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This paper considers a class of systems called density systems. For such systems, the derivative of a quadratic function depends on some function termed the density function. The latter function is used to define the properties of the space affecting the behavior of the systems under consideration. The role of density systems in control law design is shown. Control systems are constructed for plants with known and unknown parameters. The theoretical results are illustrated by numerical simulation. [ABSTRACT FROM AUTHOR]

This paper handles with the Hadamard and the Caputo-Hadamard fractional derivative and stability of related systems without and with delay. Firstly, the derivative inequalities are obtained, which is indispensable in applying the theorems derived in this paper. Then, for systems without delay, we get the stability results by using the Lyapunov direct method and for systems with delay, we explore two useful inequalities to verify the stability. Examples are presented with numerical simulations to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

This paper considers mechanical systems with linear velocity forces and highly non-linear positional forces containing distributed-delay terms. Asymptotic stability conditions of system equilibria are proved using Lyapunov's direct method and the decomposition method. The developed approaches are applied to the monoaxial stabilization of a solid body. The theoretical outcomes are confirmed by computer simulation results. [ABSTRACT FROM AUTHOR]

In this paper, an uncertain SIR (spreader, ignorant, stifler) rumor spreading model driven by one Liu process is formulated to investigate the influence of perturbation in the transmission mechanism of rumor spreading. The deduced process of the uncertain SIR rumor spreading model is presented. Then an existence and uniqueness theorem concerning the solution is proved. Moreover, the stability of uncertain SIR rumor spreading differential equation is proved. In addition, the influence of different parameters on rumor spreading is analyzed through numerical simulation. This paper also presents a paradox of stochastic SIR rumor spreading model. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of a general differential equation with neutral type are investigated. Under certain assumptions, the stability of positive equilibrium and the existence of Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. And global existence of positive periodic solutions is established by using the global Hopf bifurcation result of Krawcewicz et al. Finally, by taking neutral Nicholson's blowflies model and neutral Mackey–Glass model as two examples, some numerical simulations are carried out to illustrate the analytical results. [ABSTRACT FROM AUTHOR]

Fractional differential equations sufficiently depict the nature in view of the symmetry properties, which portray physical and biological models. In this paper, we present a proficient collocation method based on cubic trigonometric B-Splines (CuTBSs) for time-fractional diffusion equations (TFDEs). The methodology involves discretization of the Caputo time-fractional derivatives using the typical finite difference scheme with space derivatives approximated using CuTBSs. A stability analysis is performed to establish that the errors do not magnify. A convergence analysis is also performed The numerical solution is obtained as a piecewise sufficiently smooth continuous curve, so that the solution can be approximated at any point in the given domain. Numerical tests are efficiently performed to ensure the correctness and viability of the scheme, and the results contrast with those of some current numerical procedures. The comparison uncovers that the proposed scheme is very precise and successful. [ABSTRACT FROM AUTHOR]

In this paper, we study the reasonability of linearized approximation and Hopf bifurcation control for a fractional-order delay Bhalekar–Gejji (BG) chaotic system. Since the current study on Hopf bifurcation for fractional-order delay systems is carried out on the basis of analyses for stability of equilibrium of its linearized approximation system, it is necessary to verify the reasonability of linearized approximation. Through Laplace transformation, we first illustrate the equivalence of stability of equilibrium for a fractional-order delay Bhalekar–Gejji chaotic system and its linearized approximation system under an appropriate prior assumption. This semianalytically verifies the reasonability of linearized approximation from the viewpoint of stability. Then we theoretically explore the relationship between the time delay and Hopf bifurcation of such a system. By introducing the delayed feedback controller into the proposed system, the influence of the feedback gain changes on Hopf bifurcation is also investigated. The obtained results indicate that the stability domain can be effectively controlled by the proposed delayed feedback controller. Moreover, numerical simulations are made to verify the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a simple network consisting of only two nodes and two paths. The first node, which is called the source, has three competing firms that send their quantities of load via the two paths to the second node, called the destination node. The static game that describes the reaction among the three firms is constructed. The Nash equilibrium point of the static game is discussed. Assuming a gradient firm based rule we investigate the dynamic game which has the same Nash equilibrium as in the static game. The local stability conditions of the Nash equilibrium are obtained in terms of the reactivity parameters among the firms and the nonlinear costs functions adopted by those firms. The obtained results are supported by a numerical simulation that in turn gives routes where Nash equilibrium may lose its stability. The simulation shows that Nash equilibrium loses its stability via flip and fold bifurcations and then chaos exists. [ABSTRACT FROM AUTHOR]

The hybrid vibration isolation, which takes advantages of both the passive and active approaches, has been an important solution for space missions. The objective of this paper is to design a vibration isolation platform for payloads on spacecrafts with the robust, wide bandwidth, and multi-degree-of-freedom (MDOF). The proposed solution is based on a parallel mechanism with six voice-coil motors (VCMs) as the actuators. The linear active disturbance resistance control (LADRC) algorithm is used for the active control. Numerical simulation results show that the vibration isolation platform performs effectively over a wide bandwidth, and the resonance introduced by the passive isolation is eliminated. The system robustness to the uncertainties of the structure is also verified by simulation. [ABSTRACT FROM AUTHOR]

The paper examines the influence of bluff body's shape on vortex shedding stability in a liquid flow. It shows that the expansion of stable vortex existence domain enables the extension of velocity measurement range for vortex flowmeters. Numerical simulation of eddy flows generated by various bluff body shapes is undertaken with the appropriate comparative analysis of flow characteristics. A bluff body with the best flow parameters enabling the improvement of flowmeter's measurement performance is selected. [ABSTRACT FROM AUTHOR]

This paper deals with adhesive frictionless normal contact between one elastic flat solid and one stiff solid with rough surface. After computation of the equilibrium solution of the energy minimization principle and respecting the contact constraints, we aim at studying the stability of this equilibrium solution. This study of stability implies solving an eigenvalue problem with inequality constraints. To achieve this goal, we propose a proximal algorithm which enables qualifying the solution as stable or unstable and that gives the instability modes. This method has a low computational cost since no linear system inversion is required and is also suitable for parallel implementation. Illustrations are given for the Hertzian contact and for rough contact. [ABSTRACT FROM AUTHOR]

The dq0 transformation is increasingly used today to model distributed sources, complex loads, renewable generators, and power electronics-based devices. This paper presents a dynamic model of long transmission lines that is based entirely on dq0 quantities, and demonstrates how such a model may be integrated with emerging dq0 models of large-scale networks. The model is first developed in the frequency domain and then converted to the time domain, using a state-space representation which inputs and outputs are dq0 signals. The proposed approach may be used to evaluate the stability and dynamic behavior of power systems that include long transmission lines, taking advantage of the dq0 reference frame inherent benefits. This is demonstrated on the basis of a 7-bus network, which shows how long transmission lines influence the network dynamics and stability. The proposed models and examples are provided as a part of an open-source software. [ABSTRACT FROM AUTHOR]

In this paper, the Hopf bifurcation control for a Lotka-Volterra predator-prey model with two delays is studied by using a hybrid control strategy. By analyzing the associated characteristic equation, its local stability and the existence of Hopf bifurcation with respect to both delays are established. In addition, the onset of an inherent bifurcation is delayed. Based on the normal form theory and the center manifold theorem, explicit formulas are derived to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solution. Numerical simulation results confirm that the hybrid controller is efficient in controlling Hopf bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, a predator-prey model with Holling type-I functional response and multi state impulsive feedback control is established, where the intensity of pesticide spraying and the release amount of natural enemies are linearly dependent on the given threshold in the second impulse. Firstly, the existence of order-1 periodic solution of the system is investigated by successor functions and Bendixson theorem of impulsive differential equations, then the stability of periodic solutions is proved by the analogue of the Poincaré criterion. Furthermore, in order to reduce the actual total cost and obtain the best economic benefit, the optimal economic threshold is obtained, which provides the optimal strategy for the practical application. Finally, numerical simulations for specific examples are carried out to illustrate the feasibility of the above conclusions. [ABSTRACT FROM AUTHOR]

This paper presents the small-signal stability analysis of an 11-kW open-loop inverter-fed induction motor drive, including the effect of inverter dead-time. The analysis is carried out using an improved small-signal model of the drive that has been reported in literature recently, and is used to demonstrate small-signal instability in a higher-power-level motor. Through small-signal stability analysis, the region of oscillatory behaviour is identified on the voltage versus frequency plane ( V- f plane), considering no-load. These predictions using the improved model are also compared against predictions of a standard model of an inverter-fed induction motor including dead-time effect. The oscillatory behaviour of the 11-kW motor drive is also studied through extensive time-domain numerical simulations and actual measurements over wide ranges of operating conditions. Both the simulation and experimental results confirm the validity of the predictions by the improved analytical model. Further, these results establish that the analysis is valid for both sine-triangle pulse-width modulation (PWM) and conventional space vector PWM. [ABSTRACT FROM AUTHOR]

The paper is concerned with a delayed diffusive predator-prey system where the growth of prey population is governed by Allee effect and the predator population consumes the prey according to Beddington-DeAngelis type functional response. The situation of bi-stability and the existence of two coexisting equilibria for the proposed model system are addressed. The stability of the steady state together with its dependence on the magnitude of time delay has been obtained. The conditions that guarantee the occurrence of the Hopf bifurcation in presence of delay are demonstrated. Furthermore, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Finally, some numerical simulations have been carried out in order to validate the assumptions of the model. [ABSTRACT FROM AUTHOR]

In this paper, the delayed finance model of enterprise operation is improved. The stability is investigated, and a Hopf bifurcation is demonstrated. Applying the normal form theory and the center manifold argument, some concrete expressions to judge the properties of the bifurcating periodic solutions are given. Computer simulations are performed to prove the correctness of theoretical analysis. Finally, a simple conclusion is included. [ABSTRACT FROM AUTHOR]

In this paper, a genetic oscillator model with time delay is discretized by the Euler method. The discrete oscillator model is discussed by using Neimark-Sacker bifurcation theory. The direction and the stability of the Neimark-Sacker bifurcation has been studied using the center manifold theorem and normal form theory. Numerical simulations illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study a predator-prey-mutualist system with digestion delay. First, we calculate the threshold value of delay and prove that the positive equilibrium is locally asymptotically stable when the delay is less than the threshold value and the system undergoes a Hopf bifurcation at the positive equilibrium when the delay is equal to the threshold value. Second, by applying the normal form method and center manifold theorem, we investigate the properties of Hopf bifurcation, such as the direction and stability. Finally, some numerical simulations are carried out to verify the main theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we extend the SIR model with vaccination into a fractional-order model by using a system of fractional ordinary differential equations in the sense of the Caputo derivative of order $\alpha\in(0,1]$. By applying fractional calculus, we give a detailed analysis of the equilibrium points of the model. In particular, we analytically obtain a certain threshold value of the basic reproduction number $R_{0}$ and describe the existence conditions of multiple equilibrium points. Moreover, it is shown that the stability region of the equilibrium points increases by choosing an appropriate value of the fractional order α. Finally, the analytical results are confirmed by some numerical simulations for real data related to pertussis disease. [ABSTRACT FROM AUTHOR]

This paper investigates the dynamics of an HTLV-I infection model with intracellular delay and immune activation delay. The primary objective of the study is to consider the effect of the time delay on the stability of the infected equilibrium. Two sharp threshold parameters $\Re_{0}$ and $\Re_{1}$ are identified as the basic reproduction number for viral infection and for CTLs response, respectively, which determine the long time behaviors of the viral infection. In particular, our mathematical analysis reveals that a Hopf bifurcation occurs when immune activation delay passes through a critical value. Using the normal form theory and center manifold arguments, the explicit formulae which determine the stability, the direction, and the period of bifurcating periodic solutions are derived. Numerical simulations are given to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the logistic equation with piecewise constant argument of generalized type. We analyze the stability of the trivial fixed point and the positive fixed point after reducing the equation into a nonautonomous difference equation. We also discuss the existence of bounded solutions for the reduced nonautonomous difference equation. Then we investigate the stability of the positive fixed point by means of Lyapunov's second method developed for nonautonomous difference equations. We find conditions formulated through the parameters of the model and the argument function. We also present numerical simulations to validate our findings. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of predator-prey model with Holling-II functional response. Firstly, by using linearization method, we prove the stability of nonnegative equilibrium points. Secondly, we obtain the existence, direction, and stability of Hopf bifurcation by using Poincare–Andronov Hopf bifurcation theorem. Finally, we demonstrate the validity of our results by numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, by taking two important network environment factors (namely point-to-group worm propagation and benign worms) into consideration, a mathematical model with multiple delays to model the worm prevalence is presented. Sufficient conditions for the local stability of the unique endemic equilibrium and the existence of a Hopf bifurcation are demonstrated by choosing the different combinations of the three delays and analyzing the associated characteristic equation. Directly afterward, the stability and direction of the bifurcated periodic solutions are investigated by using center manifold theorem and the normal form theory. Finally, special attention is paid to some numerical simulations in order to verify the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper an SIR deterministic mathematical model for co-infection of pneumonia and meningitis is proposed. We use a seven compartment model by using ordinary differential equations. The positivity of future solution of the model, the invariant region, and disease-free as well as endemic equilibrium are studied. To study the stability of the equilibria, a basic reproduction number is obtained by using the next generation matrix. The robustness of the model is also investigated. To identify the effect of each parameter on the expansion or control of the diseases, sensitivity analysis is performed. The effects of treating pneumonia infected only, meningitis infected only, and co-infected individuals have been identified by using the numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors of a discrete-time prey-predator model with Allee effect on the prey population are investigated. The existence and topological classification of the fixed points of the model are analyzed. It is shown that the model can undergo a Neimark-Sacker bifurcation at the unique positive fixed point by choosing a as a bifurcation parameter. The conditions of the existence for Neimark-Sacker bifurcation and the direction of bifurcation via bifurcation theory are presented. Also, some numerical simulations are presented to support of the analytical finding. Then bifurcation diagrams and phase portraits of the model are given. [ABSTRACT FROM AUTHOR]

In this paper we present a mathematical model of malaria transmission. The model is an autonomous system, constructed by considering two models: a model of vector population and a model of virus transmission. The threshold dynamics of each model is determined and a relation between them established. Furthermore, the Lyapunov principle is applied to study the stability of equilibrium points. The common basic reproduction number has been determined using the next generation matrix and its implication for malaria management analyzed. Hence, we show that if the threshold dynamics quantities are less than unity, the mosquitoes population disappears leading to malaria disappearance; but if they are greater than unity, mosquitoes population persists and malaria also.Finally, numerical simulations are carried out to support our mathematical results. [ABSTRACT FROM AUTHOR]

A two-species commensal symbiosis model involving Allee effect and one party can not survive independently is proposed and studied in this paper. Sufficient conditions which ensure the local and global stability of the boundary equilibrium and the positive equilibrium are obtained, respectively. Numeric simulations show that with the increasing of Alee effect, the system takes much longer time to reach its stable steady-state solution, though the Allee effect has no influence on the final density of the species. The Allee effect has instable effect on the system, however, such effect is controllable. [ABSTRACT FROM AUTHOR]