scrambling algorithms and a diffusion algorithm are proposed. And a high-quality encryption scheme that can be applied to a wide range of color images is proposed. The scheme is found to have excellent security after verification by various security analyses and comparison with other literatures. This paper provides a basis for the superiority of fractional-order chaotic systems and provides new methods in the field of image encryption. • We construct a fractional-order 5D hyperchaotic system based on memristor. • The system is chaotic only to fractional-order and has a rich and complex dynamical behavior. • We propose a possible mechanism for the generation of attractor coexistence. • Designed new encryption algorithms and proposed a new color image encryption scheme [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behavior in a spatially heterogeneous reaction–diffusion SIS epidemic model with general nonlinear incidence and Dirichlet boundary condition is investigated. The well-posedness of solutions, including the global existence, nonnegativity, ultimate boundedness, as well as the existence of compact global attractor, are first established, then the basic reproduction number R 0 is calculated by defining the next generation operator. Secondly, the threshold dynamics of the model with respect to R 0 are studied. That is, when R 0 < 1 the disease-free steady state is globally asymptotically stable, and when R 0 > 1 the model is uniformly persistent and admits one positive steady state, and under some additional conditions the uniqueness of positive steady state is obtained. Furthermore, some interesting properties of R 0 are established, including the calculating formula of R 0 , the asymptotic profiles of R 0 with respect to diffusion rate d I , and the monotonicity of R 0 with diffusion rate d I and domain Ω. In addition, the bang–bang-type configuration optimization of R 0 also is obtained. This rare result in diffusive equation reveals that we can control disease diffusion at least at one peak. Finally, the numerical examples and simulations are carried out to illustrate the rationality of open problems proposed in this paper, and explore the influence of spatial heterogeneous environment on the disease spread and make a comparison on dynamics between Dirichlet boundary condition and Neumann boundary condition. [ABSTRACT FROM AUTHOR]