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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]

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

Density-dependent selection is a universal feature in the evolution of populations, and such an adaptive behavior can change the survival strategy of species during evolution. In this paper, we investigate the role of adaptive behavior in biodiversity in the system of cyclic competition. By incorporating a density-dependent mechanism into intraspecific competition, which is well-known as a key mechanism leading to biodiversity, we studied how such adaptive intraspecific competition can affect biodiversity and collective behavior during the evolution of cyclically competing species. Microscopically, we found that species can coexist strongly and are spirally entangled or collectively united by presenting two distinct pattern formations on spatially extended systems. While the adaptive mechanism can always promote species coexistence in a mean-field manner for particular sensitivity to the group scale, corresponding spatial dynamics exhibit nonmonotonic features for the robustness of extinction at moderately high mobility regime over the critical mobility when the associated mean-field system exhibits asymptotically stable heteroclinic cycles. The findings can shed light on a new aspect of the collective behavior of coexisting populations which may indicate the possibility of changing the survival strategies of each group to maintain the coexistence of cyclically competing populations. • The effect of the adaptive intraspecific competition is studied. • Species biodiversity is promoted at moderately high mobility values. • Anomalous viability accompanies the change of new pattern formation for coexistence. • The role of adaptive intraspecific competition is studied quantitively. • Critical degrees of adaptive intraspecific competition are discussed. [ABSTRACT FROM AUTHOR]

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

We study generalised rock–paper–scissors models with an arbitrary odd number N ≥ 5 of species, among which n are weak, with 2 ≤ n ≤ (N − 1) / 2. Because of the species' weakness, the probability of individuals conquering territory in the cyclic spatial game is low. Running stochastic simulations, we study the role of unevenness in the rock–paper–scissors game in spatial patterns and population dynamics, considering diverse models where the weak species are in different positions in the cyclic game order. Studying systems with five and seven species, we discover that the individuals' spatial organisation arising from the pattern formation process determines the stability of the cyclic game with multiple weak species. Our outcomes show that the presence of species unbalances the spatial distribution of organisms of the same species bringing consequences on territorial dominance, with the predominant species being determined by the position in the cyclic game order. Our simulations elucidate that, in general, the further apart the regions inhabited by different weak species are, the less the coexistence between the species is jeopardised. We show that if multiple weak species occupy adjacent spatial domains, the unevenness in the cyclic game is reinforced, maximising the chances of biodiversity loss. Our discoveries may also be helpful to biologists in comprehending systems where weak species unbalance biodiversity stability. • Stochastic simulations of the rock–paper–scissors models with five and seven species were performed. • The asymmetry in the spatial patterns due to the presence of multiple weak species is investigated. • The effects of species' weakness on population dynamics are quantified. • The impact of multiple weak species in jeopardising biodiversity is studied. [ABSTRACT FROM AUTHOR]

Organisms may respond to local stimuli that benefit or threaten their fitness. The adaptive movement behaviour may allow individuals to adjust their speed to maximise the chances of being in comfort zones, where death risk is minimal. We investigate spatial cyclic models where the rock-paper-scissors game rules describe the nonhierarchical dominance. We assume that organisms of one out of the species can control the mobility rate in response to the information obtained from scanning the environment. Running a series of stochastic simulations, we quantify the effects of the movement strategy on the spatial patterns and population dynamics. Our findings show that the ability to change mobility to adapt to environmental clues is not reflected in an advantage in cyclic spatial games. The adaptive movement provokes a delay in the spatial domains occupied by the species in the spiral waves, making the group more vulnerable to the advance of the dominant species and less efficient in taking territory from the dominated species. Our outcomes also show that the effects of adaptive movement behaviour accentuate whether most individuals have a long-range neighbourhood perception. Our results may be helpful for biologists and data scientists to comprehend the dynamics of ecosystems where adaptive processes are fundamental. • Stochastic simulations of the spatial rock-paper-scissors model with organisms of one out of the species performing adaptive movement strategies are performed. • Individuals can adjust their mobility to run away from hostile environments and stay in comfort zones. • The impact of the adaptive movement strategy on pattern formation is described and the characteristic length of the typical single-species spatial domain is quantified. • The mean species densities and selection risks are calculated based on the organisms' perception range and responsiveness strength. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers a class of social foraging swarms with a nutrient profile (or an attractant/repellent) and an attraction–repulsion coupling function, which is chosen to guarantee collision avoidance between individuals. The paper also studies non-identical interaction ability or efficiency among different swarm individuals for different profiles. The swarm behavior is a result of a balance between inter-individual interplays as well as the interplays of the swarm individuals (agents) with their environment. It is proved that the individuals of a quasi-reciprocal swarm will aggregate and eventually form a cohesive cluster of finite size for different profiles. It is also shown that the swarm system is completely stable, that is, every solution converges to the set of equilibrium points of the system. Moreover, all the swarm individuals will converge to more favorable areas of the profile under certain conditions. For general non-reciprocal swarms, numerical simulations show that more complex self-organized rotation may occur in the swarms. [Copyright &y& Elsevier]

• We introduce the effect of changing environmental carrying capacity on evolution of asymmetric rock-paper-scissors game. • According to assumptions on environmental gradients, the system can exhibit various survival states including multistability of single species survival. • Symmetry-breaking of competition rates and environmental carrying capacity can be significant factors to yield rich behavior of species survival in systems of cyclic competition. • Considering ecological factors is found to be an important issue on understanding mechanisms of evolution among cyclically competing species in the perspective of maintaining coexistence and promoting biodiversity. Asymmetrical rock-paper-scissors (RPS) competition has been perceived as a crucial factor in shaping species biodiversity, and understanding this ecological issue in a multi-species paradigm is rather difficult because community dynamics usually depend on distinct factors such as abiotic environments, biotic interactions and symmetry-breaking phenomenon. To address this problem, we employ a Lotka-Volterra competitive system consisting of both symmetrical, asymmetrical interactions and abiotic environment components. We discover that that asymmetrical RPS competition in heterogeneous environments can yield much richer dynamical behaviors, compared to the symmetrical and asymmetrical competition in homogeneous environments. While it is observed that species coexistence outcomes and/or oscillatory solutions are maintained as in the case of homogeneous environments, the nonuniformity in the environmental carrying capacities may lead to extra dynamics with regards to the appearance of survival states; for instance, coexistence of any two-species and single-species persistence states, which are not evident in the previous modelling studies. By means of bifurcation analysis, various salient features of the dynamical systems, including the emergence of certain attractors (e.g., different steady states, stable limit cycles and heteroclinic cycles) and co-dimension one bifurcations (e.g., transcritical and supercritical Hopf bifurcations) are realized in this ecological model. Overall, this modelling work provides a novel attempt to simultaneously encompass not only symmetry-breaking phenomenon through RPS competition, but also heterogeneity in the environments. This framework can provide additional insights to better understand various mechanisms underlying the effects of distinct ecological processes on multi-species communities. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the well known He’s variational iteration method (VIM) is used to construct solitary wave solutions for Boussinesq equation (BE). The chosen initial solution (trial function) can be in soliton form with some unknown parameters which can be determined in the solution procedure. [Copyright &y& Elsevier]

Abstract: In this paper, a three dimensional autonomous system which is similar to the Lorenz system is considered. By choosing an appropriate bifurcation parameter, we prove that a Hopf bifurcation occurs in this system when the bifurcation parameter exceeds a critical value. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is presented by applying the normal form theory. Finally, an example is given and numerical simulations are performed to illustrate the obtained results. [Copyright &y& Elsevier]

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

In this paper, fuzzy SP-irresolute, fuzzy SP-irresolute open and fuzzy SP-irresolute closed functions between fuzzy topological spaces in Sˇostak sense are defined. Their properties and the relationships between these functions and other functions introduced previously are investigated. Next fuzzy SP-connectedness is introduced and studied with the help of r-fuzzy strongly preopen sets. [Copyright &y& Elsevier]

Cyclic dominance is observed in predator-prey interactions, the mating strategy of side-blotched lizards, the overgrowth of marine sessile organisms and competition in microbial populations and many other natural systems. Rock-Paper-Scissor(RPS) is a popular game which demonstrates cyclic dominance. In this paper, we investigate replicator dynamics of RPS-game under logistic growth functions with Allee effect. The results obtained are compared with the case of no Allee effect. Due to Allee effect the number of stable attractors increases in a certain parameter region. The obtained result can be interpreted biologically that diversity of an ecological system increases due to Allee effect. [ABSTRACT FROM AUTHOR]

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

This is a call for papers that should contribute to the unification of behavioral sciences and team management, focusing on the biological origin of cooperation and swarm intelligence, moving from biology to psychology and from sociology to political science, with the help of the theoretical tools of complex networks. This issue should shed light into the origin of ergodicity breaking and contribute to establishing a connection, still lacking theoretical support, between complexity properties that are expected to be correlated. Examples are: non-Poisson renewal events and multi-fractality; complexity matching and chaos synchronization; criticality and extended criticality of small size systems. Although the emphasis is on systems of small size, and especially on the search of the size maximizing both information transport and cooperation emergence, special attention will be devoted to the interaction between small groups and their larger societies. [ABSTRACT FROM AUTHOR]

• Being the first extensive numerical study of the dynamics of rock-paper-scissors type models with a total number of species (NS) between 3 and 12 having one or more (weak) species characterised by a reduced predation probability. • The demonstration, using lattice based spatial stochastic simulations with random initial conditions large enough for coexistence to prevail, that parity effects are significant in rock-paper-scissors models, specially if the number of species is smaller or equal to 8. • The verification that, despite the significant dispersion observed among individual models, weak species have on average higher abundances strong ones if the reduced predation probability is sufficiently smaller than unity, with the exception being of the four species case. We investigate the impact of parity on the abundance of weak species in the context of the simplest generalization of the rock-paper-scissors model to an arbitrary number of species — we consider models with a total number of species (N S) between 3 and 12, having one or more (weak) species characterized by a reduced predation probability (by a factor of P w with respect to the other species). We show, using lattice based spatial stochastic simulations with random initial conditions, large enough for coexistence to prevail, that parity effects are significant. We find that the performance of weak species is dependent on whether the total number of species is even or odd, especially for N S ≤ 8 , with odd numbers of species being on average more favourable to weak species than even ones. We further show that, despite the significant dispersion observed among individual models, a weak species has on average a higher abundance than a strong one if P w is sufficiently smaller than unity — the notable exception being the four species case. [ABSTRACT FROM AUTHOR]

• The rock-paper-scissors model is presented for two-person non-zero sum game in which the strategies of rock mutate to scissors and paper. • The evolutionary dynamics of the population of species rock, paper and scissors is studied when the patches are connected through random walk. • When the patches are coupled, the state of synchronization and stability is observed. • When mutation is allowed, the limit cycle converges to stable state and the transition from one phase to another phase is observed. • The replicator-mutator equations are solved analytically as well as numerically. Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist. [ABSTRACT FROM AUTHOR]

• We consider both proportional delay and distributed delay, which are more difficult and need more challenging to calculation than the recent works. • Through designing impulsive controller, several novel sufficient conditions are given and rigorously proved to ensure that the heterogeneous dynamic NNs and the desired trajectory achieve quasi-synchronization. • Using the generalized formula for the variation of proportional delay and distributed delay parameters, the theoretical error bounded of quasi-synchronization is estimated. In this paper, we discuss the quasi-synchronization of delayed heterogeneous dynamic neural networks based on impulsive control. The main difference of this paper with previous works on quasi-synchronization is that both proportional delay and distributed delay are considered. By establishing a novel impulsive delay inequality, combining Lyapunov theory and the concept of average impulsive interval, some necessary items for quasi-synchronization of delayed heterogeneous dynamic neural networks are obtained. Moreover, through using the generalized formulae for the variation of proportional and distributed delay parameters, the theoretical error bounded of quasi-synchronization is estimated. Finally, numerical examples are listed to explain the validity of our results. [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]

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]

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]

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]

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]

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]

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]

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]

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]

Several graph models are available today to model online social networks. These graph models are used to analyze the structural properties of the online social network, such as detecting communities, finding the influential spreader and predicting the behavior of the network. However, these models are based on deterministic single-layer graphs that may not be appropriate when online users use multiple social networks at the same time and social networks provide specific services. Moreover, because of the unknown and dynamic nature related to the behaviors and activities of online users, as well as structural and behavioral parameters, which may vary over time, stochastic multi-layer models could be applied to better capture and represent this phenomenon, as well as the dynamic nature of social networks. For example, in recommender systems, users' interests are unknown parameters and vary over time. Therefore, stochastic multilayer graph modeling can be used to develop recommender systems by considering different layers for different types of interests or preferences. In this paper, we propose a stochastic multilayer graph in which the edges are associated with random variables as a potential graph model for the analysis of online social networks. Therefore, after redefine some network measures related to stochastic multilayer graphs, we propose a Cellular Goore Game (CGG) based algorithm to computes the redefine network measures. A CGG-based algorithm computes defined network measures by learning automata from the edges of stochastic multilayer graphs. The experimental results show that the new CGG-based algorithm requires fewer samples from the edges of stochastic multilayer graphs than the standard sampling method in network measures calculation. Furthermore, the obtained results demonstrate that, from an evaluation perspective, the CGG-based algorithm provides superior results in terms of Kolmogorov-Smirnov (KS-test), Pearson Correlation Coefficient (PCC), Normalized Root Mean Square Error (NRMSE) and Kullback–Leibler divergence (KL-test). • The paper opens up a new horizon by introducing stochastic multilayer graphs as a model for real social networks. • This paper defines some new network measures for stochastic multilayer graphs. • We present a novel algorithm for estimating stochastic multilayer graph measurements based on the Cellular Goore Game (CGG). • The algorithm works via distributed computing to estimate stochastic multilayer graph measurements with learning automata. • The simulation results show that the proposed model outperforms a similar model in modeling real-world multilayer graphs. [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]

For uncertain fractional-order nonlinear systems (UFONS) with unknown control coefficients and intermittent actuator faults, the asymptotic tracking control problem is investigated in this paper. Firstly, to weaken the influence of virtual control coefficients and intermittent actuator faults, a smooth fractional-order projection operator-based adaptive compensation mechanism is presented. Additionally, a fractional-order nonlinear filter is constructed to replace the fractional-order derivative of virtual control functions approximately, which not only avoids the issue of complexity explosion existed in backstepping control frame, but fully compensates the effects of boundary errors caused by the employed filter in spite of the unknown virtual control coefficient. By constructing a fractional Lyapunov function from the property of projection operator, it is proved that all signals in the closed-loop system are bounded, and the asymptotic tracking control object is achieved. Definitively, a simulation study is presented to verify the availability of the presented method. • This paper investigates the issue of asymptotic tracking control design for fractional-order nonlinear systems with unknown control coefficients and actuator faults. • Firstly, in order to weaken the influence of virtual control coefficient, an adaptive compensation mechanism based on the smooth fractional-order projection operator design method is proposed. • Additionally, a fractional-order nonlinear filter is constructed to approximately replace the virtual control functions as well as its fractional-order derivative, which not only avoids the inherent complexity explosion problem under the framework of backstepping, but also fully compensates the effect of boundary error caused by the introduced filter when the virtual control coefficient is unknown. [ABSTRACT FROM AUTHOR]

The research on cascading failures in static networks has become relatively mature, and an increasing number of scholars have started to explore the network scenarios where mobile nodes and static nodes coexist. In order to enhance the resilience of static-mobile networks against cascading failures, an algorithm based on Q-learning for optimizing the path of the mobile node is proposed in this paper. This paper proposes a Q-learning-based algorithm for optimizing the path of the mobile node. To achieve this objective, a cascading failure model is established based on sequential interactions between mobile nodes and static nodes in this study. In this model, the motion paths of the mobile node are generated by the Q-learning algorithm. Based on this approach, extensive experiments are conducted, and the results demonstrate the following findings: 1) By increasing the adjustable load parameters of static nodes in the network, the occurrence of cascading failures is delayed, and the frequency of cascading failures is decreased. 2) Increasing the adjustable load parameter, capacity parameter, and network size of static nodes contributes to the network's resilience against cascading failures. 3) As the communication radius of the mobile node increases, the scale of failures in the static network initially increases and then decreases. 4) Different trajectories of the mobile node have a significant impact on network robustness, and paths generated based on Q-learning algorithm exhibit significantly improved network robustness compared to Gaussian-Markov mobility trajectories. The Q-learning algorithm is compared to the Ant Colony Optimization algorithm in terms of execution time, path length, and network robustness, and the Q-learning algorithm demonstrating favorable performance. These experimental results can be valuable for theoretical research on cascading failures in static-mobile networks. • A novel methodology is proposed to optimize the path of the mobile node, effectively enhancing the network's resilience against cascading failures. • The motion trajectories of the mobile node are generated using a Q-learning-based algorithm. • The paths generated based on the Q-learning algorithm exhibit significantly improved network robustness compared to Gaussian-Markov mobility trajectories. • As the communication radius of the mobile node increases, the scale of failures in the static network initially increases and then decreases. [ABSTRACT FROM AUTHOR]

Numerous studies have explored the relationship between cooperation and inter-individual trust. However, the understanding of the dynamic interaction process of trust among networked agents with bounded rationality remains limited. To bridge this gap, this paper develops an evolutionary game model that incorporates a trust perspective considering emotional scaling. Trust is quantified based on individuals' limited memory of interaction experiences, acting as a perturbation factor that influences cooperative strategies. The role of emotional factors, particularly guilt and anger, in trust formation is emphasized. Extensive numerical simulations reveal that introducing trust greatly enhances cooperation in the system, especially in networks with lower randomness. Trust-based cooperative decision-making demonstrates evident historical path dependence characteristics. Notably, the effects of emotions differ. Guilt effectively motivates defectors to switch strategies, thereby enhancing cooperation. In contrast, the influence of anger on cooperation diminishes as it becomes intertwined with other factors during the trust-building process. The empirical results of higher returns for low-trust groups have also been validated in this paper. Overall, this work contributes to a deeper understanding of the mechanisms underlying cooperation formation. • A trust-based evolutionary game model with emotional scaling is developed. • The emphasis is on the interaction between network structure and emotions. • Trust-based cooperation exhibits clear historical path dependence. • Guilt and anger emotions have distinct impacts on the emergence of cooperation. [ABSTRACT FROM AUTHOR]

Abstract: Recently, Alaca et al. [Alaca et al., Chaos, Solitons & Fractals 2006;29:1073–9] proved some fixed point theorems in intuitionistic fuzzy metric spaces by a strong definition of Cauchy sequence (see [George and Veeramani, Fuzzy Sets Syst 1994;64:395–9] and [Veeramani and Vasuki, Fuzzy Sets Syst 2003;135:409–13]), also the intuitionistic fuzzy metric space has extra conditions (see [Gregori et al., Chaos, Solitons & Fractals, 2006;28:902–5]). In this paper, we consider generalized intuitionistic fuzzy metric spaces i.e., -fuzzy metric spaces and prove the fuzzy version of Banach and Edelstein contraction theorems in these spaces for modified definition of Cauchy sequence. [Copyright &y& Elsevier]

Complex systems in ecological science can be generally defined by either the number of different species or the structure among species having many relations, and understanding the given interaction structure is essential to predict the evolution of ecosystems. In this paper, we propose a multi-species system whose competition can occur cyclically. By exploiting the generalized system of cyclic competition among seven species, we explore how species biodiversity can appear when the generalized system is established by possessing the underlying mechanism of rock–paper–scissors (RPS) and rock–paper–scissors–lizard–spock (RPSLS) games. Through Monte-Carlo simulations, similar to the RPSLS system having the phase transition in biodiversity from five to one containing the three species survival in the middle, the model for seven species also exhibits similar cascading features in the biodiversity as mobility increases, validated by measuring the survival probability. We also found that not every cyclic structured system among seven species exhibits a common cascading feature in the transition in biodiversity. It is revealed that such a characteristic may require sufficient structures of RPS-like subgroups. Our findings may provide insights into the biodiversity of cyclically competing species and the link to predict biodiversity associated with the interaction structure in the microscopic framework. • Spatiotemporal evolution of cyclically competing seven species is investigated. • Species biodiversity depending on mobility exhibits the similar pattern of cascading dynamics. • Different pathways of cyclic competition among seven species can present different biodiversity. • Predicting biodiversity can be possible based on the competition structure among species. [ABSTRACT FROM AUTHOR]

• We introduce population flow and its effect on biodiversity in evolutionary dynamics of rock-paper-scissors game. • According to assumptions on population flows, the system can exhibit various survival states including persistent coexistence and multistability of single group survival. • Population flow can change the carrying simplex for evolutions of the system. • The basin structure for multistability may be spirally entangled and discontinuous. • The coexistence state can exhibit oscillatory dynamics according to the magnitude of population flow. Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups. [ABSTRACT FROM AUTHOR]

Introducing memristors into the traditional chaotic system can generate multiscroll chaotic attractors, expanding possibilities for information processing and chaotic applications. This paper first proposes a novel multi-segment memristor model based on a multi-segment linear function. Then, based on the Sprott-B system, one-directional memristive multiscroll chaotic attractors (1D-MMSCAs), 2D-MMSCAs, and 3D-MMSCAs are produced separately, with different numbers of novel memristors introduced. The dynamic behavior of the MMSCAs is analyzed in terms of equilibrium points, Lyapunov exponents and bifurcations, coexisting attractors, and complexity. Lyapunov exponent and bifurcation analysis reveal rich dynamic behavior of the MMSCAs, including period-doubling bifurcations, bursts of chaos, and transient of chaos. The MMSCAs exhibit dynamic phenomena such as coexisting attractors, multistability, and super multistability under different initial conditions. Furthermore, the existence and feasibility of the MMSCAs are verified through circuit simulation. Coexisting attractors generation circuits that can change the initial values of arbitrary state variables are designed. Using an improved Euler algorithm and the STM32 microcontroller, the MMSCAs are digitally implemented, expanding the application scope. Comparative results with other multi-scroll chaotic attractors (MSCAs) demonstrate the advantages of the proposed MMSCAs, including controllable scroll number and direction, simple implementation circuits, and rich dynamic behavior. Finally, the MMSCAs are applied to finite-time synchronization. Simulation results show that the two proposed synchronization schemes in this paper require less time to achieve complete synchronization compared to other synchronization schemes. This characteristic enhances the efficiency and practicality of the proposed synchronization strategy in real-world applications. [ABSTRACT FROM AUTHOR]

Stefan melting problems involve the tracking of a sharp melt front during the heat conduction controlled melting of a solid. A feature of this problem is a jump discontinuity in the heat flux across the melt interface. Time fractional versions of this problem introduce fractional time derivatives into the governing equations. Starting from an appropriate thermodynamic balance statement, this paper develops a new sharp interface time fractional Stefan melting problem, the memory-enthalpy formulation. A mathematical analysis reveals that this formulation exhibits a natural regularization in that, unlike the classic Stefan problem, the flux is continuous across the melt interface. It is also shown how the memory-enthalpy formulation, along with previously reported time fractional Stefan problems based on a memory-flux, can be derived by starting from a generic continuity equation and melt front condition. The paper closes by mathematically proving that the memory-enthalpy fractional Stefan formulation is equivalent to the previous memory-flux formulations. A result that provides a thermodynamic consistent basis for a widely used and investigated class of time fractional (memory) Stefan problems. • A new time fractional Stefan problem is presented, the memory-enthalpy formulation. • The problem is obtained from an appropriate thermodynamic balance statement. • We prove that this formulation exhibits a natural regularization of the problem. • A comparison with the previous memory-flux formulation is made. • We prove that the memory-enthalpy formulation is equivalent to the memory-flux one. [ABSTRACT FROM AUTHOR]

The paper presents a study of issues related to the identification of a non-linear mathematical model describing dynamics of the temporomandibular joint (TMJ) disc. Based on the tests of real disks, a non-linear model was built and verified, and then numerical simulations were carried out, the purpose of which was to analyze the behavior of the model for various excitation conditions. They include, among others, plotting a multi-colored map of distribution of the largest Lyapunov exponent based on which the areas of occurrence of periodic and chaotic motion zones are identified. Bifurcation diagrams of steady states for sample sections of the Lyapunov map and phase flows of periodic and chaotic solutions are generated. For the same sections, numerical simulations are performed to identify coexisting solutions. These studies are carried out using diagrams showing the number of coexisting solutions and their periodicity. The research presented in the paper shows a very good match between the results of computer simulations and the data recorded in the laboratory experiment. Due to the very strong damping occurring in the system, the chaotic attractors resemble quasi-periodic solutions with their geometric shape. Strong damping also significantly affects multiple solutions, which are relatively rare in the analyzed model. Most of the chaotic responses and multiple solutions occur in the range of low amplitude values of the dynamic load affecting the tissues of the articular disc. The obtained results of numerical experiments clearly indicate that in the range of low frequency values of the external load acting on the system, single periodic solutions with a periodicity of 1 T dominate. With the increase of the load amplitude, the area of occurrence of such solutions increases. [Display omitted] • A novel non-linear mathematical model of the TMJ disk has been proposed. • The model was very well adjusted to the results of experimental studies. • The behavior of the model for chaotic and periodic motion zones was tested. • The presence of coexisting solutions was confirmed. [ABSTRACT FROM AUTHOR]

This paper presents a new approach for solving fractional delay differential equations of variable order using the spectral element method. The proposed method overcomes the limitations of traditional spectral methods, such as poor approximation in long intervals and inefficiency in high degrees. By introducing a variable order differentiation matrix and using basic Lagrangian functions to approximate the solution in each element, the method achieves high accuracy and efficiency. A penalty method is also applied to minimize the jump of fluxes at interface points, and the effectiveness of this approach is analyzed. Finally, three benchmark problems are solved, and the convergence analysis demonstrates the effectiveness and efficiency of the proposed method. In essence, this paper offers a significant contribution to the literature on fractional differential equations and their numerical solution methodologies. • New approach for solving fractional delay differential equations using spectral element method. Overcomes limitations of traditional methods. • Introduces variable order differentiation matrix and Lagrangian functions for high accuracy. Penalty method minimizes flux jumps at interface points. • Rigorous convergence analysis of proposed method. • Effectiveness demonstrated with practical examples and discretization. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive security analysis on an improved chaos-based image encryption algorithm. The initial algorithm, proposed by Li et al., involves permutation related to the sum of plaintext pixel values and diffusion associated with 9 specific pixel values in the permuted image. However, a thorough analysis conducted by Liu et al. reveals two major flaws in it: firstly, the 9 specific pixel values are not involved in the diffusion process; secondly, the permutation method exhibits significant vulnerabilities. In response to these shortcomings, Liu et al. proposed targeted improvements on it, which include incorporating a permutation step for the 9 specific pixels and enhancing the original permutation method. In this study, we analyze the improved algorithm and discover that it still possesses security vulnerabilities, rendering it susceptible to chosen-plaintext attack. By constructing three categories of special plaintexts, one can decipher the equivalent permutation and diffusion. Theoretical analysis and experimental results provide strong evidence for the effectiveness of our analysis in this paper. [ABSTRACT FROM AUTHOR]

A close relation has recently emerged between two of the most fundamental concepts in physics and mathematics: chaos and supersymmetry. In striking contrast to the semantics of the word 'chaos', the true physical essence of this phenomenon now appears to be a spontaneous order associated with the breakdown of the topological supersymmetry (TS) hidden in all stochastic (partial) differential equations, i.e. , in all systems from a broad domain ranging from cosmology to nanoscience. Among the low-hanging fruits of this new perspective, which can be called the supersymmetric theory of stochastic dynamics (STS), are theoretical explanations of 1/f noise and self-organized criticality. Central to STS is the physical meaning of TS breaking order parameter (OP). In this paper, we discuss that the OP is a field-theoretic embodiment of the 'butterfly effect' (BE) – the infinitely long dynamical memory that is definitive of chaos. We stress that the formulation of the corresponding effective theory for the OP would mark the inception of the first consistent physical theory of the BE. Such a theory, potentially a valuable tool in solving chaos-related problems, would parallel the well-established and successful field theoretic descriptions of superconductivity, ferromagnetism and other known orders arising from the spontaneous breakdown of various symmetries of nature. • The paper provides a concise review of a theory that reveals that dynamical chaos is a spontaneous order. • It is shown that the corresponding order parameter describes the butterfly effect. • It is discussed how this theory lays the foundation for the first consistent physical theory of the butterfly effect. • It is argued that in some cases, the butterfly effect may admit a holographic field-theoretic description. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and develop two novel classifications of enriched (q , θ) -contractions on Banach spaces. The paper includes illustrative examples to demonstrate these concepts and establishes the convergence of Krasnoselskii's iteration method when applied to approximate the fixed point of such enriched (q , θ) -contractions. Additionally, the paper explores the application of these concepts in constructing the fractals of the corresponding Hutchinson–Barnsley operators. The above construction is illustrated by some examples. These discoveries provide new fixed-point solutions for iterated function systems under various contractive conditions. Finally, as an application of our main result, the existence of the solution to the problem of fourth order differential equation is presented. Furthermore, the findings not only validate but also enhance and expand upon multiple established conclusions in the existing literature. [ABSTRACT FROM AUTHOR]