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In this paper we construct, by using the theory of exponential dichotomies, a Melnikov-type function by which we can detect the existence of homoclinic orbits for the perturbed systems x = g ( x ) + ɛ h ( t , x , ɛ ) . Our result of this paper may be complementary to that of K.J. Palmer[3].

In this paper the derivation algebra of Lie algebra ∑ ¯ of characteristic two is determined. Using this result we obtain the necessary and sufficient condition under that ∑ ¯ of characteristic two is the restreted Lie algebra. Finally we prove that ∑ ¯ of characteristic two isn't isomorphic the Lie algebras of characteristic two which are know n by authors of this paper. But it still is a filtered deformation of the Lie algebra of H-type i.e. Its associated grated algebra Gr ∑ ¯ is isomorphic to H ( 2n + 2. r ).

In this paper, the bifurcation characteristics of the forced van der Pol oscillator on a specific parameter plane, including intermediate parameter regions, are investigated. The successive arrangement of the dominant mode-locking regions, where a single subharmonic solution with the rotation number, 1 (2k + 1) , exists, and the transitional zones between them are depicted. The transitional zones are explicitly proposed to be classified into two groups according to the different global characters: (1) the simple transitional zones, in which coexistence of two mode-locked solutions with rotation numbers 1 (2k ± 1) appear; (2) the complex transitional zones, in which the sub-zones with the mode-locked solutions, whose rotation numbers are rational fractions between 1 (2k + 1) and 1 (2k − 1) , and the quasi-periodic solutions exist. The emphasis of this paper is to study the evolution of the global structures in the transitional zones. A complex transitional zone generally evolves from a Farey tree, when the forcing amplitude is small, to a chaotic regime, when forcing amplitude is sufficiently large. It is of great interest that the sub-zone with a rotation number, 1 2k , which has the largest width within a complex transitional zone, can usually intrude into the dominant regions of 1 (2k − 1) before it completely vanishes. Moreover, the features of overlaps of mode-locking sub-zones and the number of coexistence of different attractors are also discussed.

This paper generalizes conditional simulation technique of uni-variate Gaussian random fields by the stochastic interpolation proposed by Hoshiya, to multi-variate random fields. The kriging estimation of multi-variate Gaussian fields is proposed, and basic formulation for conditional simulation of multi-variate random fields is established. For the particular case of uncorrelated components of multi-variate field, the formulation reduces to that of uni-variate field given by Hoshiya. The paper also provides proofs of some important properties of the estimation error vector, which guarantee that the conditional simulation of the multi-variate field can be implemented by separately computing its kriging estimate and simulating the error vector. An analytical example of two-variate field is elucidated and some numerical results are discussed.

Considering the physico-mathematical problem set at the title of this paper, knowing also that it covers actual circumstances of dynamical derailments of vehicles with large safety consequences, one has to face unusual modelling difficulties. Mainly two results are awaited: (1) the description of the mechanisms allowing derailment and (2) an evaluation of the risk. Obviously (and fortunately), dynamical derailments resulting of random association of track geometry with chaotic oscillations of the wagons are rare, almost impossible to produce experimentally on purpose and consequently difficult to describe. However, the available experimental field is not empty but presents lots of time histories of chaotic signals (forces) which cannot be studied on a deterministic basis. As these vehicles are quite non-linear mechanical systems, the only possibility of their modelling is numerical. Thus, there is only one chance to be effective, it is to develop numerical models using simplifying hypothesis in order to yield the shortest CPU times so as to be able to test and adjust the effects of modelling assumptions and to cover the largest possible field, including influence of wear modifications of the rolling profiles. This task has been done (the development lasted 10 years) and our method, which we do not pretend to be the only one, is explained in this paper focusing on physical assumptions rather than on already published mathematical developments. Among numerous difficulties, we had to face chaotic results of the computations. They appeared when vs/time numerical results of long computations were found depending, for instance, on the type of the micro-processor or on the way of factorising the Fortran statements. We are now convinced that there is no solution to this difficulty in the direction of increasing numerical precision because it comes from the nature of the physical problem itself. We could not prove it, but, as all experimental signals were different from each other, one has to admit that the physical reality that is to be modelled cannot be observed deterministically and, consequently, that, even if it would be possible to develop a deterministic model, it would be impossible to validate it.

Discrete time dynamical systems on discrete state spaces called lattices are the subject of this paper. These dynamical systems have properties that differ from discrete time dynamical systems on continuous state spaces. These differences include the existence of discrete integrals as constants of motion along orbits. Computer arithmetic, on lattices of floating point numbers, is used typically to evaluate orbits of discrete dynamical systems defined on continuous state spaces, i.e. the iteration of maps on continuous spaces. For chaotic dynamical systems, in which orbits diverge rapidly, computer arithmetic is not suited to this purpose. Calculations based on exact arithmetic performed on lattices are developed in this paper. New dynamical properties are found for discrete dynamical systems on discrete state spaces by restricting to embedded lattices discrete dynamical systems on continuous state spaces. The new results include theorems on existence of discrete integrals for discrete time dynamical systems on lattices, entropy of orbits described by probability distributions, and decompositions of lattices by the dynamics into orbits. A new viewpoint emerges when one takes a countable infinite union of lattices. This is a way in which the computable number field can be approached. The nature of dynamics changes when the discrete state space is infinite. Chaos appears naturally in this setting and in an interesting way.

The problem of conditional simulation of random fields gained a significant interest recently due to its applications to urban earthquake monitoring. In this paper, for the first time in the literature, the method of conditional simulation of non-Gaussian random fields is developed. It combines previous techniques of iterative procedure of unconditional simulation of non-Gaussian fields, and the procedure of conditional simulation of Gaussian random fields. To contrast the agreement between the simulated correlation function and targeted correlation function, the numerical error is decomposed into two parts, namely, into simulation error and mapping error. Simulation error can be reduced by increasing number of samples while mapping error is eliminated by the suitable iteration procedure. In this paper univariate and time-independent random fields are considered. Numerical example shows that the correlation structure and probability distribution of the simulated random field have excellent agreements with given correlation structure and probability distribution, respectively.

This paper deals with the order of magnitude of the partial sums of the spherical harmonic series and its convergence rate in Bessel potential spaces. The partial results obtained in the paper are the analogue of those on the circle.

This paper concerns analytic integrated C-semigroups, analytic C-semigroups and analytic integrated semigroups with nondensely defined generators. The main purpose of this paper is to establish some generation theorems for these semigroups, which improve and perfect the related results given by Da Prato, deLaubenfels, Neubrander and others. Moreover, several perturbation theorems arc also included.

In this paper, we introduce the concepts of cut ratio and covered ratio of domains. Further, we give an explicit estimation of the constant in the general Ahlfors’ covering theorem and Ahlfors'fundamental theorem on a sphere. Because of the importance of these theorems, it is no doubt that these rstimations is significant. Applying these yesults, we improve a series of theorems, prove that there exists a sequence of filling circles for algebroidal function, and establish a new normal criterion. (These works will be done in other papers.)

This paper discusses the property of semimartingales of B-valued R. R. C. process W, and the relation between this property and critical stopping time σ w ˙ of W after giving the definition of σ w ˙ and the structural approach to σ w ˙ . A conclusion is drawn from this paper that the process W being a non–semimartingale is characterized by P ( σ w ˙ ∞ ) > 0 . Finally, two kinds of vector valued processes with R-control process are discussed, and they are shown to be non–semimartingales by the use of Theorem 2 in this paper.

In previous papers, the stationary distributions of a class of discrete and continuous-time random graph processes with state space consisting of the simple and directed graphs on N vertices were studied. In this paper, the random graph process is extended one important step further by allowing interaction of edges. Similarly, We obtain the expressions of the stationary distributions and prove that the process is ergodic under different conditions.

This paper presents a new method to approach the problem of finding empirical curves in some cases, especially when the smoothness of the curves is of paramount importance and the equations of them are troublesome to be expressed in terms of known mathematical formulas. As an example, a smooth numerical model of a part of the observed trajectory of the Jupiter in the celestial sphere is given. Thus this method is to fit a smooth numerical model to the given data, instead of fitting a curve. The theoretical background of this method lies mainly in a theory called the method of sequence of circular rates which gives out some fundamental relations between a smooth curve and an arbitrary finite set of points lying on it. But some new ideas are also introduced to meet the purpose of the present paper.

In this paper, by Laplace transform version of the Trotter-Kato approximation theorem and the integrated C-semigroup introduced by Myadera, the authors obtained some Trotter-Kato approximation theorems on exponentially bounded C-semigroups, where the range of C (and so the domain of the generator) may not be dense. The authors deduced the corresponding results on exponentially bounded integrated semigroups with nondensely generators. The results of this paper extended and perfected the results given by Lizama, Park and Zheng.

By making use of the differential inequalities, in this paper we study the uniqueness of solutions of the two kinds of the singularly perturbed boundary value problems for the nonlinear third order ordinary differential equation with a small parameter ɛ>0: ɛ y ″ = f ( x , y , y ′ , y ″ , ɛ ) , y ′ i ) ( 0 , ɛ ) = a i ( ɛ ) , y ( 1 , ɛ ) = β ( ɛ ) , y ′ ( 1 , ɛ ) = γ ( ɛ ) , where i = 1, 2; αi(ɛ), β(ɛ) and Λ(ɛ) are functions defined on (0, ɛ0], while ɛ0>0 is a constant. This paper is the continuation of our works [4, 6].

This paper is concerned with the existence-uniqueness and the asymptotic behavior of the solution of a coupled system of reaction diffusion equations which arose from biochemistry. Under Dirichlet boundary condition, the existence and the bifurcation of nonegative steady-state solutions of the system will be studied. The principal methods used in this paper are the monotone method ond the topological degree method.

In this paper, a new proof of D'Alembert formula for Cauchy problems of wave equation is proposed. And some D'Alembert-type formulas for Cauchy problem with discontinuous coefficients or semi-infinite initial boundary value problem are proposed. Based on above discussion, the characteristics methods for the direct and inverse problems of the stress wave equation for uniform layered elastic medium are investigated. The aim of this paper is to stress the significance and application of the characteristics concept and approach.

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.

This paper investigates the structures and properties of the spatial Julia set generated by a fractional complex Lotka-Volterra system with noise. The influence of several types of dynamic noise upon the system’s Julia set is quantitatively analyzed through the Julia deviation index. Then, the symmetry of the Julia set is discussed and the symmetrical structure destruction caused by noise is studied. Numerical simulations are presented to further verify the correctness and effectiveness of the main theoretical results.

In this work, we introduce ABC-Caputo operator with ML kernel and its main characteristics are discussed. Viral diseases models for AIDS and Zika are considered, and finally, as third model, the macroeconomic model involving ABC fractional derivatives is investigated, respectively. It is presented that the AB Caputo derivatives satisfy the Lipschitz condition along with superposition property. The numerical methods for solving the fractional models are presented by means of ABC fractional derivative in a detailed manner. Finally the simulation results obtained in this paper according to the suitable values of parameters are also manifested.

In this paper, a competition model which describes the spatial interaction among three species in nonlinear fashion is considered. In the model, the standard time derivative is replaced with the Atangana-Baleanu fractional operator in the sense of Caputo. Linear stability analysis which serves as a guide in the choice of parameters when numerically simulating the full system is also examined. The existence and uniqueness of solutions are studied via a fixed point theorem. Different numerical approximation techniques are introduced. Numerical results presented in one and two dimensions revealed some spatiotemporal Turing patterns such as stripes and spots.

Researchers around the world are still wondering about the real origin and causes of hidden oscillating regimes and hidden attractors exhibited by some non-linear complex models. Such models are characterized by a dynamic with a basin of attraction that does not contain neighborhoods of equilibrium points. In this paper, we show that hidden oscillating regimes and hidden attractors can also exist in systems resulting from a combination with fractional differentiation. We apply a fractional derivative with Mittag–Leffler Kernel to a dynamical system with an exponential non-linear term and analyzed the resulting model both analytically and numerically. The combined model, which has no equilibrium points is however shown to display complex oscillating trajectories that culminate in chaos. Numerical simulations show some bifurcation dynamics with respect to the derivative order β and prove that the observed chaotic behavior persists as β varies. These observations made here allow us to say that the fractional model under study belongs to the category of systems with hidden oscillations.

This paper considers mathematical analysis and numerical treatment for fractional reaction-diffusion system. In the model, the first-order time derivatives are modelled with the fractional cases of both the Atangana-Baleanu and Caputo-Fabrizio derivatives whose formulations are based on the notable Mittag-Leffler kernel. The main system is examined for stability to ensure the right choice of parameters when numerically simulating the full model. The novel Adam-Bashforth numerical scheme is employed for the approximation of these operators. Applicability and suitability of the techniques introduced in this work is justified via the evolution of the species in one and two dimensions. The results obtained show that modelling with fractional derivative can give rise to some Turing patterns.

This paper aims to study the critical sectional area (CSA) of surge chamber considering the nonlinearity of head loss of diversion tunnel and steady output of turbine. Firstly, three basic equations for hydropower station with surge chamber are established. Four mathematical models for the derivation of CSA of surge chamber are constructed. Then, the stability of hydropower station with surge chamber is analyzed by Hopf bifurcation. Based on the critical stable state of hydropower station, the formulas for the CSA of surge chamber are derived. Finally, the verification and comparison of different CSAs are conducted. The correctness and rationality of obtained formulas are explained. The results indicate that, under load increase operation condition, both the nonlinearity of head loss of diversion tunnel and nonlinearity of steady output of turbine can reduce the value of CSA of surge chamber and are favorable for the stability of hydropower station. Under load decrease operation condition, the rules are opposite. Under both load increase and load decrease operation conditions, the effect of the nonlinearity of head loss of diversion tunnel on CSA of surge chamber is much more significant than that of the nonlinearity of steady output of turbine. The formula for CSA of surge chamber considering both the nonlinearity of head loss of diversion tunnel and nonlinearity of steady output of turbine can be expressed as an amplification coefficient times of the Thoma formula. That formula has a higher precision than Thoma formula.

The purpose of the paper is to provide an efficient method for the continuously monitored geometric Asian options under the double Heston model. By introducing two small parameters, we slightly modify the double Heston model. With singular and regular perturbation techniques, we derive the first-order asymptotic expansions for pricing geometric Asian options with fixed and floating strikes and provide the convergence of the asymptotic formulae. We also provide the Greeks of geometric Asian options. Numerical results verify the efficiency of the pricing method. We calibrate the modified model to real markets and examine the impacts of two-factor volatilities on geometric Asian option prices.

In this paper, a design of Poincare maps and time–piecewise–constant state–feedback control laws for the stabilization of limit cycles in periodically–forced, non–autonomous, nonlinear dynamical systems is achieved. Our methodology is based mainly on the linearization of the nonlinear dynamics around a desired period–m unstable limit cycle. Thus, this strategy permits to construct an explicit mathematical expression of a controlled Poincare map from the structure of several local maps. An expression of the generalized controlled Poincare map is also developed. To make comparisons, we design three different time–piecewise–constant control laws: an mT–piecewise–constant control law, a T–piecewise–constant control law and a T n –piecewise–constant control law. As an illustrative application, we adopt the chaotic Duffing oscillator. By applying the designed piecewise–constant control laws to the Duffing oscillator, the system is stabilized on its desired period–m limit cycle and hence the chaotic motion is controlled.

In this paper we are established the existence of positive solutions (EPS) and the Hyers-Ulam (HU) stability of a general class of nonlinear Atangana-Baleanu-Caputo (ABC) fractional differential equations (FDEs) with singularity and nonlinear p-Laplacian operator in Banach’s space. To find the solution for the EPS, we use the Guo-Krasnoselskii theorem. The fractional differential equation is converted into an alternative integral structure using the Atangana-Baleanu fractional integral operator. Also, HU-stability is analyzed. We include an example with specific parameters and assumptions to show the results of the proposal.

This paper investigates a novel version for the nonlinear 2D telegraph equation involving variable-order (V-O) time fractional derivatives in the Atangana–Baleanu–Caputo sense with Mittag–Leffler non-singular kernel. A meshfree method based on the moving least squares (MLS) shape functions is proposed for the numerical solution of this class of problems. More precisely, the V-O fractional derivatives in this model are approximated by the finite difference scheme at first. Then, the θ-weighted method is utilized to derive a recursive algorithm. Next, the solution of the problem is expanded in terms of the MLS shape functions with undetermined coefficients. Eventually, by substituting this expansion and its partial derivatives into the original equation, solution of the problem in each time step is reduced to the solution of a linear system of algebraic equations. Several numerical examples are investigated to show the applicability, validity and accuracy of the presented method. The achieved numerical results reveal that the established method is high accurate in solving such V-O fractional models.

In this paper, the third order partial differential equation defined by Caputo fractional derivative with Atangana–Baleanu derivative has been investigated. The stability estimates are proved for the exact solution. Difference schemes for Crank–Nicholson finite difference scheme method is constructed. The stability of difference schemes for this problem is shown by Von Neumann method (Fourier analysis method). Numerical results with respect to the exact solution confirm the accuracy and effectiveness of the technique. The reproducing kernel function for the problem has been found.

In this paper we analyse random walk on a fractal structure, specifi- cally fractal curves, using the recently develped calculus for fractal curves. We consider only unbiased random walk on the fractal stucture and find out the corresponding probability distribution which is gaussian like in nature, but shows deviation from the standard behaviour. Moments are calculated in terms of Euclidean distance for a von Koch curve. We also analyse Levy distribution on the same fractal structure, where the dimen- sion of the fractal curve shows significant contribution to the distrubution law by modyfying the nature of moments. The appendix gives a short note on Fourier transform on fractal curves., Comment: 15 pages, 4 figures

In this paper, the classical Duffing oscillator under dual frequency excitations is studied by the homotopy analysis method(HAM). Analytical study of the low-order approximations is firstly conducted and the saddle node(SN) bifurcation boundary for the initial guess solution is obtained. The maximum value bifurcation plot of the high order approximations with the bifurcation parameters f1 and λ1 are obtained and compared with the numerical solutions based on the Runge–Kutta method. The results show that the initial guess solution can qualitatively reflect the trend of the numerical solution, and the high order approximations agree well with the numerical solutions. The maximum value bifurcation plots of high order approximations show periodic and quasi-periodic solutions, which agree well with the numerical ones.

In this paper, we propose a fractional form of two-dimensional generalized mythical bird, butterfly wings and paradise bird maps involving the fractional conformable derivative of Khalil’s and Atangana’s type, the Liouville–Caputo and Atangana–Baleanu derivatives with constant and variable-order. We obtain new chaotical behaviors considering numerical schemes based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. Also, the dynamics of the proposed maps are investigated numerically through phase plots considering combinations of these derivatives and mixed integration methods for each map. The numerical simulations show very strange and new behaviors for the first time in this manuscript.

In this paper, we incorporate a nonlinear incidence rate and a logistic growth rate into a SIR epidemic model for a vector-borne disease with incubation time delay and Holling type II saturated treatment. We compute the basic reproduction number and show that it completely determines the local stability of the disease-free equilibrium. Sufficient conditions for the existence of backward bifurcation and Hopf bifurcation are also established. Furthermore, we determine the direction and stability of the Hopf bifurcation around the endemic equilibrium by means of the center manifold theory. Our study reveals that the model admits a Bogdanov–Takens bifurcation when the time delay and the maximal disease transmission rate are varied. Numerical simulations are presented to illustrate the dynamics of the model and to study the effects caused by varying the treatment rate and delay parameters.

In this paper, a mathematical model which considers population dynamics among cancer stem cells, tumor cells, healthy cells, the effects of excess estrogen and the body’s natural immune response on the cell populations was considered. Fractional derivatives with power law and exponential decay law in Liouville–Caputo sense were considered. Special solutions using an iterative scheme via Laplace transform were obtained. Furthermore, numerical simulations of the model considering both derivatives were obtained using the Atangana–Toufik numerical method. Also, random model described by a system of random differential equations was presented. The use of fractional derivatives provides more useful information about the complexity of the dynamics of the breast cancer competition model.

Recently, Alaca et al. [Alaca C, Turkoglu D, Yildiz C. Fixed points in intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals 2006;29:10738] proved fuzzy Banach fixed point theorem in intuitionistic fuzzy metric spaces and Saadati [Saadati R. Notes to the paper “fixed points in intuitionistic fuzzy metric spaces” and its generalization to L -fuzzy metric spaces. Chaos, Solitions & Fractals 2008;35:80–176] extended it in generalized fuzzy metric spaces. The purpose of this paper is to give a correct proof of the main result in Saadati [Saadati R. Notes to the paper “fixed points in intuitionistic fuzzy metric spaces” and its generalization to L -fuzzy metric spaces. Chaos, Solitions & Fractals 2008;35:80–176].

In this paper, we provide an epidemic SIR model with long-range temporal memory. The model is governed by delay differential equations with fractional-order. We assume that the susceptible is obeying the logistic form in which the incidence term is of saturated form with the susceptible. Several theoretical results related to the existence of steady states and the asymptotic stability of the steady states are discussed. We use a suitable Lyapunov functional to formulate a new set of sufficient conditions that guarantee the global stability of the steady states. The occurrence of Hopf bifurcation is captured when the time-delay τ passes through a critical value τ*. Theoretical results are validated numerically by solving the governing system, using the modified Adams-Bashforth-Moulton predictor-corrector scheme. Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model. In some cases, the phase portrait gets stretched as the order of the derivative is reduced.

Latterly, variable carrying capacity in the predator–prey population models has been widely studied to provide more realistic understanding about population dynamics. In the present paper, we discuss the bio-economic harvesting of an intraguild fishery model where the logistic carrying capacities of both predator and prey fish populations are variable based on the shared biotic resource. Distinct from most fishery models, we incorporate the independent harvesting strategies on the fisheries with different harvesting efforts by assuming them to have different economic values. In our model, the prey fish population is assumed to be infected directly by an anthropogenic toxicant from the environment while the predator is infected indirectly when they feed on prey. We investigate the local stability of trivial equilibria with Routh–Hurwitz criterion and the global stability of non-trivial equilibrium by constructing a Lyapunov function. Bifurcation and numerical analyses are presented to show distinctive result that a Hopf bifurcation occurs with respect to harvesting parameter instead of resource enrichment parameter as in the literature. Bionomic equilibrium is explored with several possibilities and restrictions. The nontrivial bionomic equilibrium is found to depend critically on the resource density. Finally, the optimal harvesting policy of the three dimensional model is derived by utilizing Pontryagin Maximum Principle. From this study, harvesting parameter is a crucial factor for the stability of the intraguild system. The objective is to obtain the optimal sustainable yield that provides maximum monetary profit while conserving the marine ecosystem.

This paper focuses on providing a novel high-order algorithm for the numerical solutions of fractional order Volterra integro-differential equations using Atangana–Baleanu approach by employing the reproducing kernel approximation. For this purpose, we investigate couples of Hilbert spaces and kernel functions, as well as, the regularity properties of Atangana–Baleanu derivative, and utilize that the representation theorem of its solution. To remove the singularity in the kernel function, using new Atangana–Baleanu approach the main operator posses smoothing solution with a better regularity properties and the reproducing kernel algorithm is designed for the required equation. The convergence properties of the proposed algorithm are also studied which proves that the new strategy exhibits a high-order of convergence with decreasing error bound. Some numerical examples of single and system formulation illustrate the performance of the approach. Summary and some notes are also provided in the case of conclusion and highlight.

Suggested independently with different definitions, fractal derivative and conformable derivative are α proportional. They have been applied in quit a few problems in many field of sciences in the last few years with great success. However, some researchers have pointed out some criticisms and even concluded that they were flawed. In this paper, while confirm the validity of the conformable and fractal derivatives and we present their applications to chaotic attractors. We considered a general non-linear Cauchy problem where the differential operator is that of fractal and conformable and present the derivation of conditions for which the existence and the uniqueness of the exact solution are reached. Several examples are considered, solved and numerical simulations depicting real world observations.

An epidemic system of HIV/AIDS transmission is examined in this paper. The classical time derivative is modelled with the Atangana-Baleanu nonlocal and nonsingular fractional operator in the Caputo sense. Mathematical analysis which shows that both the disease free equilibrium state and endemic equilibrium are locally asymptotically stable. A viable numerical approximation technique of Atangana-Baleanu operator is also given. Some numerical simulation results obtained for different instances of fractional order γ are reported to justify the theoretical results.

In this paper, we define the weighted Caputo–Fabrizio fractional derivative of Caputo sense, and study related linear and nonlinear fractional differential equations. The solution of the linear fractional differential equation is obtained in a closed form, and has been used to define the weighted Caputo–Fabrizio fractional integral. We study main properties of the weighted Caputo–Fabrizio fractional derivative and integral. We also, apply the Banach fixed point theorem to establish the existence of a unique solution to the nonlinear fractional differential equation. Two examples are presented to illustrate the efficiency of the obtained results.

In this review paper, we are mainly concerned with the numerical methods for solving fractional equations, which are divided into the fractional differential equations (FDEs), time-fractional, space-fractional and space-time-fractional partial differential equations (FPDEs), fractional integro-differential equations (FIDEs) and delay fractional differential and/or fractional partial differential equations (DFDE/DFPDEs). The concept of the variable-order fractional operators will also be reviewed. At the same time, the techniques for improving the accuracy and computation and storage are also introduced.

In this paper we investigate an investment problem with partial reversibility proposed by Abel and Eberly [4] in a finite horizon. In this model, a firm can purchase capital at a given price and sell capital at a lower price. This problem can be categorized into a singular control problem and can be formulated as a Hamilton–Jacobi–Bellman(HJB) equation. Based on theoretical results in [10] and the Mellin transform techniques, we derive the coupled integral equations satisfied by the optimal investment and disinvestment boundaries, respectively. By using the recursive integration method, we solve numerically the integral equations and present the optimal investment boundary and disinvestment boundary.

In the present paper we analysis the Modified Chua attractor using new concept of fractional differentiation with non-local and non-singular kernel. A new numerical scheme that was recently suggested was used for the Volterra equation with Atangana–Baleanu fractional integral, Caputo–Fabrizio integral and finally Riemann–Liouvile integral. The numerical solution obtained from the new numerical scheme let no doubt than to believe that the new numerical scheme is very efficient and converges toward exact solution very rapidly. Applicability and suitability of the scheme is justified when applied to solve some novel chaotic system with fractional order. Existence and uniqueness of the Volterra type is presented. We presented some numerical simulations for different values of fractional order.

In this research paper, we plan to study the relation among concentration of greenhouse gases, human population and environmental tax. For this purpose, a deterministic nonlinear mathematical model is proposed and analyzed in regard to the boundedness and persistence of its solutions, equilibria and their stabilities. Whenever, a policy is implemented, an automated delay comes into existence because outcomes of application of any policy always take some time to become visible. To see the effects of delay, analyses for the stability and direction of Hopf bifurcation for delay system are done. Critical value of the delay parameter is calculated theoretically and numerically, and then verified graphically. For verifications and descriptions of analytical findings, numerical simulations are performed. Graphical comparisons for different values of various parameters provides us some realistic and interesting results. Overall, model analysis shows that the increasing level of greenhouse gases can be controlled by applying environmental tax and imposing some restrictions on corruption.

The paper provides a theoretical analysis of some local bifurcations in the fractional Chen system. Contrary to the integer-order case, basic bifurcation properties of the fractional Chen system are shown to be qualitatively different from those described previously for the fractional Lorenz system. Further, the fractional Hopf bifurcation in the Chen system is expressed rigorously with respect to general entry parameters. Based on these observations, some particularities of the fractional dynamics of the Chen system are documented and its chaotic behavior for low derivative orders is discussed.