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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]

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]

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]

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]

• 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]

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]

Some 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]

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]

• 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]

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

Some errors exist in the above paper. [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]

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]

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]

• 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]

• 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]

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]

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]

Abstract: In a recent paper (Asker, 2007) [1] a dynamic Cournot oligopoly game is proposed and it is claimed that this model represents competition among firms that exploit a common access natural resource. According to the author’s claim, the feature that relates the model with renewable natural resource harvesting is given by the presence of a particular cost function where the total cost of each fisherman is proportional to the square of the own quantity of harvesting and inversely proportional to the total harvesting quantity. In contrast, the usual function used in the literature on the exploitation of natural resources (such as fisheries) is inversely proportional to the available resource stock, and not to the total harvesting. This, in some sense, assumes exactly the opposite (as the available resource is inversely proportional to the total harvesting). So, we believe that the paper (Asker, 2007) [1] contains an error which is probably due to a misunderstanding or a misreading and misinterpretation of the (well-established) literature on bioeconomic modelling, but nevertheless misleading to researchers interested in bioeconomic modelling. The aim of this short note is to explain the mistake and to summarize the correct derivation and interpretation of the cost function. Our goal is to avoid the propagation of a subtle (but nevertheless misleading) error. [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 derive a finite population spatial replicator dynamic for arbitrary games. • We analyze the effects of nite populations on generalized rock-paper- scissors. • We show the finite population spatial replicator dynamic destroys a trav- eling wave. We derive both the finite and infinite population spatial replicator dynamics as the fluid limit of a stochastic cellular automaton. The infinite population spatial replicator is identical to the model used by Vickers and our derivation justifies the addition of a diffusion to the replicator. The finite population form generalizes the results by Durett and Levin on finite spatial replicator games. We study the differences in the two equations as they pertain to a one-dimensional rock-paper-scissors game. [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]