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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.

The classical Morse–Sard theorem claims that for a mapping v : R n → R m + 1 of class C k the measure of critical values v ( Z v , m ) is zero under condition k ≥ n − m . Here the critical set, or m-critical set is defined as Z v , m = { x ∈ R n : rank ∇ v ( x ) ≤ m } . Further Dubovitskiĭ in 1957 and independently Federer and Dubovitskiĭ in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the C k category. Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v : R n → R d belongs to the Holder class C k , α , 0 ≤ α ≤ 1 , then for every q > m the identity H μ ( Z v , m ∩ v − 1 ( y ) ) = 0 holds for H q -almost all y ∈ R d , where μ = n − m − ( k + α ) ( q − m ) . Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa. The result is new even for the classical C k -case (when α = 0 ); similar result is established for the Sobolev classes of mappings W p k ( R n , R d ) with minimal integrability assumptions p = max ⁡ ( 1 , n / k ) , i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools).

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.

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.

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.

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.

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.

This is a chiefly expository paper on the subject of the title which, in our view, has not received a detailed treatment in the literature which is commensurate with its importance. We expect the results presented here to be useful in a number of contexts. For example, several of them will be applied in a forthcoming paper by the authors., Comment: 25 pages

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.

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.

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 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.

Reduction theory has played a major role in the study of Hamiltonian systems. Whilst the Hamilton–Jacobi theory is one of the main tools to integrate the dynamics of certain Hamiltonian problems and a topic of research on its own. Moreover, the construction of several symplectic integrators relies on approximations of a complete solution of the Hamilton–Jacobi equation. The natural question that we address in this paper is how these two topics (reduction and Hamilton–Jacobi theory) fit together. We obtain a reduction and reconstruction procedure for the Hamilton–Jacobi equation with symmetries, even in a generalized sense to be clarified below. Several applications and relations to other reduction of the Hamilton–Jacobi theory are shown in the last section of the paper. It is remarkable that as by-product we obtain a generalization of the Ge–Marsden reduction procedure [18] and the results in [17] . Quite surprisingly, the classical ansatze available in the literature to solve the Hamilton–Jacobi equation (see [2] , [19] ) are also particular instances of our framework.

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 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 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.

The paper addresses the mathematical analysis of the fishery model in the context of the fractional derivative operator. We use the Caputo–Fabrizio derivative in the investigations. We first prove the fishery model is biologically well definite by proposing the existence and the uniqueness of its solution. The main objective of this paper is to study the dynamics of the predator and the prey in the fishery model when the fractional-order derivative is used. Notably, we analyze the impact of the fractional-order derivative on the dynamics of the fishery model explicitly. To answer this issue, we introduce a new numerical scheme based on the discretization of the fractional integral associated with the Caputo–Fabrizio derivative. The numerical simulations of the solutions of the fractional model are intended to illustrate the numerical scheme presented in our paper. We finish by analyzing the local and global asymptotic stability of the equilibrium points using the Jacobian matrix and the Lyapunov direct method. The Lyapunov function is constructed using standard construction. We notice here that the solutions of the fractional fishery model with different values of the orders describes a cycle when we depict them in three dimensional spaces in time, and furthermore the marine reserves ensure the sustainability of fractional system.

Uncertain differential equation (UDE) has been widely applied in the financial market, and many option pricing formulas are derived based on UDE. But the existing literature don’t consider the parameter estimation of UDE, and the parameters are just assumed as some given constants. This paper will present an approach to estimate the unknown parameter of UDE from discretely sampled data via α-path method for the first time. And the concepts of forecast value and confidence interval are introduced to predict the future value in a UDE. As five special UDEs, the parameters estimations of arithmetic Liu process, geometric Liu process, uncertain Ornstein-Uhlenbeck process, uncertain mean-reverting process, and uncertain exponential Ornstein-Uhlenbeck process are also derived, and the corresponding numerical examples are given. Furthermore, this paper proposes a means to estimate unknown parameters for the general geometric Liu process, and as an application, this method will be successfully used in Alibaba stock.

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.

Stability in distribution, implying the existence of the invariant probability measure, is an important measure of stochastic hybrid system. However, the effect of Levy jumps on the stability in distribution is still unclear. In this paper, we consider a n-species competitive Lotka–Volterra model with Levy jumps under regime-switching. First, we prove the existence of the global positive solution, obtain the upper and lower boundedness. Then, asymptotic stability in distribution as the main result of our paper is derived under some sufficient conditions. Finally, numerical simulations are carried out to support our theoretical results and a brief discussion is given.

In this paper we report for the first time on the binary generalized synchronization, when for the certain values of the coupling strength two unidirectionally coupled dynamical systems generating the aperiodic binary sequences are in the generalized synchronization regime. The presence of the binary generalized synchronization has been revealed with the help of both the auxiliary system approach and the largest conditional Lyapunov exponent calculation. The mechanism resulting in the binary generalized synchronization has been explained. The finding discussed in this paper gives a strong potential for new applications under many relevant circumstances.

In this paper, we consider a time fractional-order derivative for a diffusive mussel–algae model. The existence of pattern formation was the subject of interest of many previous research works in the case of the diffusive mussel–algae model. Examples include the Turing instability, Hopf bifurcation, Turing-Hopf bifurcation, and others. The presence of the time–fractional–order derivative never been investigated in this model. Next to it ecological relevant, it can generate some important patterns. One of these patterns is produced by the presence of the Turing-Hopf bifurcation. Therefore, our main interest is to analyze the effect of the time fractional–order derivative on the spatiotemporal behavior of the solution, which never been achieved for the mussel-algae model. Besides, Turing–Hopf was studied exclusively on the classical reaction-diffusion systems, where it was also considered for the diffusive mussel-algae model. Thus, our paper puts the fist steps on proving the existence of this type of codimension bifurcation on the diffusive systems with time fractional–order–derivative systems. Further, a suitable numerical simulations are used for confirming the theoretical obtained results.

In this paper, we present analytical solutions for linear and nonlinear neutral Caputo-fractional pantograph differential equations. An attractive new method we called the Laplace-Residual power series method, is introduced and used to create series solutions for the target equations. This method is an efficient simple technique for finding exact and approximate series solutions to the linear and nonlinear neutral fractional differential equations. In addition, numerical and graphical results are also addressed at different values of α to show the behaviors of the Laplace-Residual power series solutions compared with other methods such as Two-stage order-one Runge-Kutta, one-legθ, variational iterative, Chebyshev polynomials, Laguerre wavelet, Bernoulli wavelet, Boubaker polynomials, Hermit wavelet, Proposed and Pricewise fractional-order Taylor methods. Finally, several examples are also considered and solved based on this method to show that our new approach is simple, accurate, and applicable. Maple software is used to calculate the numerical and symbolic quantities in the paper.

In this paper, we propose a novel method for obtaining the solution of the fractional differential equation in the class of second-grade fluids models. The technique described in this paper is called the double integral method. The method generates, in general, an approximate solution for the fractional diffusion equations, the energy equations, or the heat equations. In our study, we use the generalized fractional derivative in Caputo–Liouville’s sense. For the illustrations of our method, we propose the graphical representations of the approximates solutions obtained by using the double integral method. We propose interpretations and physical discussions of the solutions obtained with the double integral method.

In this paper, we present an application of the homotopy perturbation transform method to compute the approximate analytical solution of the nonlinear fractional order computer virus propagation (CVP) model. The fractional derivatives are used in Caputo sense. The proposed approximate method generates the numerical solution in the shape of a rapid convergent series by utilizing the provided initial conditions. The main purpose of the paper is to analyze the effect of variation of fractional order α on the meeting time of susceptible, infected and recovered computers. Moreover, the local stability analysis of the fractional order computer virus model is also presented using Routh–Hurwitz stability criterion.

Let f be a polynomial in n variables over some number field and Z a subscheme of affine n-space. The notion of motivic oscillation index of f at Z was initiated by Cluckers in [7] and Cluckers-Mustaţǎ-Nguyen in [12] . In this paper we elaborate on this notion and raise several questions. The first one is stability under base field extension; this question is linked to a deep understanding of the density of non-archimedean local fields over which Igusa's local zeta function of f has a pole with given real part. The second one is around Igusa's conjecture for exponential sums with bounds in terms of the motivic oscillation index. Thirdly, we wonder if the above questions only depend on the analytic isomorphism class of singularities. By using various techniques as the GAGA theorem, resolution of singularities and model theory, we can answer the third question up to a base field extension. Next, by using a transfer principle between non-archimedean local fields of characteristic zero and positive characteristic, we can link all three questions with a conjecture on weights of l-adic cohomology groups of Artin-Schreier sheaves associated to jet polynomials. This way, we can answer all questions positively if f is a polynomial ‘of Thom-Sebastiani type’ with non-rational singularities. As a consequence, we prove Igusa's conjecture for arbitrary polynomials in three variables and polynomials with singularities of A − D − E type. In an appendix, we answer affirmatively a recent question of Cluckers-Mustaţǎ-Nguyen in [12] on poles of maximal order of twisted Igusa's local zeta functions.

This paper focuses on the Dysthe equation which is a higher order approximation of the water waves system in the modulation (Schrodinger) regime and in the infinite depth case. We first review the derivation of the Dysthe and related equations. Then we study the initial-value problem. We prove a small data global well-posedness and scattering result in the critical space L 2 ( R 2 ) . This result is sharp in view of the fact that the flow map cannot be C 3 continuous below L 2 ( R 2 ) . Our analysis relies on linear and bilinear Strichartz estimates in the context of the Fourier restriction norm method. Moreover, since we are at a critical level, we need to work in the framework of the atomic space U S 2 and its dual V S 2 of square bounded variation functions. We also prove that the initial-value problem is locally well-posed in H s ( R 2 ) , s > 0 . Our results extend to the finite depth version of the Dysthe equation.

In this paper we prove spectral multiplier theorems for abstract self-adjoint operators on spaces of homogeneous type. We have two main objectives. The first one is to work outside the semigroup context. In contrast to previous works on this subject, we do not make any assumption on the semigroup. The second objective is to consider polynomial off-diagonal decay instead of exponential one. Our approach and results lead to new applications to several operators such as differential operators, pseudo-differential operators as well as Markov chains. In our general context we introduce a restriction type estimates a la Stein-Tomas. This allows us to obtain sharp spectral multiplier theorems and hence sharp Bochner-Riesz summability results in some situation. Finally, we consider the random walk on the integer lattice Z n and prove sharp Bochner-Riesz summability results similar to those known for the standard Laplacian on R n .

Let $m\ge 1$, in this paper, our object of investigation is the regularity and and continuity properties of the following multilinear strong maximal operator $${\mathscr{M}}_{\mathcal{R}}(\vec{f})(x)=\sup_{\substack{R \ni x R\in\mathcal{R}}}\prod\limits_{i=1}^m\frac{1}{|R|}\int_{R}|f_i(y)|dy,$$ where $x\in\mathbb{R}^d$ and $\mathcal{R}$ denotes the family of all rectangles in $\mathbb{R}^d$ with sides parallel to the axes. When $m=1$, denote $\mathscr{M}_{\mathcal{R}}$ by $\mathcal {M}_{\mathcal{R}}$.Then, $\mathcal {M}_{\mathcal{R}}$ coincides with the classical strong maximal function initially studied by Jessen, Marcinkiewicz and Zygmund. We showed that ${\mathscr{M}}_{\mathcal{R}}$ is bounded and continuous from the Sobolev spaces $W^{1,p_1}(\mathbb{R}^d)\times \cdots\times W^{1,p_m}(\mathbb{R}^d)$ to $W^{1,p} (\mathbb{R}^d)$, from the Besov spaces $B_{s}^{p_1,q} (\mathbb{R}^d)\times\cdots\times B_s^{p_m,q}(\mathbb{R}^d)$ to $B_s^{p,q}(\mathbb{R}^d)$, from the Triebel-Lizorkin spaces $F_{s}^{p_1,q}(\mathbb{R}^d)\times\cdots\times F_s^{p_m,q}(\mathbb{R}^d)$ to $F_s^{p,q}(\mathbb{R}^d)$. As a consequence, we further showed that ${\mathscr{M}}_{\mathcal{R}}$ is bounded and continuous from the fractional Sobolev spaces $W^{s,p_1}(\mathbb{R}^d)\times \cdots\times W^{s,p_m}(\mathbb{R}^d)$ to $W^{s,p}(\mathbb{R}^d)$ for $0, Comment: 49 pages

In this paper we prove classification results for gradient shrinking Ricci solitons under two invariant conditions, namely nonnegative orthogonal bisectional curvature and weakly PIC1, without any curvature upper bound. New results on ancient solutions for the Ricci and Kahler-Ricci flow are obtained. Applications to Kahler manifolds with almost nonnegative orthogonal bisectional curvature are derived as consequences. The main new feature is that no curvature upper bound is assumed.

In this paper, the second in a series, we continue to study the generalized Lame equation with the Treibich-Verdier potential y ″ ( z ) = [ ∑ k = 0 3 n k ( n k + 1 ) ℘ ( z + ω k 2 | τ ) + B ] y ( z ) , n k ∈ Z ≥ 0 from the monodromy aspect. We prove the existence of a pre-modular form Z r , s n ( τ ) of weight 1 2 ∑ n k ( n k + 1 ) such that the monodromy data ( r , s ) is characterized by Z r , s n ( τ ) = 0 . This generalizes the result in [17] , where the Lame case (i.e. n 1 = n 2 = n 3 = 0 ) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations Δ u + e u = 16 π δ 0 and Δ u + e u = 8 π ∑ k = 1 3 δ ω k 2 on a flat torus has the same number of even solutions. This result is quite surprising from the PDE point of view.

This paper investigates the dynamic behavior of a diffusive plankton model with defensive and offensive effects in two cases. For the single compartment model, we first derive the sufficient conditions for the stability and Hopf bifurcation of coexisting equilibrium, which implies that the changes of defense and offense can cause oscillation of planktonic population. Then the properties of Hopf bifurcation are discussed by center manifold theorem. For the spatially extended model, we obtain the sufficient conditions for Turing instability and Hopf bifurcation. It is observed that spatial patterns put in place, under the interaction of diffusion, defense and offense. Finally, some numerical simulations are carried out to support the analytical results.

In this paper, modeling the heat flow equation with fractional-fractal differentiation is considered. This problem has opened a new viewpoint for modeling the classical and the fractional differentiation. We presented the existence of positive solution of the new model using the fixed-point approach and we established the uniqueness of the positive solution. Finally, we provide an example to illustrate one of the main results.

In this paper, the existence results of the solution and stability are focused for the variable fractional order differential equation. In view of the definitions of three kinds of Caputo variable fractional order operator, the sufficient condition of the solution existence for the variable fractional order differential system is obtained by use of Arzela–Ascoli theorem. Moreover, some criterions of the Mittag–Leffler stability and asymptotical stability are proposed for the variable fractional order differential system according to the Fractional Comparison Principle.

This paper is concerned with an operational matrix method based on the shifted Legendre cardinal functions for solving the nonlinear variable-order time fractional Schrodinger equation. The variable-order fractional derivative operator is defined in the Atangana–Baleanu–Caputo sense. Through the way, a new operational matrix of variable-order fractional derivative is derived for the shifted Legendre cardinal functions and used in the established method. More precisely, the unknown solution is separated into the real and imaginary parts, and then these parts are expanded in terms of the shifted Legendre cardinal functions with undetermined coefficients. These expansions are substituted into the main equation and the generated operational matrix is utilized to extract a system of nonlinear algebraic equations. Thereafter, the yielded system is solved to obtain an approximate solution for the problem. The precision of the established approach is examined through various types of test examples. Numerical simulations confirm that the suggested approach is high accurate in providing satisfactory results.

We discuss the smallness of the set of attractors for iterated function systems. This paper is an attempt to measure the difference between the family of IFS attractors and a broader family, the set of attractors for the weak iterated function systems. We prove that the IFS attractors form a σ-porous subset of K([0, 1]d), the family of compact subsets of a metric space [0, 1]d. We also show that weak IFS attractors form a first category set.