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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.

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.

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., L -fuzzy metric spaces and prove the fuzzy version of Banach and Edelstein contraction theorems in these spaces for modified definition of Cauchy sequence.

In this paper, it is shown that von Neumann continuous geometry may be regarded as the first attempt towards formulating a general quantum spacetime geometry akin to that of Cantorian spacetime E(∞) and noncommutative geometry.

The purpose of this short paper is to present a new form of the so called early exercise premium for the American type derivatives. The decomposition we derived consists of the corresponding European derivative and a derivative with a stochastic maturity. In different particular cases we reach to the well known form for the American put option where the underlying asset is driven by a Brownian motion or a Levy process.

This short paper outlines the basic topological features of two new fuzzy Kahler manifolds in duality. Relation to E-infinity theory of the compactified Klein modular space in connection with ’t Hooft holographic principles is briefly discussed.

The Lie symmetry and invariants for a generalized Birkhoffian system on time scales are studied, which include exact invariants and adiabatic invariants. First, the generalized Pfaff-Birkhoff principle on time scales is established, and by using Dubois-Reymond lemma the generalized Birkhoff’s equations on time scale are derived. Secondly, the determining equations of Lie symmetry for the generalized Birkhoffian system on time scales are established. We prove that if the Lie symmetry satisfies the structural equation, it leads to a conserved quantity, which is an exact invariant of the system. Again, the perturbation of Lie symmetry under the action of small disturbance is considered, the determining equations and the structural equations of disturbed system are established, and the adiabatic invariants led by the Lie symmetry perturbation for the generalized Birkhoffian system on time scales are given. Because of the arbitrariness of selecting time scales and the generality of the generalized Birkhoffian system, the results of this paper are of universal significance. The results of this paper contain the corresponding results for Birkhoffian system on time scales and classical generalized Birkhoffian system as its special cases. At the end of the paper, an example is given to illustrate the validity of the method and the results.

The economic system is an irreversible entropy increase process which is constructed by many elements and is far away from the equilibrium point; and affected by various parameters change, it is quite common that its motion state appears chaotic phenomenon due to instability. The extremely complex and not completely random aperiodic motion form of chaotic phenomenon is strongly sensitive to initial conditions. The development of nonlinear science, especially the emergence and development of chaos and fractal theory, has gradually become a powerful tool for economists to study the complexity, uncertainty and nonlinearity of social economic systems; and some visionary economists began to apply the research results of nonlinear science to economics, which has produced nonlinear economics. On the basis of summarizing and analyzing previous research works, this paper first obtains the non-degenerate equilibrium point of some typical fractional-order chaotic economic systems and transforms the equilibrium points of those systems to the origin through coordinate transformation, and then analyzes the Jacobi matrixes of new systems obtained through coordinate translation, and the parameter conditions of bifurcation in the economic systems are finally given and the numerical simulation of the fractional-order chaotic economic system evolution is carried out through bifurcation diagram, phase diagram and time series diagram. The study results of this paper provide a reference for the further study of the evolution of fractional-order chaotic economic systems with non-degenerate equilibrium points.

In this paper, we study the existence and the numerical estimates of solutions for a specific types of fractional differential equations. The nonlinear part of the problem, however, presupposes certain hypotheses. Particularly, for exact localization of the parameter, the existence of a non-zero solution is established, which requires the sublinearity of the nonlinear part at the origin and infinity. We also take into consideration several theoretical and numerical examples. One of the main novelties of this paper is to use the variational method to investigate the properties of solutions of boundary values problems involving Atangana–Baleanu fractional derivative for the first time.

A fractional compartmental mathematical model for the spread of the COVID-19 disease is proposed. Special focus has been done on the transmissibility of super-spreaders individuals. Numerical simulations are shown for data of Galicia, Spain, and Portugal. For each region, the order of the Caputo derivative takes a different value, that is not close to one, showing the relevance of considering fractional models., This is a preprint of a paper whose final and definite form is published by 'Chaos Solitons Fractals' (ISSN: 0960-0779). Paper Submitted 04/June/2020; Revised 22/July/2020 and 02/Aug/2020; Accepted 04/Jan/2021

Research findings have shown that evolution equations containing non-integer order derivatives can lead to some useful dynamical systems which can be used to describe important physical scenarios. This paper deals with numerical simulations of multicomponent symbiosis systems, such as the parasitic predator-prey model, the commensalism system, and the mutualism case. In such models, we replace the classical time derivative with either the Caputo fractional derivative or the Atangana-Baleanu fractional derivative in the sense of Caputo. To guide in the correct choice of parameters, we report the models linear stability analysis. Numerical examples and results obtained for different instances of fractional power α are provided for non-spatial models as well as the spatial case in one and two dimensions in other to justify our theoretical findings which include the chaotic phenomena, spatiotemporal and oscillatory patterns, multiple steady states and other spatial pattern processes. This paper further suggest an alternative approach to incessant killing of wildlife animals for pattern generation and decorative purposes.

This paper is concerned with the spatiotemporal behaviors of a Gierer–Meinhardt system in discrete time and space form. Through the linear stability analysis, the parametric conditions are gained to ensure the stability of the homogeneous steady state of the system. Based on the bifurcation theory, as well as center manifold theorem, we derive the critical parameter values of the flip, Neimark–Sacker and Turing bifurcation respectively. Besides, the specific parameter expression to form patterns are also determined. In order to identify chaos among regular behaviors, we calculate the Maximum Lyapunov exponents. The results obtained in this paper are illustrated by numerical simulations. From the simulations, we can see some complex dynamics, such as period doubling cascade, invariant cycles, periodic windows, chaotic behaviors, and some striking Turing patterns, e.g. circle, mosaic, spiral, spatiotemporal chaotic patterns and so on, which can be produced by flip-Turing instability, Neimark–Sacker–Turing instability and chaos.

In a recent paper [R. Qesmi, Dynamics of an opinion model with threshold-type delay, Chaos, Solitons & Fractals 98 (2020). (https://doi.org/10.1016/j.chaos.2020.110379)], we proposed a mathematical model of threshold-type delay differential equations describing the relationship between two subpopulations with opposite opinions and the opinion spread dynamics. The study there showed the possibility of a transcritical forward and backward bifurcations of positive equilibria. In the present paper, we show that the opinion model undergoes a Hopf bifurcation through which one of the bifurcation branches loses the stability and periodic solutions appear. One of the important consequences of the obtained dynamics is that the consensus of the both opinions could be lost by maintaining the balance between the time taken for an individual to become convinced of the outsider opinion, which need be short, and the number of individuals converted to the local opinion which need be low. Finally, we provide numerical simulations to illustrate and support our theoretical results. results.

This paper investigates the stability and stabilization problem of variable-order fractional nonlinear dynamic systems with impulsive effects (VO-IFNDS) via a linear feedback controller. New inequalities on the VO Caputo fractional derivatives are established in this paper, which plays an essential role in the study of the stability theory of VO-IFNDS. Based on utilizing S-procedure and analytical technique, several sufficient criteria on Mittag-Leffler stability and asymptotical stability of VO-IFNDS are presented by means of the extension of the Lyapunov direct method. Finally, numerical examples are given to show the efficiency of the proposed method.

This paper introduces self-similarity inherent in planar Milich-Jennings centered flip graphs derived from the Narayana sequence. We show that self-similarity found in a Narayana sequence yields a connected spanning subgraph with a centered flip. This paper has several main results (1) Every Narayana sequence constructs a flip graph, (2) Every Narayana sequence is self-similar and (3) Every Pascal 3-triangle has a free group presentation.

Taking the unbalance development inside a region into consideration, it will be better to express the regional electricity consumption (EC) value as an interval number to preserve more complete information. However, there are rare papers about this interesting topic. For the small amount of EC data, grey interval prediction model is employed in this paper. However, the existing models are almost all integer-order grey interval models and based on area coordinate conversion method. In order to fill this gap, and to obtain more accurate forecasting results, a conformable fractional non-homogenous discrete grey model (CFNDGM(1,1, α )) is proposed, and on the basis of CFNDGM model, two conformable fractional grey interval models are built. One is based on area coordinate conversion method (CFNDGM_AC), and the other is with information decomposition conversion method (CFNDGM_ID). The mathematical relationship of the two types of grey interval models is firstly given in this paper. It indicates that which type of model has better performance depends on the characteristics of original data. To assess the two fractional grey models, annal EC values in southern Jiangsu are taken as an example, and other four grey interval models are taken for comparison. Results show that CFNDGM_ID has the best performance among six models in both simulation and prediction. Then CFNDGM_ID is chosen to predict EC of southern Jiangsu in the next five years. To further improve forecasting accuracy, the optimal fitting size of CFNDGM_ID is selected. Forecasting results show that the EC in southern Jiangsu will increase in the next five years, but at a lower growth rate.

In order to understand the microstructure of pellets and improve the metallurgical properties of pellets, fractal theory is introduced to extract the edge features of pellets. Firstly, the original mineralogical phase is obtained by experiment, which is preprocessed by the histogram equalization to enhance the overall contrast. Based on the discrete Fractional Brownian Random Field Model, the algorithm is redesigned to calculate the dimension of each pixel, map the gray space of image into the dimension space, select the appropriate window size, transform and edge extraction. Comparing the algorithm in this paper with Canny operator and Laplace–Gauss operator, it is concluded that the algorithm in this paper has certain advantages in mineralogical phase edge extraction. Then, Gauss noise is added to the original gray image, and Canny operator and this algorithm are used to extract the edges of the noisy image. The numerical results of peak signal to noise ratio and root mean square error are obtained. Finally, the comparison proves that the algorithm can extract more complete edges, and has a stronger noise immunity.

The present paper shows the dynamics of Hepatitis E with optimal control. The paper is analyzed by two different aspects: first, we explore the dynamics of Hepatitis E model and then applying the optimal control analysis. Secondly, we use the most appropriate and recent fractional order derivative called the Atangana–Baleanu derivative for the dynamical analysis of Hepatitis E model. The proposed model considered is locally asymptotically stable when the threshold quantity less than one. Further, we explore the stability analysis of the model when R 0 > 1 . Then, we choose some appropriate control to formulate the optimality system. The results associated to the optimal control are obtained and discussed with different strategies. Moreover, we apply Atangana–Baleanu derivative to the proposed model and obtain the required results necessary for the fractional order model. Numerical results for the optimal control problem and Atangana–Baleanu derivative are obtained and discussed in detail. The results suggest that control variables chosen should be properly applied to get rid of the infection of Hepatitis E. The Atangana–Baleanu derivative results suggest that at any time t we can check the disease status and make a useful strategy for the early elimination of Hepatitis E from the community.

In this paper, we present and analyze a prey-predator system, in which prey species can be infected with some disease. The model presented in this paper is motivated from D. Mukherjee’s model in which he has considered an SI model for the prey species. There are substantial evidences that infected individuals have the ability to recover from the disease if vaccinated/ treated properly. In this regard, Mukherjee’s model is modified by considering SIS model for prey species. Theoretical and numerical simulations show that the recovery of infected prey species plays a crucial role in eliminating the limit cycle oscillations and thus making the interior equilibrium point stable. The possibility of Hopf bifurcation around non zero equilibrium point using the recovery rate as a bifurcation parameter, is discussed. Further, the model is extended by incorporating the harvesting of predator population. A monitory agency has been introduced which monitors the exploitation of resources by implementing certain taxes for each unit biomass of the predator population. The main purpose of the present research is to explore the effect of recovery rate of prey on the dynamics of the system and to optimize the total economical net profits from harvesting of predator species, taking taxation as control parameter.

This paper develops the analytical conditions for the existence of periodic solutions of a second-order,time-delayed, discontinuous dynamical system. A sample model consists of two linear delayed sub-systems with a switching boundary. The defined G-functions for the delayed, discontinuous systems are introduced, and sufficient and necessary conditions for a flow crossing, sliding and grazing along the switch boundary are developed for such a delayed, discontinuous system. Furthermore, nine (9) regular basic mappings in phase plane and thirty-three (33) delay-related mappings for the second-order, time-delayed, discontinuous systems are classified. Constraint equations are predicted analytically for two periodic orbits with initial functions provided posteriorly. Finally, three numerical examples are illustrated to verify the existence of generalized slowly oscillating periodic orbits without and with sliding portions. This paper improves and extends motion switchability conditions at the boundary in discontinuous dynamical systems without delay.

A three factor variance model introduced by Gatheral in 2008, called the double mean reverting (DMR) model, is well-known to reflect the empirical dynamics of the variance and prices of options on both SPX and VIX consistently with the market. One drawback of the DMR model is that calibration may not be easy as no closed form solution for European options exists, not like the Heston model. In this paper, we still use the double mean reverting nature to extend the Heston model and study the pricing of variance swaps given by simple returns in discrete sampling times. The constant mean level of Heston’s stochastic volatility is extended to a slowly varying process which is specified in two different ways in terms of the Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ross (CIR) processes. So, two types of double mean reversion are considered and the corresponding models are called the double mean reverting Heston-OU model and the double mean reverting Heston-CIR models. We solve Riccati type nonlinear equations and derive closed form exact solutions or closed form approximations of the fair strike prices of the variance swaps depending on the correlation structure of the three factors. We verify the accuracy of our analytic solutions by comparing with values computed by Monte Carlo simulation. The impact of the double mean reverting formulation on the fair strike prices of the variance swaps are also scrutinized in the paper.

Recently fractional differential operators with non-index law properties have being recognized to have brought new weapons to accurately model real world problems particularly those with non-Markovian processes. This present paper has two double aims, the first was to prove the inadequacy and failure of index law fractional calculus and secondly to show the application of fractional differential operators with no index law properties to statistic and dynamical systems. To achieve this, we presented the historical construction of the concept of fractional differential operators from Leibniz to date. Using a matrix based on the fractional differential operators, we proved that, fractional operators obeying index law cannot model real world problems taking place in two states, more precisely they cannot describe phenomena taking place beyond their boundaries, as they are scaling invariant, more precisely our results show that, mathematical models based on these differential operators are not able to describe the inverse memory, meaning the full history of a physical problem cannot be described accurately using these derivatives with index law properties. On the other hand, we proved that, differential operators with no index-law properties are scaling variant, thus can describe situations taking place in different states and are able to localize the frontiers between two states. We present the renewal process properties included in differential equation build out of the Atangana–Baleanu fractional derivative and counting process, which is connected to its inter-arrival time distribution Mittag–Leffler distribution which is the kernel of these derivatives. We presented the connection of each derivative to a statistical family, for instance Riemann–Liouville–Caputo derivatives are connected to the Pareto statistic, which has no well-defined average when alpha is less than 1 corresponding to the interval where fractional operators mostly defined. We established new properties and theorem for the Atangana–Baleanu derivative of an analytic function, in particular we proved that, they are convolution of the Mittag–Leffler function with the Riemann–Liouville–Caputo derivatives. To see the accuracy of the non-index law derivative to in modeling real chaotic problems, 4 examples were considered, including the nine-term 3-D novel chaotic system, King Cobra chaotic system, the Ikeda delay system and chaotic chameleon system. The numerical simulations show very interesting and novel attractors. The king cobra system with the Atangana–Baleanu presented a very novel attractor where at the earlier time we observed a random walk and latter time we observed the real sharp of the cobra. The Ikeda model with Atangana–Baleanu presented different attractors for each value of fractional order, in particular we obtain a square and circular explosions. The results obtained in this paper show that, the future of modeling real world problem relies on fractional differential operators with non-index law property. Our numerical results showed that, to not model physical problems with fractional differential operators with non-singular kernel and imposing index law in fractional calculus is rightfully living with closed eyes without ever taking a risk to open them.

It has been shown that natural interval extensions (NIE) can be used to calculate the largest positive Lyapunov exponent (LLE). However, the elaboration of NIE are not always possible for some dynamical systems, such as those modelled by simple equations or by Simulink-type blocks. In this paper, we use rounding mode of floating-point numbers to compute the LLE. We have exhibited how to produce two pseudo-orbits by means of different rounding modes; these pseudo-orbits are used to calculate the Lower Bound Error (LBE). The LLE is the slope of the line gotten from the logarithm of the LBE, which is estimated by means of a recursive least square algorithm (RLS). The main contribution of this paper is to develop a procedure to compute the LLE based on the LBE without using the NIE. Additionally, with the aid of RLS the number of required points has been decreased. Eight numerical examples are given to show the effectiveness of the proposed technique.

Focusing on the unpredictability of person-to-person contacts and the complexity of random variations in nature, this paper will formulate a stochastic SIR epidemic model with nonlinear incidence rate and general stochastic noises. First, we derive a stochastic critical value R 0 S related to the basic reproduction number R 0 . Via our new method in constructing suitable Lyapunov function types, we obtain the existence and uniqueness of an ergodic stationary distribution of the stochastic system if R 0 S > 1 . Next, via solving the corresponding Fokker-Planck equation, it is theoretically proved that the stochastic model has a log-normal probability density function when another critical value R 0 H > 1 . Then the exact expression of the density function is obtained. Moreover, we establish the sufficient condition R 0 C 1 for disease extinction. Finally, several numerical simulations are provided to verify our analytical results. By comparison with other existing results, our developed theories and methods will be highlighted to end this paper.

Semi-Markov processes established a rich framework for many real-world problems. Semi-Markov processes include Markov processes, Markov chains, renewal processes, Markov renewal processes, Poisson processes, birth and death processes, and etc. Since semi-Markov processes describe more complex mathematical models of various objects, in many cases the obtained mathematical models are investigated numerically. But in some cases it is possible to study such mathematical models using analytical methods. In the presented paper, the authors chose to do the second way. In this paper we study the semi-Markov random walk processes with negative drift and positive jumps. The random variable or the moment, at which the process for the time reaches in zero level is introduced. An integral equation for the Laplace transform of conditional distribution of this random variable is obtained. In this paper length of jump is given by the gamma distribution with parameters α and β resulting in a fractional order integral equation. In the class of gamma distributions, the resulting general integral equation of convolution type is reduced to a fractional order differential equation with constant coefficients. And also, the exact solution of the resulting fractional differential equation with constant coefficients has been found. Finally, using form of Laplace transform the expectation and variance of the random variable are found.

In this paper, we replace the classical differential operators with the fractal-fractional differential operators corresponding to the power law, exponential decay, and the generalized Mittag-Leffler kernels. These operators have two parameters created: the first is a fractal dimension and the second is a fractional order. The numerical schemes are combination of the Lagrange interpolating polynomial and theory of fractional calculus. In the case of δ = k = 1 the numerical solutions for the proposed models are found to be in an excellent agreement with the finite difference methods. We investigate the effects of the fractal-fractional order on the oscillations in the Fractal-Fractional Brusselator Chemical Reaction (FFBCR). All calculations in this paper were done using the mathematica package.

The present work is devoted to giving new insights into the extended Malkus-Robbins (EMR) dynamo. Firstly, based on differential geometry method, i.e. Kosambi-Cartan-Chern (KCC) theory, the paper investigates the Jacobi stability of the equilibrium and periodic orbit by the eigenvalues of the deviation curvature tensor. The deviation vector is applied to analyze the trajectory behaviors near the equilibrium and periodic orbit. Secondly, the zero-zero-Hopf bifurcation is investigated. The paper obtains the conditions that two periodic solutions appear at the bifurcation point and discusses their stability. Finally, on the global dynamics, the ultimate bound sets of the system are estimated. Numerical simulations are given to verify and visualize the corresponding theoretical results.

Firstly, by combining Mawhin’s continuation theorem of coincidence degree theory with Lyapunov functional method and by using inequality techniques, a sufficient condition on the existence of periodic solutions for inertial BAM neural networks is obtained. Secondly, a novel sufficient condition which can ensure the global asymptotic stability of periodic solutions of the system is obtained by using Lyapunov functional method and by using inequality techniques. In our paper, the assumption for boundedness on the activation functions in existing paper is removed, the conditions in inequality form in existing papers are replaced with novel conditions, and the prior estimate method of periodic solutions is replaced with Lyapunov functional method. Hence, our result on global asymptotic stability of periodic solutions for above system is less conservative than those obtained in existing paper and more novel than those obtained in existing papers.

Randomness is a common phenomenon in nonlinear systems. And conditions to reach synchronization are more complex and difficult when chaotic systems have random parameters. So in this paper, an adaptive scheme for synchronization of chaotic system with random parameters by using the fuzzy impulsive method and combining the properties of Wiener process and Ito differential is investigated. The main concepts of this paper are applying fuzzy method to approximate the nonlinear part of system, then using Ito differential to study the Wiener process of random parameters of chaotic system, and realizing synchronization under fuzzy impulsive method. The stability is analyzed by Lyapunov stability theorem. At the end of the paper, numerical simulation is presented to illustrate the effectiveness of the results obtained in this paper.

In this paper we compute topological entropy with prescribed accuracy for different economic models, showing the existence of a topologically chaotic regime for them. In order to make the paper self-contained, a general overview on the topological entropy of continuous interval maps is given. More precisely, we focus on piecewise monotone maps which often appear as dynamical models in economy, but also in population growth and physics. Our main aim is to show that when topological entropy can be approximated up to a given error, it is a useful tool which helps to analyze the chaotic dynamics in one dimensional models.

A periodic predator-prey model has been introduced in [5] to study the effect of water level on persistence or extinction of fish populations living in an artificial lake. By using the continuation theorem of Mawhin’s coincidence degree theory, the authors give sufficient conditions for the existence of at least one positive periodic solution. In this paper we study the problem in the general case. We begin by analyzing the invariance, permanence, non-persistence and the globally asymptotic stability for the system. Most interestingly, under additional conditions, we find that the periodic solution obtained in [5] is unique. Finally, in order to make the model system more realistic, we consider the special case when the periodicity in [5] is replaced by almost periodicity. We obtain conditions for existence, uniqueness and stability of a positive almost periodic solution. The methods used in this paper will be comparison theorems and Lyapunov functions. An example is employed to illustrate our result.

This paper surveys the latest advances of the first author's group on the three new methodologies of fractional and fractal derivatives modeling to meet the increasing and challenging demands in scientific and engineering communities. Firstly, the structural fractal was proposed as a generalization of the Euclidean distance. Using the structural metric, the structural derivative approach was derived as a significant extension of the global fractional calculus and the local fractional derivative approaches to tackle the perplexing modeling problems. The classical derivative describes the change rate of a certain physical variable with respect to time or space, which rarely takes into account the significant influence of mesoscopic time-space metric of a complex system on its physical behaviors. The structural function plays a central role in this new strategy as a kernel transform of underlying time-space structural metric of physical systems. Secondly, we employed the fundamental solution or probability density function of statistical distribution which can describe the problem of interest to construct the implicit calculus governing equation. The ‘implicit’ suggests that the explicit calculus expression of this governing equation is difficult to derive and not required. The fundamental solution or potential function of calculus governing equation and corresponding boundary conditions are sufficient to do numerical simulation. We call this strategy the implicit calculus equation modeling. Thirdly, based on the implicit calculus equation modeling approach, we introduced the concept of fundamental solution on fractal and consequently defined the fractal differential operator to describe various mechanical behaviors of fractal materials. Fractal calculus operator significantly extends the application scope of the classical calculus modeling approach under the framework of continuum mechanics. This is also a step-forward advance of the fractal derivative proposed earlier by the first author. To demonstrate the structural derivative application, we applied the inverse Mittag-Leffler function as the structural function to model ultraslow diffusion of a random system of two interacting particles. On the other hand, this paper uses the fractional Riesz potential as the fundamental solution to establish the implicit calculus equation of fractional Laplacian modeling the power law behaviors of steady heat conduction in multiple phase material. Finally, by using the singular boundary method, we made numerical simulation of the fractal Laplacian equation for phenomenological modeling potential problems in fractal media. Numerical experiments show that all the three new methodologies are feasible mathematical tools to describe complex physical behaviors.

This paper presents a new chaotic system that exhibits a two wing (2W) chaotic attractor in its integer order dynamics, three-wing (3W) and four-wing (4W) chaotic attractors in its fractional-order (FO) dynamics, and an eight-wing (8W) attractor in its nonautonomous fractional dynamics. An interesting feature of the proposed system is that two distinct periodic orbits coexist with a strange attractor that gradually evolves into a 4W attractor. The asymmetry, dissimilarity and topological structure of this proposed system with respect to those available in literature, manifest increased irregularity, which in turn indicate more chaos. Besides, the authors have drawn its comparison with various well-known fractional-order chaotic systems (FOCS)s to prove its enhanced features in terms of higher Lyapunov Exponent, fractional order orbital velocities, bandwidth, density, range of dynamical behaviour, etc. A control scheme is proposed to enable switching hybrid synchronisation between the 8W nonautonomous FOCS and the 4W autonomous FOCS, using the former as master and the latter as slave. This work throws light on the potential practical applicability of the proposed system by designing a circuit using minimum circuit components possible, thus signifying the objectives of the paper are finally achieved.

This paper presents a synopsis of fractional calculus tools for characterising respiratory mechanics. A discussion on power law impedance, viscoelasticity and tissue heterogeneity is made based on morphological and structural properties of lungs. Although targeted towards respiratory system applications, these tools can serve as a basis for modelling other biological structures as well. The paper underlines the importance of characterising the viscoelastic properties of the respiratory tissue and its time-varying properties as a function of disease progress. Perspectives are suggested for developing models able to mimic disease progression and understand treatment effects for better therapeutic management.

In this paper, we study the effect of stochastic noise on the virus infection model with nonlytic immune response. Firstly, the mathematical model describing the virus infection with nonlytic immune response is presented. The basic reproduction number is derived and the stability of disease-free state E0 and disease state E1 are analysed. Then the threshold conditions for extinction and persistence of the virus are derived by the rigorous theoretical proofs. It is found that when the noise is large enough, the virus will die out without constraint. When the noise is small, the virus will become extinct under the condition R 0 * 1 and persistence under R 0 * * > 1 . Besides, the upper bound and lower bound for persistence have been given. At last, some numerical simulations are carried out to support our results. The conclusion of this paper could help provide the theoretical basis for the further study of the virus infection.

Yilmaz et al. (Fluct. Noise Lett. 17, 1830002, 2018) investigated the stochastic phenomenological bifurcations of a generalized Chua circuit driven by Skew-Gaussian distributed noise. They proved it is possible to decrease the number of scrolls by properly choosing the stochastic excitation, manipulating the skewness and noise intensity parameters. Based on the latter, this paper proposes an extension of skew-gaussian noise based on the family of Scale Mixtures of Skew-normal (SMSN) distributions, which includes the skew- t , the skew-gaussian, and the gaussian noises as particular cases. The Lorenz, Generalized Lorenz, Proto–Lorenzand Rossler attractors driven by skew- t distributed noise are considered. Results show that the chaotic regime’s behavior is influenced by the freedom parameter degrees of skew- t noise, increasing the noise variance. This paper concludes that noise intensity increases by rescaling the skew- t distribution at zero mean, rather than by increasing the asymmetry parameter.

Urban spatial system is a pre-fractal system, which has a sensitive reaction zone similar to the self-organized critical state (SOC) of complex systems on different spatial scales. In this paper, theoretical analysis includes the interpretation of multifractal dimension spectrum (f(a)-a(q) fitted curve) and its parameters. According to q-value distribution area, the fitted curve is divided into interval A (typical multifractal system distribution), interval B (atypical multifractal system distribution) and interval C (mediocre distribution). Furthermore, this paper launches a case study of the four-year period of construction land in Zhengzhou since the reform and opening up to reveal the spatial-temporal evolution features and operational mechanism of this region from a broader dimension. Analysis shows that Zhengzhou's development has long been in a spatial development trend dominated by chaos (interval B) and mediocrity (interval C), and the influence of q-level with spatial multifractal feature (interval A) on each spatial scale is relatively weak. Until the 2010s, multifractal feature tends to be strengthened, and there is a tendency to dominate regional spatial evolution. The significance of this study is that the combination and evolution of three intervals of the f(a)-a(q) fitting curve can be used to reflect the trend and regularity of regional spatial multifractal system and spatial-temporal succession relationship between them, so as to deeply understand the inherent and complete development and evolution of the geographical objects.

This paper is concerned with a SIR model for the spread of an epidemic amongst a population of individuals with random additive perturbations of the transmission rate. Recently, many papers are devoted to the case of the Gaussian white noise perturbation. However, this model violates the condition of positivity of the transmission rate. In the paper we consider three models of the random perturbation which do not change this condition. The two of them are the telegraphic noise, trichotomous noise and the third is the bounded noise. Explicit conditions of the amost sure asymptotic stability of disease-free equilibrium state are obtained in the case of the first two models. An efficient numerical procedure is proposed for the construction of stability charts in the case of bounded noise. The effect of random perturbations on the stability behavior of disease-free equilibrium is discussed. Some transient mean-square properties of the SIR stochastic epidemic model are also presented.

In a published paper of Journal of Chaos, Solitons and Fractals, some miss prints were found. One is about the calculation of the solution and also about the expression of the solution using Green’s function of the fractional differential equation studied in this paper; The others are for the properties for Green’s function, these miss prints affected the deriving of the main results in Khan’s paper, some corrections for Khan’s paper should be needed. In this paper, we make some corrections and give the correct proof proceeding for the results, a new example is given to validate part of the proven results.

The theory of classical ergodic Markov genetic diffusion process, Feller genetic diffusion, and the genetic drift diffusion with and without selection and mutation for a single unlinked locus with two alleles, Kimura models, is studied. These genetic problems are numerically investigated on the long run. The approximates solutions of these models are described as the conditional probabilities to find the specific gene with frequencies between zero and one at generation T . As a surprise, we notice that the summation of these approximate solutions lose their unity rapidly as the number of generations increase than five, i.e. T > > 5 . The earlier biologists did due the reason to the migration, immigration and death of individuals. Till now, there is no mathematical proof to this phenomenon. In this paper, we solve this problem by extending the first order time derivative to the time fractional derivative, to study the effect of the memory on these models. This extension ensures that the summations of the approximate solutions, i.e. of the Non-Markov cases, are one at any number of generations. The time evolution of the approximate solutions are simulated and compared with other published papers, for different values of generations and for the Markov and Non-Markov cases with different values of the fractional order β . The convergence of the approximate descrete solutions and the reversibility property of these stochastic processes for both the Markov cases and Non-Markov cases are also numerically simulated and discussed.

Focusing on the results of Rajasekar (2020) and the continuous dynamics of stochastic differential equation (SDE) developed by Mao (1997), a stochastic SIRSI epidemic model with saturation incidence rate and logistic growth is investigated in this paper. First, we propose and prove that the unique solution of stochastic model is globally positive. By constructing some suitable Lyapunov functions, the sufficient condition R 0 h > 1 is obtained for the unique stationary distribution which has ergodicity property. Next, by solving the corresponding Fokker-Planck equation, we derive the approximate probability density function around the quasi-endemic equilibrium of the stochastic system. The above stationary distribution and density function can reveal all statistical properties of the disease persistence. In addition, by comparison with other existing articles, our developed theoretical results and some numerical simulations are introduced at the end of this paper.

In this paper, we propose a new method to improve the safety of secure communication. This method uses the generalized dislocated lag projective synchronization and function projective synchronization to form a new generalized dislocated lag function projective synchronization. Moreover, this paper takes the examples of fractional order Chen system and Lu system with uncertain parameters as illustration. As the parameters of the two systems are uncertain, the nonlinear controller and parameter update algorithms are designed based on the fractional stability theory and adaptive control method. Moreover, this synchronization form and method of control are applied to secure communication via chaotic masking modulation. Many information signals can be recovered and validated. Finally, simulations are used to show the validity and feasibility of the proposed scheme.

This paper presents the influence of two kinds of variable growth hypernetworks on the scaling law of hypernetworks. Our model can be degenerated to the original evolving model. On one hand, we establish the variable growth of hyperedges evolving hypernetworks model and obtain the stationary average hyperdegree distribution of the model by employing the Poisson process theory and the continuous method. The theoretical analyses in this paper agree with the conducted numerical simulations. The results show that the transient average hyperdegree distribution of the hypernetwork follows the scale-free law. However, the stationary average hyperdegree distribution does not, which indicates that the number of hyperedges at each time step affects the scaling law of hypernetworks. On the other hand, we establish the new nodes variable growth hypernetworks model and obtain analytical solutions of the transient average hyperdegree distribution. We simulate to compare the different node growth rate effect on the hyperdegree distribution characteristics. The results show that the growth rate does not affect the scaling law properties of the hypernetworks. Only when λ = 1 can we obtain analytical solutions of stationary average hyperdegree distribution of the hypernetwork which is not scale-free law.

This paper is concerned with a class of neutral-type Hopfield neural networks with multi-proportional delays and leakage delays. Using the differential inequality theory, a set of sufficient conditions which guarantee that all solutions of neutral-type Hopfield neural networks with multi-proportional delays and leakage delays converge exponentially to zero vector are derived. Computer simulations are carried out to verify our theoretical findings. The obtained results of this paper are new and complement some previous studies.

This review paper aims at answering a basic question on the sign of Lyapunov exponents. A few recent papers reported hyperchaotic system having the sign of Lyapunov exponents as (+, +, −, −). Such (+, +, −, −) sign of Lyapunov exponents is in contradiction with the well known (+, +, 0, −) sign of Lyapunov exponents for a 4-D hyperchaotic system. This paper thus discusses various issues related to Lyapunov exponents and proves that the reported sign of (+, +, −, −) Lyapunov exponents is actually (+, +, 0, −) or (+, 0, −, −). This clarification is very important and essential since Lyapunov exponents are the only quantitative measure for the existence of hyperchaos. Three different algorithms are used for calculating the Laypunov exponents to prove the actual sign of Lyapunov exponents for a hyperchaotic system.

The paper explores an eco-epidemiological model with weak Allee in predator, and the disease in the prey population. We consider a predator-prey model with type II functional response. The curiosity of this paper is to consider different competition coefficients within the prey population, which leads to the emergent carrying capacity. We perform the local and global stability analysis of the equilibrium points and the Hopf bifurcation analysis around the endemic equilibrium point. Further we pay attention to the chaotic dynamics which is produced by disease. Our numerical simulations reveal that the three species eco-epidemiological system without weak-Allee induced chaos from stable focus for increasing the force of infection, whereas in the presence of the weak-Allee effect, it exhibits stable solution. We conclude that chaotic dynamics can be controlled by the Allee parameter as well as the competition coefficients. We apply basic tools of non-linear dynamics such as Poincare section and maximum Lyapunov exponent to identify chaotic behavior of the system.

This paper presents an effective and efficient kernel approach to recognize image set which is represented as a point on extended Grassmannian manifold. Several recent studies focus on the applicability of discriminant analysis on Grassmannian manifold and suffer from not obtaining the inherent nonlinear structure of the data itself. Therefore, we propose an extension of Grassmannian manifold to address this issue. Instead of using a linear data embedding with PCA, we develop a non-linear data embedding of such manifold using kernel PCA. This paper mainly consider three folds: 1) introduce a non-linear data embedding of extended Grassmannian manifold, 2) derive a distance metric of Grassmannian manifold, 3) develop an effective and efficient Grassmannian kernel for SVM classification. The extended Grassmannian manifold naturally arises in the application to recognition based on image set, such as face and object recognition. Experiments on several standard databases show better classification accuracy. Furthermore, experimental results indicate that our proposed approach significantly reduces time complexity in comparison to graph embedding discriminant analysis.

The calculation of pullback by the analysis of strain on pipe shows that it involves many disciplines. It is hard to build a mechanical model that is very consistent with the actual working conditions and with the general applicability. After summarizing the calculation method of domestic and foreign authorities or national standards, the paper derives a set of its own formulas. Based on statistical data of actual projects, the paper develops a formula more suitable for ultra-long pullback by comparing the computing result with the actual measured drag force, that is for engineering design and site construction personnel reference. The calculated value of ultra-long pullback coincides well with the engineering value. Three components of pullback resistance, all occupy a high proportion in back-dragging load, and the contribution of the pullback load weights back Trawler changes dynamically. Thus proving that the method is a reasonable, simple, reliable and effective method to predict pipeline status, and makes a useful exploration for the research to improve the existing prediction models in the future.

A novel model referred to as two-dimensional continuous 3 × 3 order hidden Markov model is put forward to avoid the disadvantages of the classical hypothesis of two-dimensional continuous hidden Markov model. This paper presents three equivalent definitions of the model, in which the state transition probability relies on not only immediate horizontal and vertical states but also immediate diagonal state, and in which the probability density of the observation relies on not only current state but also immediate horizontal and vertical states. The paper focuses on the three basic problems of the model, namely probability density calculation, parameters estimation and path backtracking. Some algorithms solving the questions are theoretically derived, by exploiting the idea that the sequences of states on rows or columns of the model can be viewed as states of a one-dimensional continuous 1 × 2 order hidden Markov model. Simulation results further demonstrate the performance of the algorithms. Because there are more statistical characteristics in the structure of the proposed new model, it can more accurately describe some practical problems, as compared to two-dimensional continuous hidden Markov model.

This paper aims to the three-dimensional continuous chaotic system and shape of the chaotic attractor by utilizing the basic theory of plane curves in classical differential geometry, the continuous controller is synthesized for the master–slave synchronization in shape. This means that the slave system can possess the same shape of state trajectory with the master system via the continuous controller. The continuous controller is composed of three sub-controllers, which respectively correspond to the master–slave synchronization in shape for the three projective curves of the chaotic attractor onto the three coordinate planes. Moreover, the proposed shape synchronization technique as well as application of control scheme to secure communication is also demonstrated in this paper, where numerical simulation results show the proposed control method works well.