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• Fractional-order reaction-diffusion system. • Taylor approximation techniques. • Fourier spectral method in high dimension. • Mathematical analysis and numerical simulations of noninteger-order dynamics. 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. [ABSTRACT FROM AUTHOR]

• Fractional-order reaction-diffusion system with Caputo-Fabrizio and Atangana-Baleanu Caputo operators. • Finite difference approximation in one and two dimensions. • Fourier spectral method of approximation. • Mathematical analysis and numerical simulations of time-fractional reaction-diffusion model Locally asymptotically stable. 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. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to study the dynamics of a monopolistic firm in a continuous-time framework. The firm is assumed to be boundedly rational and to experience time delays in obtaining and implementing information on output. The dynamic adjustment process is based on the gradient of the expected profit. The paper is divided into three parts: we examine delay effects on dynamics caused by one-time delay and two-time delays in the first two parts. Global dynamics and analytical results on local dynamics are numerically confirmed in the third part. Four main results are demonstrated. First, the stability switch from stability to instability occurs only once in the case of a single delay. Second, the alternation of stability and instability can continue if two time delays are involved. Third, the occurence of Hopf bifurcation is analytically shown if stability is lost. Finally, in a bifurcation process, there are a period-doubling cascade to chaos and a period-halving cascade to the equilibrium point in the case of two time delays if the difference between the two delays is large. [Copyright &y& Elsevier]

Highligths • Construct a novel TRBG architecture. • Present chaotic system based true random bit generation using fixed-point arithmetic to realize on FPGA board. • FPGA implementation of the proposed TRBG scheme with a few number of logical operations. • Verifying randomness of generated bit by carrying out NIST tests. Abstract The aim of this paper is to present a true random bit generator (TRBG) based on a memristive chaotic circuit and its implementation on Field Programmable Gate Array (FPGA) board. The proposed TRBG architecture makes use of a memristive canonical Chua's oscillator and a logistic map as entropy sources, while the XOR function is used for post-processing. The optimal parameter set for the chaotic systems has been chosen by carrying out numerical simulations of the system and adopting the scale index parameter to determine the degree of non-periodicity of the obtained bit streams. The proposed TRBG system has been then modeled and co-simulated on the Xilinx System Generator (XSG) platform and implemented on the Xilinx Kintex-7 KC705 FPGA Evaluation Board, obtaining experimental results in agreement with the expectations. Finally, the system has been validated with statistical analysis by using the NIST 800.22 statistical test suite. [ABSTRACT FROM AUTHOR]

Abstract: Tang et al. proposed a novel method for obtaining S-boxes based on the well-known two-dimensional chaotic Baker map. Unfortunately, some mistakes exist in their paper. The faults are corrected first in this paper and then an extended method is put forward for acquiring cryptographically strong S-boxes. The new scheme employs a three-dimensional chaotic Baker map, which has more intensive chaotic characters than the two-dimensional one. In addition, the cryptographic properties such as the bijective property, the nonlinearity, the strict avalanche criterion, the output bits independence criterion and the equiprobable input/output XOR distribution are analyzed in detail for our S-box and revised Tang et al.’s one, respectively. The results of numerical analysis show that both of the two boxes can resist several attacks effectively and the three-dimensional chaotic map, a stronger sense in chaotic characters, can perform more smartly and more efficiently in designing S-boxes. [Copyright &y& Elsevier]

Abstract: Detailed analysis of solitary and periodic stationary solutions of the reduced Ostrovsky equation is presented. The full Ostrovsky equation (u t + c 0 u x + αuu x + βu xxx ) x = γu was derived in 1978 for oceanic waves affected by Earth rotation, and its reduced form without small-scale dispersion, (u t + c 0 u x + αuu x ) x = γu, was pointed out in his original paper and then, derived later by many authors in different physical contexts. However, some aspects of stationary solutions to the last equation remained unclear so far. An attempt to introduce clarity to this problem is undertaken in this paper. [Copyright &y& Elsevier]

This paper concerned with a delayed diffusive eco-epidemiological model with fear effect. First, we discuss the existence and boundedness of the solution of the system. Then we give some conditions for the existence and stability of the nonnegative equilibria, and Turing instability. Furthermore, we choose the delay as bifurcation parameter to study Hopf bifurcation. Finally, we present some numerical simulations to verify our theoretical results. By mathematical and numerical analyses, we find that the fear can prevent the occurrence of limit cycle oscillation and increase the stability of the system, and the diffusion can induce the spatial pattern in the system. • We propose a delayed diffusive eco-epidemiological model with fear effect. • We have studied Hopf bifurcation and Turing bifurcation of the model. • We have discussed the global stability of the equilibria. • The results show that system can generate a wide variety of spatial patterns. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct and analyze a mathematical model for the response of T-cytotoxic lymphocytes to the Ebola virus using the Herz-tuckwill model with nonlinear Conformable order differential equations which derive from real biological data, then study the global stability of the equilibria using an appropriate Lyapunov function and the LaSalle invariance principle. Furthermore, we demonstrate the impact of the non-integer order of the model compared with the integral order. Finally consider a numerical simulation that justifies the biological hypotheses and the theory results. The numerical results provided that, the non-integer order has a great impact on the treatments of Ebola virus in this model. [ABSTRACT FROM AUTHOR]

Abstract: This paper concerns a special class of Moran sets, the λ-Moran sets, and obtains the necessary and sufficient conditions to describe the thickness and thinness of λ-Moran sets. [Copyright &y& Elsevier]

Abstract: This paper first presents a rigorous mathematical analysis for the active sliding mode control method proposed recently by Haeri and Emadzadeh. Second, the technique is applied to achieve synchronization between two coupled systems of J-J. Numerical simulations are used to verify the above analytical results. [Copyright &y& Elsevier]

Abstract: Extensive work has been done in the theory and the construction of compactly supported orthonormal wavelet bases of . Some of the most distinguished work was done by Daubechies, who constructed a whole family of such wavelet bases. In this paper, we construct a class of orthonormal wavelet bases by using the principle of Daubechies, and investigate the length of support and the regularity of these wavelet bases. [Copyright &y& Elsevier]

Abstract: In Hatir and Noiri [Hatir E, Noiri T. Decompositions of continuity and complete continuity. Acta Math Hungary 113(4);2006:281–287], δ–β-continuity has given to obtain a decomposition of continuity. In this paper, we investigate the properties of δ–β-continuous functions and discuss characterizations and the relationships with related functions. [Copyright &y& Elsevier]

Abstract: In this paper, we applied the factorization scheme for the generalized Telegraph Fisher’s equation and an exact particular solution has been found. The exact particular solution for the generalized Fisher’s equation was obtained as a particular case of the generalized Telegraph Fisher’s equation and the two-parameter solution can be obtained when . [Copyright &y& Elsevier]

Abstract: This paper presents the synchronization of chaos by designing united controller. First, this method is implemented in synchronization of a simple system, then we realize the synchronization of Lü hyperchaotic system, we also take tracking control to realize the synchronization of Lü hyperchaotic system. Comparing with results, we can find that hybrid feedback control approach is more effective than tracking control for hyperchaotic system. Numerical simulations show the united synchronization method works well. [Copyright &y& Elsevier]

Abstract: In this paper, we show that under identical conditions which guarantee the contractivity of the theoretical solutions of general nonlinear NDDEs, the numerical solutions obtained by a class of linear multistep methods are also contractive. [Copyright &y& Elsevier]

Abstract: In this paper, the variational iteration method is applied to solve fourth order boundary-value problems. Comparison with numerical solutions shows that the method is very effective and convenient. [Copyright &y& Elsevier]

Abstract: In this paper, we will develop a numerical technique for finding the eigenvalues of fourth-order non-singular Sturm–Liouville problems. We used the variational iteration methods as a basis for this technique. Numerical results and conclusions will be presented. Comparison results with others will be presented. [Copyright &y& Elsevier]

Abstract: In this paper, exact and numerical solutions are obtained for the generalized Zakharov equation (GZE) by the well known variational iteration method (VIM). This method is based on Lagrange multipliers for identification of optimal values of parameters in a functional. Using this method creates a sequence which tends to the exact solution of the problem. [Copyright &y& Elsevier]

Abstract: In this paper, a fuzzy method for interpolating of smooth curves was represented. We present a novel approach to interpolate real data by applying the universal approximation method. In proposed method, fuzzy linguistic model (FLM) applied as universal approximation for any nonlinear continuous function. Finally, we give some numerical examples and compare the proposed method with spline method. [Copyright &y& Elsevier]

Abstract: In this paper, chaos of a generalized van der Pol system with fractional orders is studied. Both nonautonomous and autonomous systems are considered in detail. Chaos in the nonautonomous generalized van der Pol system excited by a sinusoidal time function with fractional orders is studied. Next, chaos in the autonomous generalized van der Pol system with fractional orders is considered. By numerical analyses, such as phase portraits, Poincaré maps and bifurcation diagrams, periodic, and chaotic motions are observed. Finally, it is found that chaos exists in the fractional order system with the order both less than and more than the number of the states of the integer order generalized van der Pol system. [Copyright &y& Elsevier]

Abstract: The paper deals with the analysis of the dynamics of a cascade of two tubular reactors with the residence time in the first reactor differing from that in the second one. Additionally, in the system under consideration, the recycle divided into individual cascade’s stage takes place. The occurrence of different residence time in both reactors enabled one the estimation of the effect of recycle division on cascade’s dynamics. The idea of the residence time multiplication factor has been introduced. The results have been presented in the time series form and Poincaré’s maps. Depending on chosen values of parameters, one observes multiperiodics-, quasiperiodics- and chaotic behaviour. [Copyright &y& Elsevier]

Abstract: In this paper, we present and analyze a simple model for a chemostat with plasmid-bearing and plasmid-free organisms and periodically pulsed substrate. We obtain an exact periodic solution with positive concentrations of substrate and plasmid-free organism, and furthermore, we prove that the periodic solution is stable if the impulsive period is more than some critical value and meanwhile we prove that system is permanent if the impulsive period is less than some critical value. Finally, we give a brief discussion and find that periodically pulsed input destroys the equilibria of the unforced continuous system. [Copyright &y& Elsevier]

Abstract: In this paper, the chaotic behaviors of a nonlinear damped Mathieu system and of a nonlinear nano resonator system with integral orders and with fractional orders are studied. By applying numerical analyses such as phase portraits, Poincaré maps and bifurcation diagrams, the periodic and chaotic motions are observed. It is found that chaos exists both in the nonlinear damped Mathieu system and in the integral order and fractional order nano resonator systems. [Copyright &y& Elsevier]

Abstract: This paper presents some new results which we obtained recently for the study of limit cycles of nonlinear dynamical systems. Particular attention is given to small limit cycles of generalized Liénard systems in the vicinity of the origin. New results for a number of cases of the Liénard systems are presented with the Hilbert number, , for j =4, i =10,11,12,13; j =5, i =6,7,8,9; and j =6, i =5,6. Detailed proofs for the existence of limit cycles are given in four cases. [Copyright &y& Elsevier]

Abstract: In this paper, numerical solutions of the nonlinear Burgers’ equation are obtained by a method based on collocation of septic B-splines over finite elements. Applying the Von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. Numerical solutions of the modified Burgers’ equation are also obtained by making a simple change of the suggested numerical scheme for the Burgers’ equation. The accuracy of the presented method is demonstrated by two test problems. The numerical results are found to be in good agreement with the exact solutions. [Copyright &y& Elsevier]

Abstract: In this paper, a generalized periodic n-species Gilpin–Ayala competition systems with impulsive effect is investigated. The sufficient conditions are derived for the permanence of the system. We discuss the stabilities of trivial solution and bounded periodic solution. By using the method of coincidence degree, a set of sufficient conditions are derived for the existence of at least one strictly positive componentwise periodic solution, the example shows that our criteria are easily verifiable. [Copyright &y& Elsevier]

Abstract: In this paper it is presented a detailed numerical investigation of acoustic emission signals obtained from test samples of fibreglass reinforced polymeric matrix composites, when subjected to tensile and flexural tests. Various fractal indices, characteristic of the signals emitted at the different structural failures of the test samples and which satisfy non-stationary distributions, have been determined. From the results obtained for these indices, related to the Hurst analysis, detrended fluctuation analysis, minimal cover analysis and to the boxcounting dimension analysis, it has been shown they can discriminate the different failure mechanisms and, therefore, they constitute their signature. [Copyright &y& Elsevier]

Abstract: The paper deals with the mathematical–numerical analysis of the logistic Mandelbrot equation extended by the dynamic continuous term. The possibilities of generation of fractal patterns with the mathematical form, defined in such a manner, were investigated. The decay of fractal structure with time was demonstrated. [Copyright &y& Elsevier]

Abstract: In the paper the analysis of the dynamics of a cascade of two homogeneous plug-flow tubular reactors with a divided recirculating stream at each stage is considered. Two cases were numerically analyzed, viz. with and without time delay in the recirculation loop. The results are presented in the form of Feigaubaum diagrams and Poincaré maps. It is demonstrated that the analyzed system is characterized by quite complex dynamics including chaotic behavior. [Copyright &y& Elsevier]

This paper presents a new three-dimensional chaotic system with a nonlinear term. More interesting is that the system can produce different types of coexisting attractors under one sets of fixed model parameters, which include point attractor, bursting orbit, three kinds of period orbits, and two kinds of chaotic orbits. At the same time, the coexisting attractors have nested structure. In addition, riddled basins are observed on local attraction basins, which indicate that the dynamical behaviors of proposed system are extremely sensitive to the initial conditions. Finally, by setting different initial condition, the circuit can generate multitudinous coexisting attractors. The circuit implementation in Pspice supports furtherly numerical analyses and validates the mathematical model. [ABSTRACT FROM AUTHOR]

In this paper, we present mathematical analysis and numerical simulation of a three-dimensional autonomous fractional system with coexistence of multi-scroll chaotic attractors. We replaced the classical derivatives of such system with the Caputo-Fabrizio fractional derivative. This derivative combines both the exponential laws and non-singular kernels in its formulation which makes it special and useful. A two-step Adams-Bashforth scheme is derived for the approximation of the fractional derivative with exponential law and non-singular kernel. We then presented both numerical results and graphical results by considering many values of the fractional-order parameter β ∈ (0, ]. We demonstrate that the observed chaotic behavior conduct perseveres as the fractional-order parameter approaches 1. [ABSTRACT FROM AUTHOR]

Abstract: Via the semi-inverse method of establishing variational principles proposed by He, a generalized variational principle is established for Ginzburg–Landau equation. The present theory provides a quite straightforward tool to the search for various variational principles for physical problems. This paper aims at providing a more complete theoretical basis for applications using finite element and other direct variational methods. [Copyright &y& Elsevier]