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In this paper, the existence of a solution to the problem describing the small vertical vibration of an elastic string on the positive half-line is investigated in the case when both viscous and material damping coefficients are present. The result is obtained by transforming the original partial differential equation into an appropriate abstract second-order ordinary differential equation in a suitable infinite dimensional space. The abstract problem is then studied using the combination of the Kakutani fixed point theorem together with the approximation solvability method and the weak topology. The applied procedure enables obtaining the existence result also for problems depending on the first derivative, without any strict compactness assumptions put on the right-hand side and on the fundamental system generated by the linear term. The paper ends by applying the obtained result to the studied mathematical model describing the small vertical vibration of an elastic string with a nonlinear Balakrishnan–Taylor-type damping term. • Vibrating string equation on the positive half-line with viscous and material damping coefficients. • Abstract second order equation depending on the first derivative. • Linear operator generating a fundamental system. • Continuously differentiable mild solutions. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of a class of nonlinear Helmholtz equations that is essentially used in mathematical models for the theoretical and numerical analysis of scattering and radiation effects. While well-known works relate to geometrically simple domains (supporting the nonlinearity) and selected nonlinearities, the new aspects lie in the transition to more generally shaped, two- or three-dimensional objects, to more general nonlinearities (including saturation), and in the possibility of an efficient numerical approximation of the electromagnetic fields and derived quantities (such as energy, transmission coefficient, etc.). The paper describes and investigates an approach that consists in transforming the original full-space transmission problem for a nonlinear Helmholtz equation into an equivalent boundary-value problem on a bounded domain by means of a nonlocal Dirichlet-to-Neumann (DtN) operator. It is shown that the transformed nonlinear problem is equivalent to the original one and is uniquely solvable under appropriate conditions. In addition, the effect of truncation of the DtN operator on the resulting solution is investigated. It is shown that the corresponding sesquilinear form satisfies a parameter-uniform inf–sup condition, that the modified nonlinear problem has a unique solution, and that the solution error caused by the truncated DtN operator can be estimated in dependence on the truncation parameter. • The response of a penetrable obstacle under an electromagnetic field is considered. • The constitutive law of the obstacle is nonlinear. • The mathematical model is a transmission problem for a nonlinear Helmholtz equation. • Existence, uniqueness of the solution of a spatially reduced problem are proved. • A parameter-independent stability estimate and an error estimate are given. [ABSTRACT FROM AUTHOR]

The emergence of new variants, such as Omicron, has raised concerns regarding the transmission dynamics of COVID-19 and the effectiveness of vaccination strategies. This paper proposes a mathematical model to investigate the impact of vaccination and booster doses on Omicron transmission dynamics, considering various infection compartments. The model incorporates multiple compartments representing different stages of infection, including susceptible individuals, vaccinated individuals, boosted individuals, and those infected with Omicron. The infection dynamics are captured by parameters such as vaccine efficacy, vaccination with booster received efficacy, and infection rate. Using mathematical analysis and numerical simulations, we explore how different vaccination and booster strategies affect the spread of Omicron. The normalized sensitivity analysis method of R 0 is investigated to understand the importance of parameters in disease transmission. Furthermore, we assess the influence of infection compartments, such as asymptomatic and symptomatic cases, on overall transmission dynamics. • The emergence of new variants, such as Omicron is studied. • The transmission dynamics and effectiveness of vaccination strategies are reported. • The model is constructed with the impact of vaccination and booster doses. • The infection dynamics are captured with vaccine efficacy and infection rate. • The normalized sensitivity analysis method of R 0 is investigated. [ABSTRACT FROM AUTHOR]

Highlights • A computational scheme for linear fractional order systems is put forward based on the Adams method and the Newmark - β method to obtain the numerical solutions precisely and efficiently, no matter how the single degree of freedom (SDOF) and multi-degree-of-freedom (MDOF) system, or the system contains different type derivative operators at the same time. • The enhanced response sensitivity approach in this paper does not depend on the selection of initial values. • Measured data contain 10 work well and insensitive to the measurement noise. • The weight matrix can indeed decrease the influence of the measurement noise and improve the accuracy of the identification procedure. Abstract This paper presents an investigation on parameters identification of fractional order system based on the enhanced response sensitivity approach. Compared with the integer order differential operator, fractional order operator can describe the characteristics of some phenomena more accurately such as hereditary and memory properties. In order to achieve the solution of the fractional order system, a new computational scheme is presented in this paper based on Adams-type scheme and the Newmark - β method. Such an identification problem is classed as an inverse problem and can be modeled as an nonlinear optimization problem. In this paper, the enhanced response sensitivity approach is derived and used to identify the parameters of the fractional order system. To this end, the time domain response of the fractional order system is obtained by the proposed computational scheme, then the response sensitivity with respect to the parameter is achieved as well. Numerical examples illustrated the computing and identification procedure and shown the accuracy and the convergence of identification scheme. [ABSTRACT FROM AUTHOR]

Continuous stirred tank reactor (CSTR) is a common reactor in the chemical industry. The accurate observation of the concentration conversion rate of the mixture and the internal temperature of the reaction vessel is a prerequisite for obtaining the desired mixture. This paper proposes a novel observer based on residual neural networks for CSTR systems. Firstly, the mathematical model of the CSTR reaction is given, as well as a detailed description of the structure and equations of the residual neural networks and the designed observer. Then the matrix method is used for the nonlinear isolation of the residual neural networks and the theory of quadratic constraints for nonlinear activation functions of the neural networks is applied. Thus, the convergence of the proposed observer is analyzed theoretically in detail. Finally, the numerical simulations are implemented to demonstrate that the proposed residual neural network-based observer can quickly and accurately observe the state changes during the CSTR reaction. • A novel residual neural network-based observer for CSTR is proposed. • The convergence analysis of the proposed observer is presented. • The selection of hyperparameters of the residual neural networks is given. • The effectiveness of the proposed observer is verified by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model describing the interaction of malignant glioma cells, macrophages and glioma specific CD8+T cells is discussed. The biologically feasible equilibria and corresponding local stability are deduced. The bifurcations and related dynamical behaviors of this model are further studied thoroughly. The existence of transcritical bifurcation and saddle–node bifurcation is derived based on Sotomayor's theorem and Hopf bifurcation is well discussed. The codimension 2 bifurcation such as Bogdanov–Takens bifurcation is investigated using the normal form theory and center manifold theorem in more detail. Finally, numerical simulations are obtained to validate our analytical findings by varying the parameters. • The dynamics of a mathematical model describing the interaction of malignantglioma cells, macrophages and glioma specific CD8+T cells are discussed. • The co-existence of equilibria of the model are obtained by numerical simulation. • The existence of codimension 1 and codimension 2 bifurcations is investigated. • Extensive numerical simulations are obtained to validate our analytical findings. [ABSTRACT FROM AUTHOR]

The statistical characterization of the oscillator response with non-integer order damping under Gaussian noise represents an important challenge in the modern stochastic mechanics. In fact, this kind of problem appears in several issues of different type (wave propagation in viscoelastic media, Brownian motion, fluid dynamics, RLC circuit, etc. ). The aim of this paper is to provide a stochastic characterization of the stationary response of linear fractional oscillator forced by normal white noise. In particular, this paper shows a new method to obtain the correlation function by exact complex spectral moments. These complex quantities contain all the information to describe the random processes but in the considered case their analytical evaluation needs some mathematical manipulations. For this reason such complex spectral moment characterization is used in conjunction with a fractional-order state variable analysis. This kind of analysis permits to find the exact expression of complex spectral moments, and the correlation function by using the Mellin transform. Moreover, the proposed approach provides an analytical expression of the response variance of the fractional oscillator. Capability and efficiency of the present method are shown in the numerical examples in which correlation and variance of fractional oscillator response are found and compared with those obtained by Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

As a fundamental concept in describing complex system, entropy measure has been proposed to various forms, like Boltzmann–Gibbs (BG) entropy, one-index entropy, two-index entropy, sample entropy, permutation entropy etc. This paper proposes a new two-index entropy S q,δ and we find the new two-index entropy is applicable to measure the complexity of wide range of systems in the terms of randomness and fluctuation range. For more complex system, the value of two-index entropy is smaller and the correlation between parameter δ and entropy S q,δ is weaker. By combining the refined two-index entropy S q,δ with scaling exponent h ( δ ), this paper analyzes the complexities of simulation series and classifies several financial markets in various regions of the world effectively. [ABSTRACT FROM AUTHOR]

Highlights • A novel stochastic mathematical model of HGS is established by considering the deferent impacts of internal and external stochastic. • Use the stochastic averaging method to study stochastic dynamic of HGS. • Analyze the stochastic global stability and stochastic bifurcation of the system with Hamiltonian Functions. Abstract In this paper, we pay attention to analyze the stochastic global stability and the bifurcation of the hydro-turbine generator. More specifically, the above research will be completed by introducing an analysis method of stochastic dynamics- stochastic averaging method. Initially, a mathematical model of the generator with a stochastic excitation will be established in Hamiltonian system. Then, utilizing the stochastic averaging method to derive the differential expression of the Hamiltonian function. Finally, the expression, whose physical meaning represent the mechanical energy, will be used to make a detailed analysis. The maximal Lyapunov exponent and stochastic bifurcation theory will respectively be adopted to research and discuss the stochastic global stability and the bifurcation. Research indicates that with the increasing of the rotation speed, the system will gradually change from absolute stability to absolute instability. Meanwhile, at a certainly interval of rotation angle, the probability of the generator being disturbed by stochastic factors will suddenly increase. All of the above, provides a novel perspective on the research of the hydropower stations stability, which is significant to the safety and stable operation of hydropower stations. [ABSTRACT FROM AUTHOR]

Highlights • We investigate the impact of environmental factors on the dynamical transmission of cholera. • The model incorporates the commensalism relationship between bacteria and the reservoirs. • The model exhibits the forward bifurcation. • Sensitivity analysis of the model has been performed. • The aquatic reservoirs plays a significant role in the outbreak of the disease. Abstract In this paper, we investigate the impact of environmental factors on the dynamical transmission of cholera within a human community. We propose a mathematical model for the dynamical transmission of cholera that incorporates the virulence of bacteria and the commensalism relationship between bacteria and the aquatic reservoirs on the persistence of the disease. We provide a theoretical study of the model. We derive the basic reproduction number R 0 which determines the extinction and the persistence of the infection. We show that the disease-free equilibrium is globally asymptotically stable whenever R 0 ≤ 1 , while when R 0 > 1 , the disease-free equilibrium is unstable and there exists a unique endemic equilibrium point which is locally asymptotically stable on a positively invariant region of the positive orthant. The sensitivity analysis of the model has been performed in order to determine the impact of related parameters on outbreak severity. Theoretical results are supported by numerical simulations, which further suggest the necessity to implement sanitation campaigns of aquatic environments by using suitable products against the bacteria during the periods of growth of aquatic reservoirs. [ABSTRACT FROM AUTHOR]

Highlights • A nonlinear DAE of the velocity v of a falling sphere in non-Newtonian fluid will be proposed by directly learning the data. Our model successfully simulates the sustaining oscillations and abrupt increase during the sedimentation of a sphere. It presents the behavior of a chaotic system which is highly sensitive to initial conditions and experimentally nonreproducible. The normalized representation covers both the classical physical laws and the nonuniform oscillations. The data-driven idea will provide scientists with more important tools to support their discovery in the future. Abstract In this paper, we will introduce a mathematical model of nonlinear jerk equation of velocity v = η 1 η 2 v ″ + 1 v ″ + η 3 v ″ to simulate the nonuniform oscillations of the motion of a falling sphere in the non-Newtonian fluid. This differential/algebraic equation is established only by learning the experimental data with the generalized Prony method and sparse optimization method. From the numerical results, our model successfully simulates the sustaining oscillations and abrupt increase during the sedimentation of a sphere through a non-Newtonian fluid. It presents the behavior of a chaotic system which is highly sensitive to initial conditions. More statistical and physical discussions about the dynamical features of the model are provided as well. Our model can be interpreted as a nonlinear elastic system, and includes both the uniform and nonuniform oscillatory motion of the falling sphere. [ABSTRACT FROM AUTHOR]

The discrete-time (map-based) approach to modeling nonlinear dynamics of spiking activity in neurons enables highly efficient numerical simulations for capturing realistic neurobiological behavior by utilizing a large time interval between computed states (samples) of neuron activity. The design and parameter tuning of these models assumes a fixed and preset sampling rate. When change of the time step is needed, it requires revisiting stages of the model design and parameter tuning. This paper presents an approach to the design of map-models in a new form where time step is added as a control parameter and can be easily changed to vary the time scale of the model behavior, i.e. sampling rate, essentially preserving the model behavior. It also discusses modification of the noise generator models needed to support simulation of map-based neurons with the modified sampling rate. The effects caused by direct control of time scale on model dynamics and limitations of this approach are discussed. [ABSTRACT FROM AUTHOR]

This paper concerns the analytical study of the general case of a new mathematical modelling on the γ -Ricker population models with a Holling type II per-capita birth function. The generalized Lambert W functions prove to be decisive in determining upper bounds for the number of fixed points of these models. In this approach, the use of the false derivative turns out to be very effective in solving generalized Lambert W functions of exponential polynomial and rational polynomial types, with multiple roots. The relation between the cusp points structures in the 4D parameter space considered and the variation of the number of fixed points is established. [ABSTRACT FROM AUTHOR]

In this paper we consider a piecewise smooth mathematical model to chemoimmunotherapy. The positive octant of the 3D space (N , I , Q) is partitioned into 4 regions separated by the planes I = I 0 and N = N 0 . The plane I = I 0 (respectively, N = N 0 ) controls if the immunotherapy (respectively, chemotherapy) is being applied or not. As a consequence, there are 4 smooth vector fields modeling each one of the scenarios. By means of the theory developed by piecewise smooth vector fields we are able to show the kind of contact between the trajectories of the smooth vector fields with the switching planes, the sliding motion defined over the planes and even the sliding motion defined over the simultaneous tangency lines of two smooth vector fields and the switching planes. This description is very useful in the understanding of the behavior of the treatment proposed. • Cancer model considering an intermittent protocol of chemo and immunotherapy. • Local dynamics around the switching manifolds. • Existence of sliding mode. • Existence of fold–fold curves and cusp points. • Recurrent behavior around the invisible–invisible fold curves. [ABSTRACT FROM AUTHOR]

The pressure term which appears in the ROM (reduced order model) associated to the incompressible Navier–Stokes equations, in particular for the shear flows, plays an important role on the velocity. The aim of this paper is to propose a Proper Orthogonal Decomposition based reduced order model (POD–ROM) to obtain both the velocity and pressure fields for incompressible flows. Two PODs are performed, one for the velocity and the other for the pressure. Contrary to existing projection methods available in the literature, the temporal velocity and pressure coefficients are sought by minimizing the residual of the momentum equation only, without the need of a Poisson equation. For the numerical test cases considered in this paper, the proposed minimum residual projection enables to obtain accurately the pressure field, and in turn to slightly improve the velocity one. The method is tested on two fluid flows: a transient mixed-convection flow and a periodic flow around a circular cylinder. In this last case, the drag, lift and pressure coefficients, as well as the Strouhal number are properly recovered compared to those of the full model. [ABSTRACT FROM AUTHOR]

This paper considers dual-channel game model with bounded rationality, using the theory of bifurcations of dynamical system. The business objectives of retailers are assumed to be different, which is closer to reality than previous studies. We study the local stable region of Nash equilibrium point and find that business objectives can expand the stable region and play an important role in price strategy. One interesting finding is that a fiercer competition tends to stabilize the Nash equilibrium. Simulation shows the complex behavior of two dimensional dynamic system, we find period doubling bifurcation and chaos phenomenon. We measure performances of the model in different period by using the index of average profit. The results show that unstable behavior in economic system is often an unfavorable outcome. So this paper discusses the application of adaptive adjustment mechanism when the model exhibits chaotic behavior and then allows the retailers to eliminate the negative effects. [ABSTRACT FROM AUTHOR]

Highlight: [•] New results amend some oscillation criteria for dynamic equations on isolated time scales. [•] One can amend related oscillation results for differential equations using the method provided in this paper. [•] The results obtained in this paper complement oscillation theory of dynamic equations. [Copyright &y& Elsevier]

Allee effect can interact with environment stochasticity and is active when population numbers are small. Our goal of this paper is to investigate such effect on population dynamics. More precisely, we develop and investigate a stochastic single species model with Allee effect under regime switching. We first prove the existence of global positive solution of the model. Then, we perform the survival analysis to seek sufficient conditions for the extinction, non-persistence in mean, persistence in mean and stochastic permanence. By constructing a suitable Lyapunov function, we show that the model is positive recurrent and ergodic. Our results indicate that the regime switching can suppress the extinction of the species. Finally, numerical simulations are carried out to illustrate the obtained theoretical results, where a real-life example is also discussed showing the inclusion of Allee effect in the model provides a better match to the data. [ABSTRACT FROM AUTHOR]

A modified multiscale sample entropy measure based on symbolic representation and similarity (MSEBSS) is proposed in this paper to research the complexity of stock markets. The modified algorithm reduces the probability of inducing undefined entropies and is confirmed to be robust to strong noise. Considering the validity and accuracy, MSEBSS is more reliable than Multiscale entropy (MSE) for time series mingled with much noise like financial time series. We apply MSEBSS to financial markets and results show American stock markets have the lowest complexity compared with European and Asian markets. There are exceptions to the regularity that stock markets show a decreasing complexity over the time scale, indicating a periodicity at certain scales. Based on MSEBSS, we introduce the modified multiscale cross-sample entropy measure based on symbolic representation and similarity (MCSEBSS) to consider the degree of the asynchrony between distinct time series. Stock markets from the same area have higher synchrony than those from different areas. And for stock markets having relative high synchrony, the entropy values will decrease with the increasing scale factor. While for stock markets having high asynchrony, the entropy values will not decrease with the increasing scale factor sometimes they tend to increase. So both MSEBSS and MCSEBSS are able to distinguish stock markets of different areas, and they are more helpful if used together for studying other features of financial time series. [ABSTRACT FROM AUTHOR]

A dispersion-relation preserving (DRP) method, as a semi-analytic iterative procedure, has been proposed by Jang (2017) for integrating the classical Boussinesq equation. It has been shown to be a powerful numerical procedure for simulating a nonlinear dispersive wave system because it preserves the dispersion-relation, however, there still exists a potential flaw, e.g., a restriction on nonlinear wave amplitude and a small region of convergence (ROC) and so on. To remedy the flaw, a new DRP method is proposed in this paper, aimed at improving convergence performance. The improved method is proved to have convergence properties and dispersion-relation preserving nature for small waves; of course, unique existence of the solutions is also proved. In addition, by a numerical experiment, the method is confirmed to be good at observing nonlinear wave phenomena such as moving solitary waves and their binary collision with different wave amplitudes. Especially, it presents a ROC (much) wider than that of the previous method by Jang (2017). Moreover, it gives the numerical simulation of a high (or large-amplitude) nonlinear dispersive wave. In fact, it is demonstrated to simulate a large-amplitude solitary wave and the collision of two solitary waves with large-amplitudes that we have failed to simulate with the previous method. Conclusively, it is worth noting that better convergence results are achieved compared to Jang (2017); i.e., they represent a major improvement in practice over the previous method. [ABSTRACT FROM AUTHOR]

This paper is dedicated to develop a mathematical model that can simulate nonlinear phenomena of a hinged plate which places into the fluid flow (1 DOF). These phenomena are fluttering (oscillation motion), autorotation (continuous rotation) and chaotic motion (combination of fluttering and autorotation). Two mathematical models are developed for 1 DOF problem using two eminent mathematical models which had been proposed for falling plates (3 DOF). The procedures of developing these models are elaborated and then these results are compared to experimental data. The best model in the simulation of the phenomena is chosen for stability and bifurcation analysis. Based on these analyses, this model shows a transcritical bifurcation and as a result, the stability diagram and threshold are presented. Moreover, an analytical expression is given for finding the boundary of bifurcation from the fluttering to the autorotation. [ABSTRACT FROM AUTHOR]

The problem of controlling chaotic systems has drawn much attention in the last two decades. However, the controlled system may be subjected to complicated constraints and few researches on controlling chaos take constraints into consideration. Therefore, the stabilization of constrained chaotic system is solved under the frame of nonlinear optimal control in this paper. A symplectic pseudospectral method based on qusilinearizaiton techniques and the parametric variational principle is developed to solve constrained nonlinear optimal control problems with arbitrary Lagrange-type cost functional. At the beginning of the proposed method, the original nonlinear optimal control problem is converted into a series of linear-quadratic constrained optimal control problems. Then each of the converted linear quadratic problems is transformed into a standard linear complementarity problem. The proposed method is successfully applied to stabilizing constrained chaotic systems around an unstable equilibrium point or an unstable periodic orbit. Numerical simulations demonstrate that the developed method is effective and efficient, and constraints are strictly satisfied. [ABSTRACT FROM AUTHOR]

A new generation of aircraft with distributed propellers leads to operational performances superior to those exhibited by standard designs. Computational simulations and experimental tests show a reduction of fuel consumption and noise. This paper proposes an analogy between aerodynamics and electrical circuits. The model reveals properties similar to those of fractional-order systems and gives a deeper insight into the dynamics of multi-propeller coupling. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a cancer network including a protein module and a corresponding microRNA cluster that inhibits the synthesis of proteins. The existence of multiple steady states and their stability depending on the parameters are firstly determined. Bistability and dependency on the parameters, Hopf bifurcations and the corresponding properties like direction and stability of Hopf bifurcations are determined by computing the normal form on the center manifold. Then, the role of the delay in the process of synthesis of the protein is investigated. We show that the delay can stabilize the unstable equilibrium and destabilize the stable equilibrium. Some simulations are carried out to numerically illustrate the obtained theoretical results. Finally, the biological interpretation of the theoretical results is discussed. [ABSTRACT FROM AUTHOR]

Embryonic development is a precise and complex process involving the cell morphology and mechanics interacting in space and time. The difficulty in quantitatively acquiring cellular morphological and mechanical information in vivo makes mathematical modeling a challenging problem and impedes model validation. Recently, the three-dimensional time-lapse live imaging and delineated developmental programs in the roundworm Caenorhabditis elegans provide an excellent platform for establishing quantitative models. In this paper, we study two popular computational models for multicellular systems, i.e., the phase-field model and the coarse-grained model, and compare their performance in characterizing the cell morphologies, cell adhesion, and cell stiffness in a real C. elegans embryo. We show that both models can capture cell–cell contact areas and heterogeneous cell adhesion, but only the phase-field model succeeds in inferring the heterogeneous cell stiffness by fitting cell shapes or cell–cell interface curvatures. Moreover, we demonstrate that the phase-field model converges to the coarse-grained model when increasing cell surface tension to dominance, obtaining a distance-dependent isotropic intercellular force. • The phase-field model and the coarse-grained model are systematically studied for multicellular morphologies and mechanics. • Both models can capture heterogeneous cell adhesion, but only the phase-field model can capture heterogeneous cell stiffness. • The phase-field model converges to the coarse-grained model when increasing cell surface tension to dominance. [ABSTRACT FROM AUTHOR]

• The paper proposes an overview of the modeling approaches for the mathematical description and analysis of processes that combine continuous and discontinuous behavior, namely impulsive differential equations, hybrid dynamical systems, and differential equations involving Dirac delta functions. • Insights are provided on the stability and attractivity analysis of hybrid behaviors, and essential differences are highlighted to the respective stability concepts for smooth dynamical systems. • Specific phenomena are discussed which are peculiar for hybrid behaviors, like beating or Zeno phenomenon, modeling of multiple impulses at a single time instance, death and splitting of solutions, etc. • With this, the paper attempts at bringing attention of the interested researchers to the methods available in other research communities and fostering the exchange of ideas and analysis techniques. We propose an overview of the modeling approaches for the mathematical description and analysis of processes that combine continuous and discontinuous behavior, namely impulsive differential equations, hybrid dynamical systems, and differential equations involving Dirac delta functions. These classes of systems are chosen due to their dominant prevalence in physics, mathematics, and control engineering research communities. A comparison of these frameworks is provided and their applicability depending on the character of the hybrid behavior is discussed. In particular, we show that special care should be taken when equations with Dirac delta function are interpreted as impulsive differential equations. We also provide insights on the stability and attractivity analysis of hybrid behaviors, highlight their essential differences to the respective stability concepts for smooth dynamical systems, and discuss specific phenomena which are peculiar for hybrid behaviors, like beating or Zeno phenomenon, modeling of multiple impulses at a single time instance, death and splitting of solutions, etc. With this, the paper attempts at bringing attention of the interested researchers to the methods available in other research communities and fostering the exchange of ideas and analysis techniques. [ABSTRACT FROM AUTHOR]

Abstract: A new model called Naming Game with Multiple Hearers (NGMH) is proposed in this paper. A naming game over a population of individuals aims to reach consensus on the name of an object through pair-wise local interactions among all the individuals. The proposed NGMH model describes the learning process of a new word, in a population with one speaker and multiple hearers, at each interaction towards convergence. The characteristics of NGMH are examined on three types of network topologies, namely ER random-graph network, WS small-world network, and BA scale-free network. Comparative analysis on the convergence time is performed, revealing that the topology with a larger average (node) degree can reach consensus faster than the others over the same population. It is found that, for a homogeneous network, the average degree is the limiting value of the number of hearers, which reduces the individual ability of learning new words, consequently decreasing the convergence time; for a scale-free network, this limiting value is the deviation of the average degree. It is also found that a network with a larger clustering coefficient takes longer time to converge; especially a small-word network with smallest rewiring possibility takes longest time to reach convergence. As more new nodes are being added to scale-free networks with different degree distributions, their convergence time appears to be robust against the network-size variation. Most new findings reported in this paper are different from that of the single-speaker/single-hearer naming games documented in the literature. [Copyright &y& Elsevier]

This paper proposes an algorithm for group classification of a nonhomogeneous equation using the group analysis provided for the corresponding homogeneous equation. The approach is illustrated by a partial differential equation, an integro-differential equation, and a delay partial differential equation. [ABSTRACT FROM AUTHOR]

This paper presents a study on amplitude death in nonlinear coupled oscillators containing fractional derivatives. Analytical criterion for amplitude death region is obtained by eigenvalue analysis and verified by numerical results. It is found that amplitude death regions can be enlarged to a large extent by fractional derivatives. For this reason, amplitude death can be detected in fractional Stuart–Landau systems with weak coupling strength and low frequency, whereas it never appears in integer-order systems. Interestingly, the widening of amplitude death region induced by fractional derivative is shared by a variety of oscillators with different types of coupling mechanisms. An interpretation for the underlying mechanism of this phenomenon is briefly addressed, based on which we further suggest a coupling organization leading to amplitude death only in fractional oscillators. [ABSTRACT FROM AUTHOR]

A novel SIVRS mathematical model for infectious diseases spreading is proposed and investigated in this paper. In this model virus variation factors are considered in the process of epidemic spreading based on complex networks, which can describe different contact status for different agents including the susceptible, the infectious, the variant and the recovered in a network. An optimal control problem is formulated to maximize the recovered agents with the limited resource allocation and optimal control strategies over the susceptible, the infected and the variant are investigated. Then the existence of a solution to the optimal control problem is given based on Pontryagin's Minimum Principle and modified forward backward sweep technique. Numerical simulations are provided to illustrate obtained theoretical results. [ABSTRACT FROM AUTHOR]

The paper deals with Kadomtsev–Petviashvili (KP) equation in presence of a small dispersion effect. The nature of solutions are examined under the dispersion effect by using Lyapunov function and dynamical system theory. We prove that when dispersion is added to the KP equation, in certain regions, yet there exist bounded traveling wave solutions in the form of solitary waves, periodic and elliptic functions. The general solution of the equation with or without the dispersion effect are obtained in terms of Weirstrass ℘ functions and Jacobi elliptic functions. New form of kink-type solutions are established by exploring a new technique based on factorization method, use of functional transformation and the Abel’s first order nonlinear equation. Furthermore, the stability analysis of the dispersive solutions are examined which shows that the traveling wave velocity is a bifurcation parameter which governs between different classes of waves. We use the phase plane analysis and show that at a critical velocity, the solution has a transcritical bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, we consider an initial value problem of a predator-prey system with integral term. By establishing comparison principle and constructing monotone sequences, the existence and uniqueness for the solution of that problem is proved. Then we further show the uniform boundedness. Finally, some conditions of Turing bifurcation occurring are given and illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

Abstract: For engineering systems, uncertainties and time delays are two important issues that must be considered in control design. Uncertainties are often encountered in various dynamical systems due to modeling errors, measurement noises, linearization and approximations. Time delays have always been among the most difficult problems encountered in process control. In practical applications of feedback control, time delay arises frequently and can severely degrade closed-loop system performance and in some cases, drives the system to instability. Therefore, stability analysis and controller synthesis for uncertain nonlinear time-delay systems are important both in theory and in practice and many analytical techniques have been developed using delay-dependent Lyapunov function. In the past decade the magnetic and levitation (maglev) transportation system as a new system with high functionality has been the focus of numerous studies. However, maglev transportation systems are highly nonlinear and thus designing controller for those are challenging. The main topic of this paper is to design an adaptive robust controller for maglev transportation systems with time-delay, parametric uncertainties and external disturbances. In this paper, an adaptive robust control (ARC) is designed for this purpose. It should be noted that the adaptive gain is derived from Lyapunov–Krasovskii synthesis method, therefore asymptotic stability is guaranteed. [Copyright &y& Elsevier]

Abstract: This paper presents a study of the passive dynamic walking of a compass-gait biped robot as it goes down an inclined plane. This biped robot is a two-degrees-of-freedom mechanical system modeled by an impulsive hybrid nonlinear dynamics with unilateral constraints. It is well-known to possess periodic as well as chaotic gaits and to possess only one stable gait for a given set of parameters. The main contribution of this paper is the finding of a window in the parameters space of the compass-gait model where there is multistability. Using constraints of a grazing bifurcation on the basis of a shooting method and the Davidchack–Lai scheme, we show that, depending on initial conditions, new passive walking patterns can be observed besides those already known. Through bifurcation diagrams and Floquet multipliers, we show that a pair of stable and unstable period-three gait patterns is generated through a cyclic-fold bifurcation. We show also that the stable period-three orbit generates a route to chaos. [Copyright &y& Elsevier]

Abstract: The aim of this letter is to confirm the achievements in and answer all the mentioned comments in . Meanwhile some other drawbacks in the comment paper are also presented in this paper. [Copyright &y& Elsevier]

Abstract: In this paper, an adaptive algorithm is proposed for synchronization of chaotic systems with different orders. A modular adaptive control strategy is applied to make states of the slave system track those of the master, despite the unknown parameters. One of the most advantages of the modularity approach, which is applied for the first time in chaos synchronization, is its flexibility in choosing identification and control modules and designing them completely independently. In this paper, a modified recursive least square algorithm is used to identify the unknown parameters of the slave system, and the control module is designed by means of two different algorithms. First it is designed based on active control method, and then, in order to synchronize with a lower energy, we design an optimal controller. The two methods are applied on a practical case study, and the results are compared. Two different dimensional neuron models, the HR neuron model and the cable model of cylindrical cell, are considered as the master and slave systems, respectively. Simulation results confirm the effectiveness of the proposed method. [Copyright &y& Elsevier]

Travelers’ route adjustment behaviors in a congested road traffic network are acknowledged as a dynamic game process between them. Existing Proportional-Switch Adjustment Process (PSAP) models have been extensively investigated to characterize travelers’ route choice behaviors; PSAP has concise structure and intuitive behavior rule. Unfortunately most of which have some limitations, i.e., the flow over adjustment problem for the discrete PSAP model, the absolute cost differences route adjustment problem, etc. This paper proposes a relative-Proportion-based Route Adjustment Process (rePRAP) maintains the advantages of PSAP and overcomes these limitations. The rePRAP describes the situation that travelers on higher cost route switch to those with lower cost at the rate that is unilaterally depended on the relative cost differences between higher cost route and its alternatives. It is verified to be consistent with the principle of the rational behavior adjustment process. The equivalence among user equilibrium, stationary path flow pattern and stationary link flow pattern is established, which can be applied to judge whether a given network traffic flow has reached UE or not by detecting the stationary or non-stationary state of link flow pattern. The stability theorem is proved by the Lyapunov function approach. A simple example is tested to demonstrate the effectiveness of the rePRAP model. [ABSTRACT FROM AUTHOR]

In this paper, the problem on the chaotic secure communication is discussed. First a new dual channel transmission mechanism is presented and used in secure communication scheme design, then the channel-switching techniques are adopted to further improve the security of information transmission. Finally some typical numerical simulations are carried out to demonstrate the effectiveness of the proposed secure communication scheme. [ABSTRACT FROM AUTHOR]

In this paper, we study the trapping problem in the node- and edge- weighted fractal networks with the underlying geometries, focusing on a particular case with a perfect trap located at the central node. We derive the exact analytic formulas of the average weighted trapping time (AWTT), the average of node-to-trap mean weighted first-passage time over the whole networks, in terms of the network size N g , the number of copies s , the node-weight factor w and the edge-weight factor r . The obtained result displays that in the large network, the AWTT grows as a power-law function of the network size N g with the exponent, represented by θ ( s , r , w ) = log s ( s r w 2 ) when srw 2 ≠ 1. Especially when s r w 2 = 1 , AWTT grows with increasing order N g as log  N g . This also means that the efficiency of the trapping process depend on three main parameters: the number of copies s > 1, node-weight factor 0 < w ≤ 1, and edge-weight factor 0 < r ≤ 1. The smaller the value of srw 2 is, the more efficient the trapping process is. [ABSTRACT FROM AUTHOR]

This paper studies the synchronization in two mechanical oscillators coupled by impacts which can be considered as a class of state-dependent impulsively coupled oscillators. The two identical oscillators are harmonically excited in a counter phase, and the synchronous (anti-phase synchronization) and the asynchronous motions are considered. One- and two-parameter bifurcations of the system have been studied by varying the amplitude and the frequency of external excitation. Numerical simulations show that the system could exhibit complex phenomena, including symmetry and asymmetry periodic solutions, quasi-periodic solutions and chaotic solutions. In particular, the regimes in anti-phase synchronization are identified, and it is found that the symmetry-breaking bifurcation plays an important role in the transition from synchronous to asynchronous motion. [ABSTRACT FROM AUTHOR]

Our goal of this paper is of a purely theoretical question, however which would be fundamental in computational partial differential equations: Can a linear solution-structure for the equation of motion for an infinite nonlinear beam be directly manipulated for constructing its nonlinear solution? Here, the equation of motion is modeled as mathematically a fourth -order nonlinear partial differential equation. To answer the question, a pseudo -parameter is firstly introduced to modify the equation of motion. And then, an integral formalism for the modified equation is found here, being taken as a linear solution-structure. It enables us to formulate a nonlinear integral equation of second kind, equivalent to the original equation of motion. The fixed point approach, applied to the integral equation, results in proposing a new iterative solution procedure for constructing the nonlinear solution of the original beam equation of motion, which consists luckily of just the simple regular numerical integration for its iterative process; i.e., it appears to be fairly simple as well as straightforward to apply. A mathematical analysis is carried out on both natures of convergence and uniqueness of the iterative procedure by proving a contractive character of a nonlinear operator. It follows conclusively , therefore , that it would be one of the useful nonlinear strategies for integrating the equation of motion for a nonlinear infinite beam, whereby the preceding question may be answered. In addition, it may be worth noticing that the pseudo-parameter introduced here has double roles; firstly, it connects the original beam equation of motion with the integral equation, second, it is related with the convergence of the iterative method proposed here. [ABSTRACT FROM AUTHOR]

Studied in this paper are the bright–dark vector soliton solutions for a generalized coupled Hirota system which describes the propagation for the high-intensity ultrashort pulses in the optical glass fiber. Beyond the existing bilinear forms, using an auxiliary function, we obtain the improved bilinear forms and bright–dark soliton solutions under two integrable constraints through the Hirota method and symbolic computation. With the help the analytic and graphic analysis, we study the soliton properties including the amplitudes, velocities and phase shifts, and show that the interactions for the bright–dark two solitons are elastic. For the bright–dark one soliton, parametric conditions that the dark component is “black” or “gray” are obtained. For the bright–dark two solitons, we find that the bright component is affected by the dark component background parameters during such an interaction, while the dark component is not affected by the bright component background parameters. Velocities for the bright–dark two solitons are inversely proportional to the higher-order effect parameter, but amplitudes and phase shifts are independent of it. Besides, the bound-state bright–dark two solitons are also presented. [ABSTRACT FROM AUTHOR]

Under investigation in this paper is a nonautonomous nonlinear Schrödinger equation with external potentials, which can govern the dynamics of nonautonomous solitons in the nonlinear optical medium non-uniformly distributed in both the transverse and longitudinal directions. Based on the Lax pair, we present an infinite sequence of the conservation laws. Bilinear forms, bilinear Bäcklund transformations, one-, two- and N -soliton solutions under a known variable-coefficient constraint are generated via the Hirota method. With G ( t ) = 0 and R ( t ) B ( t ) being a constant, amplitude of the soliton remains unvarying during the propagation, where t is the scaled time, G ( t ) is the gain/loss coefficient, B ( t ), the group velocity dispersion coefficient, and R ( t ), the nonlinearity coefficient. If we set G ( t ) ≠ 0 or R ( t ) B ( t ) as a variable, the amplitude becomes varying. Due to the different choices of the linear oscillator potential coefficient α ( t ), periodic-, parabolic-, S - and V -type solitons are observed. Meanwhile, we find that α ( t ) has no influence on the soliton amplitude. Interaction between the two amplitude-unvarying solitons and that between the two amplitude-varying ones are displayed, respectively. The velocity of a moving soliton always keeps varying. [ABSTRACT FROM AUTHOR]

For the fold bifurcation with an eigenvalue +1, there are three types of potential solutions from saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation. In the existing analysis methods for high maps, there is a problem that for the fold bifurcation, saddle-node bifurcation and transcritical bifurcation cannot be distinguished by the center manifold without bifurcation parameter. In this paper, a parameterized center manifold has been derived to unfold the solutions of the fold bifurcation with an eigenvalue +1, which is used to reduce a general n -dimensional map to one-dimensional map. On the basis of the reduced map, the conditions of the fold bifurcations including saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation are established for general maps, respectively. We show the applications of the proposed bifurcation conditions by three four-dimensional map examples to distinguish saddle-node bifurcation, transcritical bifurcation and pitchfork bifurcation. [ABSTRACT FROM AUTHOR]

This paper studies a generalized fractional hemivariational inequality in infinite-dimensional spaces. Under the suitable assumptions, the existence result is delivered by using the temporally semi-discrete scheme and the surjectivity result for multivalued pseudomonotone operator. As an illustrative application, we propose a frictional contact model that describes the quasi-static contact between a viscoelastic body and a solid foundation. The viscoelastic constitutive equation is modeled by fractional Kelvin-Voigt law with ψ -Caputo derivative, and the frictional contact conditions are expressed as the Clarke subdifferentials of the nonconvex and nonsmooth functionals. Finally, the weak solvability of the mechanical system is obtained by using our abstract mathematical result. • Mathematical modeling of the fractional contact problem is challenging. • The existence result of a fractional hemivariational inequality is delivered. • The weak solvability of the corresponding mechanical system is obtained. [ABSTRACT FROM AUTHOR]

In this paper we present an abstract nonsmooth optimization problem for which we recall existence and uniqueness results. We show a numerical scheme to approximate its solution. The theory is later applied to a sample static contact problem that describes an elastic body in frictional contact with a foundation. This problem leads to a hemivariational inequality which we solve numerically. Finally, we compare three computational methods of solving contact mechanical problems: the direct optimization method, the augmented Lagrangian method and the primal–dual active set strategy. • Mathematical modeling of the body's behavior on the contact boundary is challenging. • Contact phenomena described by nonmonotone laws require special numerical treatment. • Nonsmooth nonconvex hemivariational problems need sophisticated methods. • Knowing the properties of algorithms for specific problems is crucial in applications. [ABSTRACT FROM AUTHOR]

Abstract: In this paper linear and nonlinear models of spacecraft attitude dynamics equations and gravity gradient moments are investigated. In addition, effects of gravity gradient moments on attitude dynamics of the satellite are studied. The purpose of this paper is to present a comparison between nonlinear and linear models of spacecraft attitude dynamics and gravity gradient moments in order to determine divergence of linear approximation from the nonlinear model. Simulation results indicate that designer of spacecraft attitude control subsystem should be meticulous in applying linear approximation of equations especially in low earth orbits. Consequently, finding an upper bound for small angle to keep the linear model valid and precise enough would be a vital part of using linear approximation. Results supported by numerical examples demonstrate various features of this study. [Copyright &y& Elsevier]

Abstract: Rolling element bearings are the key components in many rotating machinery. For efficient performance of the machine it is necessary to accurately predict the effect of various parameters and operating conditions on the machine’s behavior. This paper deals with the development of a nonlinear model of the rotor-bearing system on rolling element bearings with clearance. Clearance is an important nonlinearity which can cause bifurcations and chaos as has been shown in this paper. In this paper a detailed model for clearance is developed. In this model the inner race center and the outer race center are not assumed to be collinear when relations for deflections in the rolling element are developed. The model is non-dimensionalized and then analyzed to reveal rich nonlinear phenomena. Further, for better performance of any machine it is necessary to identify and stay out of chaotic regimes of operation. Hence, Lyapunov exponents and Poincaré mappings are used to analyze the system and determine the regions of chaotic response. [Copyright &y& Elsevier]

Abstract: Main idea of the phase plane method applied to the optimal control in nonlinear dynamical systems with triggers of a coupled singularities, and with one degree of freedom, is reconsidered. Theory and models of the heavy mass particle optimal control motion along rough curvilinear line are presented. Introduction of the paper presents a short review of the author’s previously published results containing series of special cases of optimal control in nonlinear dynamical systems with trigger of coupled singularities important for engineering applications. Task of the defined dynamical system optimal control is: by using controlling force acting on the system, transfer initial kinetic state of the nonlinear dynamics of the system into the final terminate kinetic state of the nonlinear dynamics in the minimal time. Use, previously published results in paper written by author, the mathematical description of the heavy mass particle motion along rough curvilinear line with Coulomb’s type friction in generalized form, the optimal control of such system dynamics analytically is considered. Three cases of a rough line are considered in the phase plane and analytical expressions for the phase trajectories of the mass particle dynamics along rough line loaded by controlling force are obtained. For considered cases of mass particle motion along rough lines, we can identify a member in the differential double equation, proportional to the square of the generalized velocity. This corresponds to the known case of system in the turbulent damping field. Also, for general case, as well as for a special case we separate a corresponding fictive conservative system with two alternate one side equilibrium positions defined by an angle of friction corresponding to the coefficient of Coulomb friction. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the non-linear process of light modulation performed by optical choppers is studied. The real functions of the transmitted flux, with transition portions are derived, in contrast to approximate approaches, of flux functions considered as series of rectangular impulses. An exact mathematical model of the dynamic phenomenon is thus achieved. Top-hat light beam distributions are considered. We are developing in this paper a new type of rotating wheel with windows that have circular-shaped edges. This is the general case of the classical configuration, of choppers with wings (or, equivalent, with sector-shaped windows) that we have solved in a previous study for all its possible relationships between the geometry of the wheel and the diameter of the light beam in the plane of the chopper. This relationship is also considered in this study, while the configuration of the new device allows for more degrees of freedom in the designing process. From this discussion, the various possible geometries of choppers that may generate a required function of the transmitted flux are obtained. [ABSTRACT FROM AUTHOR]