This article investigates the solvability and optimal controls of systems monitored by fractional delay evolution inclusions with Clarke subdifferential type. By applying a fixed-point theorem of condensing multivalued maps and some properties of Clarke subdifferential, an existence theorem concerned with the mild solution for the system is proved under suitable assumptions. Moreover, an existence result of optimal control pair that governed by the presented system is also obtained under some mild conditions. Finally, an example is given to illustrate our main results. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this study, the generation of smooth trajectories of the end effector of a rotating extensible manipulator arm is considered. Possible trajectories are modelled using Cartesian and polar piecewise cubic interpolants expressed as polynomial Hermite-type functions. The use of polar piecewise cubic interpolants devises continuous first-order and – in some cases – second-order derivatives and allows easy calculation of kinematics variables such as velocity and acceleration. Moreover, the manipulator equations of motion can be easily handled, and the constrained trajectory of the non-active end of the manipulator derived directly from the position of the end-effector. To verify the proposed approach, numerical simulations are conducted for two different configurations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we propose an eco-epidemiological predator–prey model, modeling the spread of infectious keratoconjunctivitis among domestic and wild ungulates, during the summer season, when they intermigle in high mountain pastures. The disease can be treated in the domestic animals, but for the wild herbivores, it leads to blindness, with consequent death. The model shows that the disease can lead infected herbivores or their predators to extinction, even if it does not affect the latter. Boundedness of solutions and equilibria feasibility are obtained. Stability around the different equilibrium points is analyzed through eigenvalues and the Routh–Hurwitz criterion. Simulations are carried out to support the theoretical results. Sensitivity with respect of some parameters is investigated. The prey vaccination as control measure is introduced and simulated, although at present, the vaccine is not yet available, but just being developed. It would then possibly eradicate the infection in the domestic animals, which are considered a disease reservoir. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we propose a space-time spectral method for solving a class of time fractional convection diffusion equations. Because both fractional derivative and spectral method have global characteristics in bounded domains, we propose a space-time spectral-Galerkin method. The convergence result of the method is proved by providing a priori error estimate. Numerical results further confirm the expected convergence rate and illustrate the versatility of our method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The aim of this work is to study *μ*-pseudo almost automorphic solutions of abstract fractional integro-differential neutral equations with an infinite delay. Thanks to some restricted hypothesis on the delayed data in the phase space, we ensure the existence of the ergodic component of the desired solution. Copyright © 2016 John Wiley & Sons, Ltd.

We propose and investigate a delayed model that studies the relationship between HIV and the immune system during the natural course of infection and in the context of antiviral treatment regimes. Sufficient criteria for local asymptotic stability of the infected and viral free equilibria are given. An optimal control problem with time delays both in state variables (incubation delay) and control (pharmacological delay) is then formulated and analyzed, where the objective consists to find the optimal treatment strategy that maximizes the number of uninfected *C**D*4^{ + } T cells as well as cytotoxic T lymphocyte immune response cells, keeping the drug therapy as low as possible. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we investigate a boundary problem with non-local conditions for mixed parabolic–hyperbolic-type equation with three lines of type changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part, we use solution of the first boundary problem with appropriate Green's function, and in hyperbolic parts, we use corresponding solutions of the Cauchy problem. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider a non-stationary Stokes system in a thin porous medium Ω_{ϵ} of thickness *ϵ* which is perforated by periodically solid cylinders of size *a*_{ϵ}. We are interested here to give the limit behavior when *ϵ* goes to zero. To do so, we apply an adaptation of the unfolding method. Time-dependent Darcy's laws are rigorously derived from this model depending on the comparison between *a*_{ϵ} and *ϵ*. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is concerned with the solvability of a fully nonlinear third-order *m*-point boundary value problem at resonance posed on the half line. The nonlinearity which depends on the first and the second derivatives satisfies a sublinear-like growth condition. Our main existence result is based on Mawhin's coincidence degree theory. An illustrative example of application is included. Copyright © 2016 John Wiley & Sons, Ltd.

Many researchers have studied simple low order ODE model problems for fluid flows in order to gain new insight into the dynamics of complex fluid flows. We investigate the existence of a global attractor for a low order ODE system that has served as a model problem for transition to turbulence in viscous incompressible fluid flows. The ODE system has a linear term and an energy-conserving, non-quadratic nonlinearity. Standard energy estimates show that solutions remain bounded and converge to a global attractor when the linear term is negative definite, that is, the linear term is energy decreasing; however, numerical results indicate the same result is true in some cases when the linear term does not satisfy this condition. We give a new condition guaranteeing solutions remain bounded and converge to a global attractor even when the linear term is not energy decreasing. We illustrate the new condition with examples. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Graph theory is a fundamental tool in the study of economic issues, and input–output tables are one of the main examples. We use the interpretation of the labour market through networks to obtain a better understanding on its overall functioning. One benefit of the network perspective is that a large body of mathematics exists to help analyze many forms of networks models. If an economic system has obtained a suitable model, then it becomes possible to utilize relevant mathematical tools, such as graph theory, to better understand the way the labour market works. This interpretation allows us to employ the concepts of coverage, invariance, orbit and the structural functions supply–demand and competition and interpret them from the point of view of circular flow. In this paper, we aim to interpret the labour market through networks that are represented by graphs and where characteristic concepts of chaos theory such as cover, invariance and orbits interact with the concept circular flow. Finally, an example of this approach to labour markets is described, and some conclusions are drawn. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is concerned with the reachable set estimation problem of singular systems with time-varying delay and bounded disturbance inputs. Based on a novel Lyapunov–Krasovskii functional that contains four triple integral terms, reciprocally convex approach and free-weighting matrix method, two sufficient conditions are derived in terms of linear matrix inequalities to guarantee that the reachable set of singular systems with time-varying delay is bounded by the intersection of ellipsoid. Finally, two numerical examples are given to demonstrate the effectiveness and superiority of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Recent studies have shown that the initiation of human cancer is due to the malfunction of some genes (such as E2F, CycE, CycD, Cdc25A, P27^{Kip1}, and Rb) at the R-checkpoint during the G_{1}-to-S transition of the cell cycle. Identifying and modeling the dynamics of these genes provide new insight into the initiation and progression of many types of cancers. In this study, a cancer subnetwork that has a mutual activation between phosphatase Cdc25A and the CycE/Cdk2 complex and a mutual inhibition between the Cdk inhibitor P27^{Kip1} and the CycE/Cdk2 complex is identified. A new mathematical model for the dynamics of this cancer subnetwork is developed. Positive steady states are determined and rigorously analyzed. We have found a condition for the existence of positive steady states from the activation, inhibition, and degradation parameter values of the dynamical system. We also found a robust condition that needs to be satisfied for the steady states to be asymptotically stable. We determine the parameter value(s) under which the system exhibits a saddle–node bifurcation. We also identify the condition for which the system exhibits damped oscillation solutions. We further explore the possibility of Hopf and homoclinic bifurcations from the saddle–focus steady state of the system. Our analytic and numerical results confirm experimental results in the literature, thus validating our model. Copyright © 2016 John Wiley & Sons, Ltd.

We first prove the uniqueness of weak solutions (*ψ*,*A*) to the 3-D Ginzburg–Landau model in superconductivity with zero magnetic diffusivity and the Coulomb gauge if
, which is a critical space for some positive constant *T*. We also prove the global existence of solutions when
and *A*_{0}∈*L*^{3}. Copyright © 2016 John Wiley & Sons, Ltd.

Our work is devoted to an inverse problem for three-dimensional parabolic partial differential equations. When the surface temperature data are given, the problem of reconstructing the heat flux and the source term is investigated. There are two main contributions of this paper. First, an adjoint problem approach is used for analysis of the Fréchet gradient of the cost functional. Second, an improved conjugate gradient method is proposed to solve this problem. Based on Lipschitz continuity of the gradient, the convergence analysis of the conjugate gradient algorithm is studied. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider the problem of the periodic solutions bifurcated from a homoclinic orbit for a pair of coupled ordinary differential equations in
. Assume that the autonomous system has a degenerate homoclinic solution *γ* in
. A functional analytic approach is used to consider the existence of periodic solution for the autonomous system with periodic perturbations. By exponential dichotomies and the method of Lyapunov–Schmidt, the bifurcation function defined between two finite dimensional subspaces is obtained, where the zeros correspond to the existence of periodic solutions for the coupled ordinary differential equations near
. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is devoted to the study of the nonlinear stability of the composite wave consisting of a rarefaction wave and a viscous contact discontinuity wave of the non-isentropic Navier–Stokes–Poisson system with free boundary. We first construct the composite wave through the quasineutral Euler equations and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding initial-boundary value problem of the non-isentropic Navier–Stokes–Poisson system. Only the strength of the viscous contact wave is required to be small. However, the strength of the rarefaction wave can be arbitrarily large. In our analysis, the domain decomposition plays an important role in obtaining the zero-order energy estimates. By introducing this technique, we successfully overcome the difficulty caused by the critical terms involved with the linear term, which does not satisfy the quasineural assumption for the composite wave. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The global stability of equilibria is investigated for a nonlinear multi-group epidemic model with latency and relapses described by two distributed delays. The results show that the global dynamics are completely determined by the basic reproduction number under certain reasonable conditions on the nonlinear incidence rate. Moreover, compared with the results in Michael Y. Li and Zhisheng Shuai, Journal Differential Equations 248 (2010) 1–20, it is found that the two distributed delays have no impact on the global behaviour of the model. Our study improves and extends some known results in recent literature. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Our interest is to quantify the spread of an infective process with latency period and generic incidence rate that takes place in a finite and homogeneous population.

Within a stochastic framework, two random variables are defined to describe the variations of the number of secondary cases produced by an index case inside of a closed population. Computational algorithms are presented in order to characterize both random variables. Finally, theoretical and algorithmic results are illustrated by several numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we prove a global well posedness of the three-dimensional incompressible Navier–Stokes equation under an initial data, which belong to the non-homogeneous Fourier–Lei–Lin space
for *σ*⩾ − 1 and if the norm of the initial data in the Lei–Lin space
is controlled by the viscosity. Copyright © 2016 John Wiley & Sons, Ltd.

This paper presents all possible exact explicit peakon, pseudo-peakon, cuspon and smooth solitary wave solutions for a nonlocal Kerr-like media. We apply the method of dynamical systems to analyze the dynamical behavior of the traveling wave solutions and their bifurcations depending on the parameters of the system. We present peakon, pseudo-peakon, cuspon soliton solution in an explicit form. We also have obtained smooth soliton. Mathematical analysis and numeric graphs are provided for those soliton solutions of the nonlocal Kerr-like media. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we consider the conductivity problem with piecewise-constant conductivity and Robin-type boundary condition on the interface of discontinuity. When the quantity of interest is the jump of the conductivity, we perform a local stability estimate for a parameterized non-monotone family of domains. We give also a quantitative stability result of local optimal solution with respect to a perturbation of the Robin parameter. In order to find an optimal solution, we propose a Kohn–Vogelius-type cost functional over a class of admissible domains subject to two boundary values problems. The analysis of the stability involves the computation of first-order and second-order shape derivative of the proposed cost functional, which is performed rigorously by means of shape-Lagrangian formulation without using the shape sensitivity of the states variables. Copyright © 2016 The Author. *Mathematical Methods in the Applied Sciences* Published by John Wiley & Sons Ltd.

In this paper, we consider a nonstationary 3-D flow of a compressible viscous and heat-conducting micropolar fluid, which is in the thermodynamical sense perfect and polytropic. The fluid domain is a subset of **R**^{3} bounded with two coaxial cylinders that present solid thermoinsulated walls. The mathematical model is set up in Lagrangian description. If we assume that the initial mass density, temperature, as well as the velocity and microrotation vectors are smooth enough cylindrically symmetric functions, then our problem has a generalized cylindrically symmetric solution for a sufficiently small time interval. Here, we prove the uniqueness of this solution. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we study the traveling wave solutions of a delayed diffusive SIR epidemic model with nonlinear incidence rate and constant external supplies. We find that the existence of traveling wave solutions is determined by the basic reproduction number of the corresponding spatial-homogenous delay differential system and the minimal wave speed. The existence is proved by applying Schauder's fixed point theorem and Lyapunov functional method. The non-existence of traveling waves is obtained by two-sided Laplace transform. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we study a natural modification of Szász–Mirakjan operators. It is shown by discussing many important established results for Szász–Mirakjan operators. The results do hold for this modification as well, be they local in nature or global, be they qualitative or quantitative. It is also shown that this generalization is meaningful by means of examples and graphical representations. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We consider a parallel fiber-reinforced periodic elastic composite that present an imperfect contact of spring type between the fiber and the matrix. We use the elliptic integral of Cauchy type for solving the plane strain local problems that arise from the asymptotic homogenization method. Several general conditions are assumed, which include that the fibers are disposed of arbitrary manner in the local cell, that all fibers present contact perfect with different constants of imperfection, and that their cross section is a smooth closed arbitrary curve. We find that there are infinity solutions for these problems, and we find relations between these solutions and effective coefficients of the composite. Finally, we obtain analytic formulae for the circular fiber case and show some numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper investigates the nonlinear boundary value problem, resulting from the exact reduction of the Navier–Stokes equations for unsteady laminar boundary layer flow caused by a stretching surface in a quiescent viscous incompressible fluid. We prove existence of solutions for all values of the relevant parameters and provide unique results in the case of a monotonic solution. The results are obtained using a topological shooting argument, which varies a parameter related to the axial shear stress. To solve this equation, a numerical method is proposed based on a rational Chebyshev functions spectral method. Using the operational matrices of derivative, we reduced the problem to a set of algebraic equations. We also compare this work with some other numerical results and present a solution that proves to be highly accurate. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper investigates the orbital stability of solitary waves for the coupled Klein–Gordon–Zakharov (KGZ) equations

where *α* ≠ 0. Firstly, we rewrite the coupled KGZ equations to obtain its Hamiltonian form. And then, we present a pair of sech-type solutions of the coupled KGZ equations. Because the abstract orbital stability theory presented by Grillakis, Shatah, and Strauss (1987) cannot be applied directly, we can extend the abstract stability theory and use the detailed spectral analysis to obtain the stability of the solitary waves for the coupled KGZ equations. In our work, *α* = 1,*β* = 0 are advisable. Hence, we can also obtain the orbital stability of solitary waves for the classical KGZ equations which was studied by Chen. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we are interested in a model derived from the 1-D Keller-Segel model on the half line *x* > as follows:

where *l* is a constant. Under the conserved boundary condition, we study the asymptotic behavior of solutions. We prove that the problem is always globally and classically solvable when the initial data is small, and moreover, we obtain the decay rates of solutions. The paper mainly deals with the case of *l* > 0. In this case, the solution to the problem tends to a conserved stationary solution in an exponential decay rate, which is a very different result from the case of *l* < 0. Copyright © 2016 John Wiley & Sons, Ltd.

The scalar spherical wave functions (SWFs) are solutions to the scalar Helmholtz equation obtained by the method of separation of variables in spherical polar coordinates. These functions are complete and orthogonal over a sphere, and they can, therefore, be used as a set of basis functions in solving boundary value problems by spherical wave expansions. In this work, we show that there exists a theory of functions with quaternionic values and of three real variables, which is determined by the Moisil–Theodorescu-type operator with quaternionic variable coefficients, and which is intimately related to the radial, angular and azimuthal wave equations. As a result, we explain the connections between the null solutions of these equations, on one hand, and the quaternionic hyperholomorphic and anti-hyperholomorphic functions, on the other. We further introduce the quaternionic spherical wave functions (QSWFs), which refine and extend the SWFs. Each function is a linear combination of SWFs and products of -hyperholomorphic functions by regular spherical Bessel functions. We prove that the QSWFs are orthogonal in the unit ball with respect to a particular bilinear form. Also, we perform a detailed analysis of the related properties of QSWFs. We conclude the paper establishing analogues of the basic integral formulae of complex analysis such as Borel–Pompeiu's and Cauchy's, for this version of quaternionic function theory. As an application, we present some plot simulations that illustrate the results of this work. Copyright © 2016 John Wiley & Sons, Ltd.

]]>An asymptotic expansion of the contrasting structure-like solution of the generalized Kolmogorov–Petrovskii–Piskunov equation is presented. A generalized maximum principle for the pseudoparabolic equations is developed. This, together with the generalized differential inequalities method, allows to prove the consistence and convergence of the asymptotic series method. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The quaternion Fourier transform (QFT) is one of the key tools in studying color image processing. Indeed, a deep understanding of the QFT has created the color images to be transformed as whole, rather than as color separated component. In addition, understanding the QFT paves the way for understanding other integral transform, such as the quaternion fractional Fourier transform, quaternion linear canonical transform, and quaternion Wigner–Ville distribution. The aim of this paper is twofold: first to provide some of the theoretical background regarding the quaternion bound variation function. We then apply it to derive the quaternion Fourier and linear canonical inversion formulas. Secondly, to provide some in tuition for how the quaternion Fourier and linear canonical inversion theorems work on the absolutely integrable function space. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we derive five new versions of Landau-type theorems for biharmonic mappings of unit disk. Moreover, we prove that all five results are sharp when the bounds are equal to 1. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we focus on a diffuse interface model named by Hele–Shaw–Cahn–Hilliard system, which describes a two-phase Hele–Shaw flow with matched densities and arbitrary viscosity contrast in a bounded domain. The diffuse interface thickness is measured by *ϵ*, and the mobility coefficient (the diffusional Peclet number) is *ϵ*^{α}. We will prove rigorously that the global weak solutions of the Hele–Shaw–Cahn–Hilliard system converge to a varifold solution of the sharp interface model as *ϵ*0 in the case of 0≤*α* < 1. Copyright © 2016 John Wiley & Sons, Ltd.

In the present work, the steady-state crack propagation in a chain of oscillators with non-local interactions is considered. The interactions are modelled as linear springs, while the crack is presented by the absence of extra springs. The problem is reduced to the Wiener-Hopf type, and solution is presented in terms of the inverse Fourier transform. It is shown that the non-local interactions may change the structure of the solution well known from the classical local interactions formulation. In particular, it may change the range of the region of stable crack motion. The conclusions of the analysis are supported by numerical results. Namely, the observed phenomenon is partially clarified by evaluation of the structure profiles on the crack line ahead. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We present an algorithm for the identification of the relaxation kernel in the theory of diffusion systems with memory (or of viscoelasticity), which is linear, in the sense that we propose a linear Volterra integral equation of convolution type whose solution is the relaxation kernel. The algorithm is based on the observation of the flux through a part of the boundary of a body.

The identification of the relaxation kernel is ill posed, as we should expect from an inverse problem. In fact, we shall see that it is mildly ill posed, precisely as the deconvolution problem which has to be solved in the algorithm we propose. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we consider a nonlinear coupled wave equations with initial-boundary value conditions and nonlinear damping and source terms. Under suitable assumptions on the damping terms and source terms and initial data in the stable set, we obtain that the decay estimates of the energy function is exponential or polynomial by using Nakao's method. By using the energy method, we obtain the blow-up result of solution with some positive or nonpositive initial energy. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we prove the local-in-time existence and a blow-up criterion of solutions in the Besov spaces for the Euler-*α* equations of inviscid incompressible fluid flows in
. We also establish the convergence rate of the solutions of the Euler-*α* equations to the corresponding solutions of the Euler equations as the regularization parameter *α* approaches 0 in
. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider a uniform elliptic nonlocal operator

- (1)

which is a weighted form of fractional Laplacian. We firstly establish three maximum principles for antisymmetric functions with respect to the nonlocal operator. Then, we obtain symmetry, monotonicity, and nonexistence of solutions to some semilinear equations involving the operator
on bounded domain,
and
, by applying direct moving plane methods. Finally, we show the relations between the classical operator − Δ and the nonlocal operator in ((1)) as *α*2. Copyright © 2016 John Wiley & Sons, Ltd.

This paper discusses uncertainty principles of images defined on the square, or, equivalently, uncertainty principles of signals on the 2-torus. Means and variances of time and frequency for signals on the 2-torus are defined. A set of phase and amplitude derivatives are introduced. Based on the derivatives, we obtain three comparable lower bounds of the product of variances of time and frequency, of which the largest lower bound corresponds to the strongest uncertainty principles known for periodic signals. Examples, including simulations, are provided to illustrate the obtained results. To the authors' knowledge, it is in the present paper, and for the first time, that uncertainty principles on the torus are studied. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In the current paper, we consider a stochastic parabolic equation that actually serves as a mathematical model describing the operation of an electrostatic actuated micro-electro-mechanical system. We first present the derivation of the mathematical model. Then after establishing the local well posedeness of the problem, we investigate under which circumstances a *finite-time quenching* for this stochastic partial differential equation, corresponding to the mechanical phenomenon of *touching down*, occurs. For that purpose, the Kaplan's eigenfunction method adapted in the context of stochastic partial differential equations is employed. Copyright © 2016 John Wiley & Sons, Ltd.

A new variant of the Adaptive Cross Approximation (ACA) for approximation of dense block matrices is presented. This algorithm can be applied to matrices arising from the Boundary Element Methods (BEM) for elliptic or Maxwell systems of partial differential equations. The usual interpolation property of the ACA is generalised for the matrix valued case. Some numerical examples demonstrate the efficiency of the new method. The main example will be the electromagnetic scattering problem, that is, the exterior boundary value problem for the Maxwell system. Here, we will show that the matrix valued ACA method works well for high order BEM, and the corresponding high rate of convergence is preserved. Another example shows the efficiency of the new method in comparison with the standard technique, whilst approximating the smoothed version of the matrix valued fundamental solution of the time harmonic Maxwell system. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We study the exact controllability of *q* uncoupled damped string equations by means of the same control function. This property is called simultaneous controllability. An observability inequality is proved, which implies the simultaneous controllability of the system. Our results generalize the previous results on the linear wave without the dampings. Copyright © 2016 John Wiley & Sons, Ltd.

The shrinkage of fossil fuel resources motivates many countries to search alternative energy sources. *Jatropha curcas* is a small drought-resistant shrub from whose seeds a high grade fuel biodiesel can be produced. It is cultivated in many tropical countries including India. However, the plant is affected by the mosaic virus (*Begomovirus*) through infected white-flies (*Bemisia tabaci*) which causes mosaic disease. Disease control is an important factor to obtain healthy crop but in agricultural practice, farming awareness is equally important. Here, we propose a mathematical model for media campaigns for raising awareness among people to protect this plant in small plots and control disease. In order to archive high crop yield, we consider the awareness campaign to be arranged in impulsive way to make people aware from infected white-flies to protect Jatropha plants from mosaic virus. The study reveals that the spread of mosaic disease can be contained or even eradicated by the awareness campaigns. To attain an effective eradication, awareness campaign should be implemented at sufficiently short time intervals. Copyright © 2016 John Wiley & Sons, Ltd.

Our aim in this paper is to study the well-posedness of a singular reaction-diffusion equation which is related with brain lactate kinetics, when spatial diffusion is taken into account. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we are concerned with the backward problem of reconstructing the initial condition of a time-fractional diffusion equation from interior measurements. We establish uniqueness results and provide stability analysis. Our method is based on the eigenfunction expansion of the forward solution and the Tikhonov regularization to tackle the ill-posedness issue of the underlying inverse problem. Some numerical examples are included to illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we establish a blow-up criterion for the three-dimentional viscous, compressible magnetohydrodynamic flows. It is shown that for the Cauchy problem and the initial-boundary-value problem with initial density allowed to vanish, the strong or smooth solution for the three-dimentional magnetohydrodynamic flows exists globally if the density, temperature, and magnetic field is bounded from above. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, the stochastic stability under small Gauss type random excitation is investigated theoretically and numerically. When *p* is larger than 0, the *p*-moment stability theorem of stochastic models is proved by Lyapunov method, Ito isometry formula, matrix theory and so on. Then the application of *p*-moment such as *k*-order moment of the origin and *k*-order moment of the center is introduced and analyzed. Finally, *p*-moment stability of the power system is verified through the simulation example of a one machine and infinite bus system. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we consider the strongly nonlinear Nernst–Planck equations coupled with the quasi-linear Poisson equation under inhomogeneous, moreover, nonlinear boundary conditions. This system describes joint multi-component electrokinetics in a pore phase. The system is supplemented by the force balance and by the volume and positivity constraints. We establish well-posedness of the problem in the variational setting. Namely, we prove the existence theorem supported by the energy and the entropy a-priori estimates, and we provide the Lyapunov stability of the solution as well as its uniqueness in special cases. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The main work is related to show the existence and uniqueness of solution for the fractional impulsive differential equation of order *α*∈(1,2) with an integral boundary condition and finite delay. Using the application of the Banach and Sadovaskii fixed-point theorems, we obtain the main results. An example is presented at the end to verify the results of the paper. Copyright © 2016 John Wiley & Sons, Ltd.

In this study, we solve an inverse nodal problem for *p*-Laplacian Dirac system with boundary conditions depending on spectral parameter. Asymptotic formulas of eigenvalues, nodal points and nodal lengths are obtained by using modified Prüfer substitution. The key step is to apply modified Prüfer substitution to derive a detailed asymptotic estimate for eigenvalues. Furthermore, we have shown that the functions *r(x)* and *q(x)* in Dirac system can be established uniquely by using nodal parameters with the method used by Wang et al. Obtained results are more general than the classical Dirac system. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we propose a new method called the fractional natural decomposition method (FNDM). We give the proof of new theorems of the FNDM, and we extend the natural transform method to fractional derivatives. We apply the FNDM to construct analytical and approximate solutions of the nonlinear time-fractional Harry Dym equation and the nonlinear time-fractional Fisher's equation. The fractional derivatives are described in the Caputo sense. The effectiveness of the FNDM is numerically confirmed. Copyright © 2016 John Wiley & Sons, Ltd.

]]>We establish a Bochner type characterization for Stepanov almost periodic functions, and we prove a new result about the integration of almost periodic functions. This result is then used together with a reduction principle to investigate the nature of bounded solutions of some almost periodic partial neutral functional differential equations. More specifically, we prove that all bounded solutions on are almost periodic. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The initial-boundary value problems for a Sobolev equation with exponential nonlinearities, classical, and nonclassical boundary conditions are considered. For this model, which describes processes in crystalline semiconductors, the blow-up phenomena are studied. The sufficient blow-up conditions and the blow-up time are analyzed by the method of the test functions. This analytical *a priori* information is used in the numerical experiments, which are able to determine the process of the solution's blow-up more accurately. The model derivation and some questions of local solvability and uniqueness are also discussed. Copyright © 2016 John Wiley & Sons, Ltd.

The paper presents a novel view on the scattering of a flexural wave in a Kirchhoff plate by a semi-infinite discrete system. Blocking and channelling of flexural waves are of special interest. A quasi-periodic two-source Green's function is used in the analysis of the waveguide modes. An additional ‘effective waveguide’ approximation has been constructed. Comparisons are presented for these two methods in addition to an analytical solution for a finite truncated system. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we prove existence and regularity results for weak solutions in the framework of anisotropic Sobolev spaces for a class of nonlinear anisotropic elliptic equations in the whole with variable exponents and locally integrable data. Our approach is based on the anisotropic Sobolev inequality, a smoothness, and compactness results. The functional setting involves Lebesgue–Sobolev spaces with variable exponents. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we first utilize the vanishing diffusivity method to prove the existence of global quasi-strong solutions and get some higher order estimates, and then prove the global well-posedness of the two-dimensional Boussinesq system with variable viscosity for *H*^{3} initial data. Copyright © 2016 John Wiley & Sons, Ltd.

In this work, we study the two-mode Korteweg–de Vries (TKdV) equation, which describes the propagation of two different waves modes simultaneously. We show that the TKdV equation gives multiple soliton solutions for specific values of the nonlinearity and dispersion parameters involved in the equation. We also derive other distinct exact solutions for general values of these parameters. We apply the simplified Hirota's method to study the specific of the parameters, which gives multiple soliton solutions. We also use the tanh/coth method and the tan/cot method to obtain other set of solutions with distinct physical structures. Copyright © 2016 John Wiley & Sons, Ltd.

]]>This paper is focused on higher order differentiation of Fourier series of functions. By means of Stokes's transformation, the recursion relations between the Fourier coefficients in Fourier series of different order (partial) derivatives of the functions as well as the general formulas for Fourier series of higher order (partial) derivatives of the functions are acquired. And then, the sufficient conditions for term-by-term differentiation of Fourier series of the functions are presented. These findings are subsequently used to reinvestigate the Fourier series methods for linear elasto-dynamical systems. The results given in this paper on the constituent elements, together with their combinatorial modes and numbering, of the sets of coefficients concerning 2*r*th order linear differential equation with constant coefficients are found to be different from the results deduced by Chaudhuri back in 2002. And it is also shown that the displacement solution proposed by Li in 2009 is valid only when the second order mixed partial derivative of the displacement vanishes at all of the four corners of the rectangular plate. Copyright © 2016 John Wiley & Sons, Ltd.

No abstract is available for this article.

]]>Dengue is a vector-borne disease transmitted from an infected human to an *Aedes* mosquito, during a blood meal. Dengue is still a major public health problem. A model for the disease transmission is presented, composed by human and mosquitoes compartments. The aim is to simulate the effects of seasonality, on the vectorial capacity and, consequently, on the disease development. Using entomological information about the mosquito behavior under different temperatures and rainfall, simulations are carried out, and the repercussions analyzed. The basic reproduction number of the model is given, as well as a sensitivity analysis of model's parameters. Finally, an optimal control problem is proposed and solved, illustrating the difficulty of making a trade-off between reduction of infected individuals and costs with insecticide. Copyright © 2014 John Wiley & Sons, Ltd.

The record of the oscillations of the electric potential of the human brain provides useful information about the mind activity at rest and during the achievement of sensory and cognitive processing tasks. The use of appropriate quantitative tools assigning numerical values to the observed variable is necessary to define good descriptors of the electroencephalogram, allowing comparisons between different recordings. In this line, we propose a numerical method for the spectral and temporal reconstruction of a brain signal. The convergence of the procedure is analyzed, providing results of the concerned approximation error. In a second part of the text, we use the methodology described to the quantification of the bioelectric variations produced in the brain waves for the execution of a test of attention related to military simulation. Copyright © 2014 JohnWiley & Sons, Ltd.

]]>A mathematical model to simulate drug delivery from a viscoelastic erodible matrix is presented in this paper. The drug is initially distributed in the matrix that is in contact with water. The entrance of water in the material changes the molecular weight, and bulk erosion can be developed depending on how fast this entrance is and how fast degradation occurs. The viscoelastic properties of the matrix also change in the presence of water as the molecular weight changes. The model is represented by a system of quasi-linear partial differential equations that take into account different phenomena: the uptake of water, the decreasing of the molecular weight, the viscoelastic behavior, the dissolution of the solid drug, and the delivery of the dissolved drug. Numerical simulations illustrating the behavior of the model are included. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this work, a modified Holling–Tanner predator–prey model is analyzed, considering important aspects describing the interaction such as the following: the predator growth function is of a logistic type, and a weak Allee effect acting in the prey growth function and the functional response is of hyperbolic type.

By making a change of variables and a time rescaling, we obtain a polynomial differential equations system topologically equivalent to the original one, in which the nonhyperbolic equilibrium point (0,0) is an attractor for all parameter values.

An important consequence of this property is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane, and the system exhibits the bistability phenomenon, because the trajectories can have different *ω*-limit sets, as an example, the origin (0,0) or a stable limit cycle surrounding an unstable positive equilibrium point.

We show that, under certain parameter conditions, a positive equilibrium may undergo saddle-node, Hopf, and Bogdanov–Takens bifurcations; the existence of a homoclinic curve on the phase plane is also proved, which breaks in an unstable limit cycle.

Some simulations to reinforce our results are also shown. Copyright © 2015 John Wiley & Sons, Ltd.

Many models of asymmetric distributions proposed in the statistical literature are obtained by transforming an arbitrary symmetric distribution by means of a skewing mechanism. In certain important cases, the resultant skewed distribution shares some properties of its symmetric antecedent. Because of this inheritance, it would be interesting to test if the symmetric generator belongs to a certain family, that is to say, testing goodness-of-fit for the symmetric component. This work proposes a test of such hypothesis. Taking into account that the normal law is perhaps the most studied distribution, as a particular case of unquestionable interest, the generalized skew-normal family is studied in detail, because the symmetric component of the distributions in this family is normal. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Interval-valued fuzzy sets are an extension of fuzzy sets and are helpful when there is not enough information to define a membership function. This paper studies the behavior of a construction method for an interval-valued fuzzy relation built from a fuzzy relation. The behavior of this construction method is analyzed depending on the used t-norms and t-conorms, showing that different combinations of them produce a big variation in the results. Furthermore, a hybrid construction method that considers weight functions and a smoothing procedure is also introduced. Among the different applications of this method, the detection of edges in images is one of the most challenging. Thus, the performance of the proposal in detecting image edges is tested, showing that the hybrid approach that combines weights and a smoothing procedure provides better results than the non-weighted methods. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Diffusion processes have traditionally been modeled using the classical parabolic advection-diffusion equation. However, as in the case of tracer transport in porous media, significant discrepancies between experimental results and numerical simulations have been reported in the literature. Therefore, in order to describe such anomalous behavior, known as non-Fickian diffusion, some authors have replaced the parabolic model with the continuous time random walk model, which has been very effective. Integro-differential models (IDMs) have been also proposed to describe non-Fickian diffusion in porous media. In this paper, we introduce and test a particular type of IDM by fitting breakthrough curves resulting from laboratory tracer transport. Comparisons with the traditional advection-diffusion equation and the continuous time random walk are also presented. Moreover, we propose and numerically analyze a stable and accurate numerical procedure for the two-dimensional IDM composed by a integro-differential equation for the concentration and Darcy's law for flow. In space, it is based on the combination of mixed finite element and finite volume methods over an unstructured triangular mesh. Copyright © 2015 John Wiley & Sons, Ltd.

]]>It is preliminarily known that *Aedes* mosquitoes be very close to humans and their dwellings also give rises to a broad spectrum of diseases: dengue, yellow fever, and chikungunya. In this paper, we explore a multi-age-class model for mosquito population secondarily classified into indoor–outdoor dynamics. We accentuate a novel design for the model in which periodicity of the affecting time-varying environmental condition is taken into account. Application of optimal control with collocated measure as apposed to widely used prototypic smooth time-continuous measure is also considered. Using two approaches, least square and maximum likelihood, we estimate several involving undetermined parameters. We analyze the model enforceability to biological point of view such as existence, uniqueness, positivity, and boundedness of solution trajectory, also existence and stability of (non)trivial periodic solution(s) by means of the basic mosquito offspring number. Some numerical tests are brought along at the rest of the paper as a compact realistic visualization of the model. Copyright © 2015 John Wiley & Sons, Ltd.

Pre-operative planning of percutaneous thermal ablations is a difficult but decisive task for a safe and successful intervention. The purpose of our research is to assist surgeons in preparing cryoablation with an automatic pre-operative path planning algorithm able to propose a placement for multiple needles in three dimensions. The aim is to optimize the tumor coverage problem while taking into account a precise computation of the frozen area. Using an implementation of the precise estimation of the ice balls, this study focuses on the optimization in an acceptable time of multiple probes positions with 5 degrees of freedom, regarding the constraint of optimal volumetric coverage of the tumor by the combined necrosis. Pennes equation was used to solve the propagation of cold within the tissues, and included in an objective function of the optimization process. The propagation computation being time-consuming, seven optimization algorithms from the literature were experimented under different conditions and compared, in order to reduce overall computation time while preserving precision. Moreover, several hybrid algorithms were tested to reduce required time for the computations. Some of these methods were found suitable for the conditions of our cryosurgery planning. We conclude that this combination of bioheat simulation and optimization can be appropriate for a use by practitioners in acceptable conditions of time and precision. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Accuracy in roads geometry is an objective to be achieved by surveyors and cartographers when they obtain their data by GPS, Lidar, or photogrammetry. Nevertheless, those capture methods are expensive. Nowadays, cheap and collaborative methods can produce big datasets, which need to be processed in order to get accuracy axis from not accurate original data. Because a roads network is composed of several points, the resulting dataset could become a large-sized file, difficult to manage, and slow in consultancy for the users. In this paper, we expose our previous solutions for estimating a representative axis and propose a novel B-spline least square method governed by a genetic algorithm. The genetic algorithm minimizes the number of knots necessary to define the B-spline representative axis while keeping the axis' original shape. We know the original shape because we have computed it using a large number of knots by an iterative and convergent method developed in a well-contrasted previous study. This paper shows that our approaches are suitable to be deployed in a web-based application in order to support collaborative digital cartography. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In epidemiology, an epidemic is defined as the spread of an infectious disease to a large number of people in a given population within a short period of time. In the marketing context, a message is viral when it is broadly sent and received by the target market through person-to-person transmission. This specific marketing communication strategy is commonly referred as viral marketing. Because of this similarity between an epidemic and the viral marketing process and because the understanding of the critical factors to this communications strategy effectiveness remain largely unknown, the mathematical models in epidemiology are presented in this marketing specific field. In this paper, an epidemiological model susceptible-infected-recovered to study the effects of a viral marketing strategy is presented. It is made a comparison between the disease parameters and the marketing application, and MATLAB simulations are performed. Finally, some conclusions are carried out and their marketing implications are exposed: interactions across the parameters suggest some recommendations to marketers, as the profitability of the investment or the need to improve the targeting criteria of the communications campaigns. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we show that a higher order Borel–Pompeiu (Cauchy–Pompeiu) formula, associated with an arbitrary orthogonal basis (called structural set) of a Euclidean space, can be extended to the framework of generalized Clifford analysis. Furthermore, in lower dimensional cases, as well as for combinations of standard structural sets, explicit expressions of the kernel functions are derived. Copyright © 2015 John Wiley & Sons, Ltd.

]]>Internet of Things is built on sensor networks, which are enabling a new variety of solutions for applications in several fields (defense, health, and industry). Wireless communications are key for taking full advantage of sensor networks but implies new challenges in the security and privacy of communications. Security in wireless sensor networks is a major challenge for extending its applications. Moreover, the restrictive processing capabilities of sensor nodes have made the encryption techniques devised for conventional networks not feasible. In such setting, symmetric-key ciphers are preferred for key management in wireless sensor network; key distribution is therefore an issue solved by means of the protocols presented in this paper. Diffie–Hellman protocol (since its development) has been in the most important cryptographic primitive being the base for plenty of security protocols. In this paper, we proposed three key exchange protocols derived from the Diffie–Hellman method, applied to elliptic curves, named ECGDH-1, ECGDH-2, and ECGDH-3. Each performs secure key distribution sharing a group key among different nodes optimizing different network parameters. The security of the methods is based upon the difficulty of solving the discrete logarithm problem on elliptic curves. Making its application feasible to restrictive environments. Copyright © 2015 John Wiley & Sons, Ltd.

]]>In this paper, we introduce a general Bayesian abstract fuzzy economy model of product measurable spaces, and we prove the existence of Bayesian fuzzy equilibrium for this model. Our results extend and improve the corresponding recent results announced by Patriche and many authors from the literature. It captures the idea that the uncertainties characterize the individual feature of the decisions of the agents involved in different economic activities. In this paper, the uncertainties can be described by using random fuzzy mappings. Further attention is needed for the study of applications of the established result in the game theory and the fuzzy economic field.Copyright © 2015 John Wiley & Sons, Ltd.

]]>*A discrete dynamical system* named *real - valued pendulum* is presented in the paper. The system is developed for modeling of particles relocations on abstract graph with formalized rules of behavior (*competition for space*) and movement plans *logistics*. Logistics is given in a formal statement by the real numbers of the unit closed interval, presented in calculus with base equals to the graph power *N*.

The central question is to study *the behavior of the pendulum depending on its architecture.*

In the previous papers, with investigation of *logistic pendulum*, we have studied *chaotic pendulum*, where the particle plan for “tomorrow” is played “today”. A particular case of logistic pendulum, such that particle plans are time shifts of *N*−ary digital representation of single generating number, is called *phase pendulum.*

In this paper, we are considering a dynamical system with generation of particles plans using a some mapping *F*, defined on the set of system states. Copyright © 2016 John Wiley & Sons, Ltd.

In the design of mathematical methods for a medical problem, one of the kernel issues is the identification of symptoms and measures that could help in the diagnosis. Discovering connections among them constitute a big challenge because it allows to reduce the number of parameters to be considered in the mathematical model. In this work, we focus on formal concept analysis as a very promising technique to address this problem. In previous works, we have studied the use of formal concept analysis to manage attribute implications. In this work, we propose to extend the knowledge that we can extract from every context using positive and negative information, which constitutes an open problem.

Based on the main classical algorithms, we propose new methods to generate the lattice concept with positive and negative information to be used as a kind of map of attribute connections. We also compare them in an experiment built with datasets from the UCI repository for machine learning. We finally apply the mining techniques to extract the knowledge contained in a real data set containing information about patients suffering breast cancer. The result obtained have been contrasted with medical scientists to illustrate the benefits of our proposal. Copyright © 2016 John Wiley & Sons, Ltd.

This paper deals with fractional differential equations with dependence on a Caputo fractional derivative of real order. The goal is to show, based on concrete examples and experimental data from several experiments, that fractional differential equations may model more efficiently certain problems than ordinary differential equations. A numerical optimization approach based on least squares approximation is used to determine the order of the fractional operator that better describes real data, as well as other related parameters. Copyright © 2015 John Wiley & Sons, Ltd.

]]>We address various topologies (de Bruijn, chordal ring, generalized Petersen, and meshes) in various ways (isometric embedding, embedding up to scale, and embedding up to a distance) in a hypercube or a half-hypercube. Example of obtained embeddings: infinite series of hypercube embeddable bubble sort and double chordal rings topologies, as well as of regular maps. Copyright © 2016 John Wiley & Sons, Ltd.

]]>In this paper, we introduce the concept of Berinde's cyclic contraction that is a generalized cyclic contraction mapping by using the idea of weak contraction mapping which is defined by Berinde and prove best proximity point theorems for such a mapping in metric space with proximally complete property, Our results improve and extend the recent results of Eldred and Veeramani and some authors. Copyright © 2016 John Wiley & Sons, Ltd.

]]>Spaces of spinor-valued homogeneous polynomials, and in particular spaces of spinor-valued spherical harmonics, are decomposed in terms of irreducible representations of the symplectic group Sp(*p*). These Fischer decompositions involve spaces of homogeneous, so-called
-monogenic polynomials, the Lie super algebra
being the Howe dual partner to the symplectic group Sp(*p*). In order to obtain Sp(*p*)-irreducibility, this new concept of
-monogenicity has to be introduced as a refinement of quaternionic monogenicity; it is defined by means of the four quaternionic Dirac operators, a scalar Euler operator
underlying the notion of symplectic harmonicity and a multiplicative Clifford algebra operator *P* underlying the decomposition of spinor space into symplectic cells. These operators
and *P*, and their Hermitian conjugates, arise naturally when constructing the Howe dual pair
, the action of which will make the Fischer decomposition multiplicity free. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, we introduce an algorithmic procedure that computes abelian subalgebras and ideals of a given finite-dimensional Malcev algebra. All the computations are performed by using the non-zero brackets in the law of the algebra as input. Additionally, the algorithm also computes the *α* and *β* invariants of these algebras, and as a supporting output, a list of abelian ideals and subalgebras of maximal dimension is returned too. To implement this algorithm, we have used the symbolic computation package MAPLE 12, performing a brief computational and statistical study for it and its implementation. Copyright © 2016 John Wiley & Sons, Ltd.

The set
of *n*-dimensional Malcev magma algebras over a finite field
can be identified with algebraic sets defined by zero-dimensional radical ideals for which the computation of their reduced Gröbner bases makes feasible their enumeration and distribution into isomorphism and isotopism classes. Based on this computation and the classification of Lie algebras over finite fields given by De Graaf and Strade, we determine the mentioned distribution for Malcev magma algebras of dimension *n*≤4. We also prove that every three-dimensional Malcev algebra is isotopic to a Lie magma algebra. For *n* = 4, this assertion only holds when the characteristic of the base field
is distinct of two. Copyright © 2016 John Wiley & Sons, Ltd.

Stochastic models that can describe real epidemiological processes can become very quickly quite complex. Approximation schemes are a useful tool to understand the qualitative behaviour of such processes. In this paper, we investigate the semiclassical approximation, performed in the context of the Hamilton–Jacobi formalism, for solutions of master equations of stochastic epidemiological systems. In a test case of a previously investigated process, the linear infection model, we can analytically solve Hamilton's equations of motion. This helps to understand generalizations to more complex epidemiological systems as needed to describe realistic cases like multi-strain systems applicable to dengue fever, for example. The connection between the semiclassical approach for epidemiological systems and the Wentzel–Kramers–Brillouin approximation in quantum mechanics is also discussed. Copyright © 2016 John Wiley & Sons, Ltd.

]]>The main aim of the paper is to research dynamic properties of a mechanical system consisting of a ball jumping between a movable baseplate and a fixed upper stop. The model is constructed with one degree of freedom in the mechanical oscillating part. The ball movement is generated by the gravity force and non-harmonic oscillation of the baseplate in the vertical direction. The impact forces acting between the ball and plate and the stop are described by the nonlinear Hertz contact law. The ball motion is then governed by a set of two nonlinear ordinary differential equations. To perform their solving, the Runge–Kutta method of the fourth order with adaptable time step was applied. As the main result, it is shown that the systems exhibit regular, irregular, and chaotic pattern for different choices of parameters using the newly introduced 0–1 test for chaos, detecting bifurcation diagram, and researching Fourier spectra. Copyright © 2016 John Wiley & Sons, Ltd.

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