Bendixson Criterion Research Articles

Existence, uniqueness, boundedness and stability of periodic solutions of a certain second-order nonlinear differential equation with damping and resonance effects

Summary In this paper, some qualitative behaviors of solutions for certain second-order nonlinear differential equation with damping and resonance effects are considered. By employing Lyapunov’s direct method, a complete Lyapunov function was used to investigate the stability of the system. Krasnoselskii’s fixed point theorem was used to establish sufficient conditions that guaranteed the existence and boundedness of a unique solution. The results show that the equilibrium point was asymptotically stable. Furthermore, a test for periodicity was conducted using the Bendixson criterion, and the results showed that the solution of the second-order nonlinear differential equation is aperiodic, which extends some results from the literature.

Dynamics of a discretized Nicholson's blowfly model with Dirichlet boundary condition

The dynamics of the classical diffusive Nicholson's blowfly model with zero Dirichlet boundary condition is less understood. A challenging question is that how the stability of the unique and implicit positive steady state changes in parameters. Numerics suggests that it could be stable or unstable, and a time periodic solution may appear in the latter case. In this paper, we make an effort toward this question by considering a discretized version of the model. To overcome the difficulty caused by the implicit steady state, we first establish some monotonicity properties, which are then used to find the parameter regions such that the positive steady state is locally stable, or unstable due to the occurrence of Hopf bifurcations. A set of sufficient conditions for the global convergence of the positive equilibrium are also obtained. Finally, the global existence of periodic solution is studied by using the global Hopf bifurcation theorem and high-dimensional Bendixson's criterion.

Nonlinear Damping Property and Limit Cycle of VSC

The synchronization stability of the system that one single VSC is connected to an infinite source via the line is studied in this letter. Based on Bendixson criterion, closed orbit (limit cycle) does not exist in the derived stable region, while may exist in the region between the conservative stable boundary and the unstable equilibrium point (UEP). Nonlinear damping effect on the system stability is analytically defined, and an unstable limit cycle appears when the positive and negative damping effect of the system over one period is mutually balanced. Numerical results verify the existence of limit cycle and nonlinear damping property of VSC.

An integer-order SIS epidemic model having variable population and fear effect: comparing the stability with fractional order

This paper investigates the dynamics of an integer-order and fractional-order SIS epidemic model with birth in both susceptible and infected populations, constant recruitment, and the effect of fear levels due to infectious diseases. The existence, uniqueness, non-negativity, and boundedness of the solutions for both proposed models have been discussed. We have established the existence of various equilibrium points and derived sufficient conditions that ensure the local stability under two cases in both integer- and fractional-order models. Global stability has been vindicated using Dulac–Bendixson criterion in the integer-order model. The forward transcritical bifurcation near the disease-free equilibrium has been investigated. The effect of fear level on infected density has also been observed. We have done numerical simulation by MATLAB to verify the theoretical results, found the impact of fear level on the dynamic behaviour of the infected population, and obtained a bifurcation diagram concerning the constant recruitment and fear level. Finally, we have compared the stability of the population in integer and fractional-order systems.

Bioeconomics fishery model in presence of infection: Sustainability and demand-price perspectives

Intense harvesting and emerging infectious diseases are potential threats to global fishery. A proper management policy equipped with the scientific understanding of species interaction is a footstep in a long-term sustainable fishery. This work performs a qualitative study of bioeconomic management of a fishery in presence of some infection. The model narrates the rate equations of the healthy fish, infected fish, fishing effort and market price, where the fishing effort is considered to be dependent on the fish price and the fish price is regulated by the demand-supply theory of the open market. Routh–Hurwitz criterion is utilized for the local stability analysis whereas the high-dimensional Bendixson criterion is used for the global stability analysis. The one and two parameters bifurcation analysis explain various switching in equilibrium states, which includes infection-free, infected and harvesting-free states. The existence conditions of the bionomic equilibrium, where both the ecological and economic equilibrium exists, has been established. The harvested fish biomass is observed to be higher at the infection-free equilibrium state compare to the infected equilibrium state under increasing infection rate, however, the outcome is opposite under increasing environmental carrying capacity. Though, the total revenue is highest at the infection-free state when demand is high. An unintuitive result is that the infection persists at a higher level if demand decreases.

Impact of Media-Induced Fear on the Control of COVID-19 Outbreak: A Mathematical Study

The COVID-19 pandemic has put the world in threat for a long time. It was first identified in Wuhan, China, in December 2019 and has been declared a pandemic by the WHO. This disease is mainly caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). So far, no vaccine or medicine has been developed for the proper treatment of this disease, so people are afraid of getting infected. The pandemic has placed many nations at the door of socioeconomic emergencies. Therefore, it is very important to predict the development trend of this epidemic, and we know mathematical modelling is a basic tool to research the dynamic behaviour of disease and predict the spreading trend of the disease. In this study, we have formulated a mathematical model for the COVID-19 outbreak by introducing a quarantine class with media-induced fear in the disease transmission rate to analyze the dynamic behaviour of this epidemic. We have calculated the basic reproduction number R0, and we observed that when R0 < 1, disease-free equilibrium is globally stable whereas if R0> 1, then the system is permanent and there exists a unique endemic equilibrium point. Global stability of the endemic equilibrium point is developed by using Li and Muldowney’s high-dimensional Bendixson criterion. Finally, some numerical simulations are performed using MATLAB to verify our analytical results.

Condition for Global Stability for a SEIR Model Incorporating Exogenous Reinfection and Primary Infection Mechanisms.

A mathematical model incorporating exogenous reinfection and primary progression infection processes is proposed. Global stability is examined using the geometric approach which involves the generalization of Poincare-Bendixson criterion for systems of n-ordinary differential equations. Analytical results show that for a Susceptible-Exposed-Infective-Recovered (SEIR) model incorporating exogenous reinfection and primary progression infection mechanisms, an additional condition is required to fulfill the Bendixson criterion for global stability. That is, the model is globally asymptotically stable whenever a parameter accounting for exogenous reinfection is less than the ratio of background mortality to effective contact rate. Numerical simulations are also presented to support theoretical findings.

Analysis of Optimal Control Problems for Hybrid Systems with One State Variable

We study a class of discounted autonomous infinite horizon optimal control problems where the state dynamics may undergo discontinuous changes when the state crosses from one region of the state space to another. Assuming the state and control space are one-dimensional, we provide structural results of the candidate optimal solutions. In particular, we show that candidate optimal state trajectories are monotonic. Further, we prove a theorem on absence of limit cycles, generalizing the Bendixson criterion to a hybrid context. Using these results, we derive necessary and sufficient conditions for the existence of tipping points for this class of problems. A tipping point is an initial condition starting from which there exist multiple candidate optimal solutions with different qualitative behaviors. Next, we specialize these results for piecewise linear quadratic models to obtain an analytic characterization of tipping points in terms of the model parameters.

Bendixson criterion in impulsive systems

Bendixson's condition on the nonexistence of periodic solutions for planar ordinary differential equations is extended to differential equations with impulse. Also, we use an example of predator‐prey model to illustrate our results.

Modelling and analyzing the epidemic of human infections with the avian influenza A(H7N9) virus in 2017 in China

In 2013, in mainland China, a novel avian influenza A(H7N9) virus began to infect humans, followed by the annual outbreaks, and had aroused severe fatality in the infected humans. After introducing the statistical characteristics including the geographical distributions of the outbreaks, a SEV‐SIRS eco‐epidemiological model is established and analyzed. In this model, the factor of virus in environment is incorporated into the model as a class; the vaccine measure in poultry is taken into account in purpose of assessing its control effect in 2017 in China; the nonmonotonic contact function is adopted to characterize the psychosocial effect. The stability of disease‐free equilibrium point (DFE) is obtained by the threshold theory; the stability of the endemic equilibrium point is gotten by the Bendixson criterion based on the geometric approach. Sensitivity analyses of system parameters indicate that the measure of vaccination in poultry can play its role but only when the vaccine rate is more than 98% can the disease control effect be effectively exerted, and the virus in environment is an extremely sensitive factor in the disease transmission and the epidemic control.

Nonlinear, Phase-Based Oscillator to Generate and Assist Periodic Motions

Oscillatory behavior is important for tasks, such as walking and running. We are developing methods for wearable robotics to add energy to enhance or vary the oscillatory behavior based on the system's phase angle. We define a nonlinear oscillator using a forcing function based on the sine and cosine of the system's phase angle that can modulate the amplitude and frequency of oscillation. This method is based on the state of the system and does not use off-line trajectory planning. The behavior of a limit cycle is shown using the Poincaré–Bendixson criterion. Linear and rotational models are simulated using our phase controller. The method is implemented and tested to control a pendulum.

Global Hopf bifurcation of a population model with stage structure and strong Allee effect

This paper is devoted to the study of a single-species population model with stage structure and strong Allee effect. By taking $τ$ as a bifurcation parameter, we study the Hopf bifurcation and global existence of periodic solutions using Wu's theory on global Hopf bifurcation for FDEs and the Bendixson criterion for higher dimensional ODEs proposed by Li and Muldowney. Some numerical simulations are presented to illustrate our analytic results using MATLAB and DDE-BIFTOOL. In addition, interesting phenomenon can be observed such as two kinds of bistability.

Global stability for a tuberculosis model with isolation and incomplete treatment

This paper deals with the global analysis of a dynamical model for the spread of tuberculosis with isolation and incomplete treatment. The model exhibits the traditional threshold behavior. We prove that when the basic reproductive number is less than unity, the disease-free equilibrium is globally asymptotically stable. When the basic reproductive number is greater than unity, the disease-free equilibrium is unstable and a unique endemic equilibrium exists which is locally asymptotically stable and globally asymptotically stable when the disease-induced death rate is equal to zero. The stability of disease-free equilibrium is derived by using Lyapunov stability theory and LaSalle’s invariant set theorem. The global stability of endemic equilibrium is proved by generalized Dulac–Bendixson criterion when the disease-induced death rate is equal to zero. Numerical simulations support our analytical results.

Stability and global Hopf bifurcation for neutral BAM neural network

In this paper, a simplified neutral BAM neural network model is considered. Some results of stability and Hopf bifurcations occurring at the zero equilibrium as the delay increases are exhibited. Global existence of periodic solutions is established by using a global Hopf bifurcation result of Wu and a Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney. Finally, some numerical simulations are carried out to support the analytical results.

Global Hopf Bifurcation Analysis of a Nicholson’s Blowflies Equation of Neutral Type

We investigate Hopf bifurcations in a delayed Nicholson’s blowflies equation of neutral type, derived from the Gurtin–MacCamy model. A key parameter that determines the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Global extension of local Hopf branches is established by combining a global Hopf bifurcation theorem with a Bendixson criterion for higher dimensional ordinary differential equations. We show that a branch of slowly varying periodic solutions and a branch of fast oscillating periodic solutions coexist for all large delays.

Global Stability Analysis and Optimal Control of a Harvested Ecoepidemiological Prey Predator Model with Vaccination and Taxation

We propose an ecoepidemiological prey predator model, where selective harvest effort on predator population is considered. Vaccination and taxation are introduced as control instruments, which are utilized to control number of susceptible prey population and protect predator population from overexploitation, respectively. Conditions which influence nonnegativity and boundedness of solutions are studied. Global stability analysis around disease-free equilibrium is discussed based on robust Bendixson criterion, which is theoretically beneficial to studying coexistence and interaction mechanism of population within harvested ecoepidemiological system. By using Pontryagin’s maximum principle, an optimal control strategy is derived to maximize the total discounted net economic revenue to society as well as protect prey population from infectious disease. Numerical simulations are carried out to show the consistency with theoretical analysis.

Global dynamics of the smallest chemical reaction system with Hopf bifurcation

The global behavior of solutions is described for the smallest chemical reaction system that exhibits a Hopf bifurcation, discovered in [12]. This three-dimensional system is a competitive system and a monotone cyclic feedback system. The Poincaré–Bendixson theory extends to such systems [2,3,6,8] and a Bendixson criterion exists to rule out periodic orbits [4].

Global existence of periodic solutions in a six-neuron BAM neural network model with discrete delays

In this paper, a six-neuron BAM neural network model with discrete delays is considered. Using the global Hopf bifurcation theorem for FDE due to Wu [Symmetric functional differential equations and neural networks with memory, Trans. Am. Math. Soc. 350 (1998) 4799–4838] and the Bendixson's criterion for high-dimensional ODE due to Li and Muldowney [On Bendixson' criterion, J. Differential Equations 106 (1994) 27–39], a set of sufficient conditions for the system to have multiple periodic solutions are derived when the sum of delays is sufficiently large.

GLOBAL EXISTENCE OF PERIODIC SOLUTIONS IN THE LINEARLY COUPLED MACKEY–GLASS SYSTEM

The dynamics of a linearly coupled Mackey–Glass system with delay are investigated. Based on the distribution of eigenvalues, we prove that a sequence of Hopf bifurcation occurs at the positive equilibrium as the delay increases and obtain the bifurcation set in the parameter plane. The explicit algorithm for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived, using the theories of normal form and center manifold. The global existence of periodic solutions is established using a global Hopf bifurcation result due to Wu [1998] and a Bendixson's criterion for higher dimensional ordinary differential equations due to [Li & Muldowney, 1993].

Bifurcation analysis in a neutral differential equation

The dynamics of a neural network model in neutral form is investigated. We prove that a sequence of Hopf bifurcations occurs at the origin as the delay increases. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using normal form method and center manifold theory. Global existence of periodic solutions is established using a global Hopf bifurcation result of Krawcewicz et al. and a Bendixson's criterion for higher dimensional ordinary differential equations due to Li and Muldowney.