Positive T-periodic Solutions Research Articles

Existence of positive periodic solutions for Liénard equation with a singularity of repulsive type

In this paper, the existence of positive periodic solutions is studied for Liénard equation with a singularity of repulsive type, x″(t)+f(x(t))x′(t)+φ(t)xμ(t)−1xγ(t)=e(t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ x''(t)+f(x(t))x'(t)+\\varphi (t)x^{\\mu}(t)-\\frac{1}{x^{\\gamma}(t)}=e(t), $$\\end{document} where f:(0,+∞)→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f:(0,+\\infty )\\rightarrow R$\\end{document} is continuous, which may have a singularity at the origin, the sign of φ(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\varphi (t)$\\end{document}, e(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$e(t)$\\end{document} is allowed to change, and μ, γ are positive constants. By using a continuation theorem, as well as the techniques of a priori estimates, we show that this equation has a positive T-periodic solution when μ∈[0,+∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mu \\in [0,+\\infty )$\\end{document}.

Bifurcation and dynamics of periodic solutions to the Rayleigh–Plesset equation: Theory and numerical simulation

In this paper, we study the oscillation of a gas-filled spherical bubble immersed in an infinite domain of incompressible liquid under the influence of a time-periodic acoustic field. The oscillation is described by the Rayleigh–Plesset equation, which is derived from the Navier–Stokes equation under the assumptions of spherical symmetry. Taking the initial radius R0, or the initial instantaneous partial pressure Pg0, of the gas-filled bubble as a parameter, we present a saddle–node bifurcation of periodic solutions to the Rayleigh–Plesset equation in the parameter space, provided the mean value of applied pressure is greater than the vapor pressure. Further, we show that the Rayleigh–Plesset equation has two classes of periodic solutions, as R0 (resp. Pg0) tends to +∞, the first class of solutions uniformly diverges to +∞ at the rate of R05 (resp. Pg05/2), while the second class converges to 0 at the rate of R0−5 (resp. Pg0−5/2). As for the case where the density of the liquid ρ is chosen as a parameter, we observe an interesting nonlinear phenomenon that, for some ρ, the Rayleigh–Plesset equation may possess four positive T-periodic solutions: two of them are stable, while the rest of them are unstable. This is quite different from the phenomenon that occurs when R0 (or Pg0) is chosen as a parameter.

On the positive periodic solutions of a class of Liénard equations with repulsive singularities in degenerate case

In this paper, we study the existence, multiplicity and dynamics of positive periodic solutions to a generalized Liénard equation with repulsive singularities. The Ambrosetti-Prodi type result is proved in the absence of the so-called anticoercivity condition. Furthermore, with s as a parameter, under some conditions on the function h, it has been shown that for any M>1 there exists sM∈R such that the equation x″+f(x)x′+h(t,x)=s has two positive T-periodic solutions u1(⋅;s) and u2(⋅;s) satisfying min⁡{u1(t;s):t∈[0,T]}>M and min⁡{u2(t;s):t∈[0,T]}<1/M for every s<sM. As a by-product of the property, we obtain sufficient conditions to guarantee the existence of positive T-periodic solutions of indefinite differential equations.

Ergodic Stationary Distribution and Threshold Dynamics of a Stochastic Nonautonomous SIAM Epidemic Model with Media Coverage and Markov Chain

A stochastic nonautonomous SIAM (Susceptible individual–Infected individual–Aware individual–Media coverage) epidemic model with Markov chain and nonlinear noise perturbations has been constructed, which is used to research the hybrid dynamic impacts of media coverage and Lévy jumps on infectious disease transmission. The uniform upper bound and lower bound of the positive solution are studied. Based on defining suitable random Lyapunov functions, we researched the existence of a nontrival positive T-periodic solution. Sufficient conditions are derived to discuss the exponential ergodicity based on verifying a Foster–Lyapunov condition. Furthermore, the persistence in the average sense and extinction of infectious disease are investigated using stochastic analysis techniques. Finally, numerical simulations are utilized to provide evidence for the dynamical properties of the stochastic nonautonomous SIAM.

Dynamical analysis and numerical simulations on a crowley-Martin predator-prey model in stochastic environment

This paper systematically investigates a stochastic Crowley-Martin predator-prey model. Firstly, we derive sufficient conditions for the existence of positive T-periodic solution for the impulsive case. Secondly, distinguished from the previous papers, asymptotical stability in probability is investigated by combining Khasminskii theory of stability with Lyapunov method. Our conclusion also improves and extends the corresponding existing ones. Thirdly, we establish the sufficient criteria for the existence of a unique ergodic stationary distribution for the Markovian switching case. Finally, two numerical examples are presented to demonstrate the effectiveness and feasibility of our analytical results and reveal the respective effect of white noises and Markovian switching.

Existence of Periodic Solutions of Seasonally Forced SEIR Models with Pulse Vaccination

In this paper, we are interested in finding the periodic oscillation of seasonally forced SEIR models with pulse vaccination. Many infectious diseases show seasonal patterns of incidence. Pulse vaccination strategy is an effective tool to control the spread of these infectious diseases. Assuming that the seasonally dependent transmission rate is a T-periodic forcing, we obtain the existence of positive T-periodic solutions of seasonally forced SEIR models with pulse vaccination by Mawhin’s coincidence degree method. Some relevant numerical simulations are presented to illustrate the effectiveness of such pulse vaccination strategy.

Positive periodic solutions to an indefinite Minkowski-curvature equation

We investigate the existence, non-existence, multiplicity of positive periodic solutions, both harmonic (i.e., T-periodic) and subharmonic (i.e., kT-periodic for some integer k≥2) to the equation(u′1−(u′)2)′+λa(t)g(u)=0, where λ>0 is a parameter, a(t) is a T-periodic sign-changing weight function and g:[0,+∞[→[0,+∞[ is a continuous function having superlinear growth at zero. In particular, we prove that for both g(u)=up, with p>1, and g(u)=up/(1+up−q), with 0≤q≤1<p, the equation has no positive T-periodic solutions for λ close to zero and two positive T-periodic solutions (a “small” one and a “large” one) for λ large enough. Moreover, in both cases the “small” T-periodic solution is surrounded by a family of positive subharmonic solutions with arbitrarily large minimal period. The proof of the existence of T-periodic solutions relies on a recent extension of Mawhin's coincidence degree theory for locally compact operators in product of Banach spaces, while subharmonic solutions are found by an application of the Poincaré–Birkhoff fixed point theorem, after a careful asymptotic analysis of the T-periodic solutions for λ→+∞.

Dynamical behavior of stochastic predator-prey models with distributed delay and general functional response

In this article, we study two stochastic predator-prey models with distributed delay and general functional response. For the nonautonomous periodic case of the system, by utilizing Khasminskii’s theory of periodic solution, we prove that the system admits a positive T-periodic solution. For the system which is perturbed by both white and colored noises, we establish sufficient conditions for the existence of positive recurrence of the solutions to the system. The existence of positive recurrence implies stochastic weak stability.

Nontrivial periodic solution of a stochastic seasonal rabies epidemic model

In this paper, we consider a stochastic nonautonomous rabies epidemic model. By using Khasminskii’s theory of periodic solutions and constructing a suitable stochastic Lyapunov function, we show that the stochastic nonautonomous epidemic model admits at least one nontrivial positive T-periodic solution under certain condition.

Periodic solutions and stationary distribution for a stochastic predator-prey system with impulsive perturbations

In this paper, we consider a stochastic predator-prey system with Beddington-DeAngelis functional response and impulsive perturbations. First, we prove that the model admits a unique global positive solution by constructing the equivalent system without impulsive perturbations. Second, we establish a sufficient condition which allows the existence of a positive periodic solution using stochastic Lyapunov functions. Finally, for the predator-prey system with Beddington-DeAngelis functional response disturbed by both white noise and telephone noise, we give the sufficient conditions for the stationary distribution which is ergodic and positive recurrent of the solution. The conclusion implies that the stochastic system has a positive T-periodic Markov process in a certain condition when the corresponding deterministic system has at least one positive T-periodic solution.

Dynamics analysis of stochastic epidemic models with standard incidence

In this paper, two stochastic SIRS epidemic models with standard incidence were proposed and investigated. For the non-autonomous periodic model, the sufficient criteria for extinction of the disease are obtained firstly. Then we show that the stochastic system has at least one nontrivial positive T-periodic solution under some conditions. For the model that are both disturbed by the white noise and telephone noise, we construct a suitable Lyapunov functions to verify the existence of a unique ergodic stationary distribution. Meanwhile, the sufficient condition for the extinction of the disease is also established. Finally, examples are introduced to illustrate the theoretical analysis.

Periodic Solution and Ergodic Stationary Distribution of Stochastic SIRI Epidemic Systems with Nonlinear Perturbations

This paper formulates two stochastic nonautonomous SIRI epidemic systems with nonlinear perturbations. The main aim of this study is to investigate stochastic dynamics of the two SIRI epidemic systems and obtain their thresholds. For the nonautonomous stochastic SIRI epidemic system with white noise, the authors provide analytic results regarding the stochastic boundedness, stochastic permanence and persistence in mean. Moreover, the authors prove that the system has at least one nontrivial positive T-periodic solution by using Lyapunov function and Hasminskii’s theory. For the system with Markov conversion, the authors establish sufficient conditions for positive recurrence and existence of ergodic stationary distribution. In addition, sufficient conditions for the extinction of disease are obtained. Finally, numerical simulations are introduced to illustrate the main results.

Asymptotic Periodicity in the Fecally-Orally Epidemic Model in a Heterogeneous Environment

To understand the influence of seasonal periodicity and environmental heterogeneity on the transmission dynamics of an infectious disease, we consider asymptotic periodicity in the fecally-orally epidemic model in a heterogeneous environment. By using the next generation operator and the related eigenvalue problems, the basic reproduction number is introduced and shows that it plays an important role in the existence and non-existence of a positive T-periodic solution. The sufficient conditions for the existence and non-existence of a positive T-periodic solution are provided by applying upper and lower solutions method. Our results showed that the fecally-orally epidemic model in a heterogeneous environment admits at least one positive T-periodic solution if the basic reproduction number is greater than one, while no T-periodic solution exists if the basic reproduction number is less than or equal to one. By means of monotone iterative schemes, we construct the true positive solutions. The asymptotic behavior of periodic solutions is presented. To illustrate our theoretical results, some numerical simulations are given. The paper ends with some conclusions and future considerations.

Periodic solutions for a class of conservative Liénard-type equations

In this paper, we study solvability of a class of second-order differential equations in a conservative Liénard form subject to periodic boundary conditions. Results on existence of non-trivial T-periodic solutions or positive T-periodic solutions are obtained respectively. Applications of the theorems are shown by examples. The results are proved by applying the coincidence degree theory for semilinear operator equations.

Existence and Stability of Periodic Solution Related to Valveless Pumping

Valveless pumping, also known as Liebau effect, can be described as the unidirectional flow of liquid in a system without valves that is caused by the asymmetry of placing of the periodically working pump. Recently, the research in this field has been reevoked, partially due to its possible application in nanotechnologies. In this paper, a configuration of one pipe and one tank is considered from the mathematical point of view. Qualitative properties of a class of nonlinear differential equations that model the assumed system configuration are investigated. New sufficient conditions for the existence of positive T-periodic solutions are given. Correspondingly, exponential stability of periodic solution is treated. Presented results are new. They extend and complement earlier ones in the literature.

Attractive singularity problems for superlinear Liénard equation

In this paper, we consider the following quasilinear Lienard equation with a singularity $$\begin{aligned} (\phi _p(x'(t)))'+f(x(t))x'(t)+g(t,x(t))=e(t), \end{aligned}$$where g has a attractive singularity at the origin and satisfies superlinear condition at $$x=+\infty $$. By using Manasevich–Mawhin continuous theorem, we prove that this equation has at least one positive T-periodic solution. We solve a difficulty to estimate it a priori bounds of a periodic solution for quasilinear Lienard equation in the case that superlinear condition. At last, example and numerical solution (phase portrait and time series portrait of the positive periodic solution of example) are given to show applications of the theorem.

Dynamical behavior of stochastic multigroup S-DI-A epidemic models for the transmission of HIV

In this paper, we study two stochastic multigroup S-DI-A epidemic models for the transmission of HIV. For the stochastic S-DI-A epidemic model with periodic coefficients, we first obtain sufficient conditions for persistence in the mean of the disease. Then in the case of persistence, we show that the model admits a positive T-periodic solution by using Khasminskii theory of periodic solution. Moreover, we establish sufficient conditions for exponential extinction of the infectious disease. For the stochastic S-DI-A epidemic model disturbed by both white and telegraph noises, we first establish sufficient conditions for persistence in the mean of the disease. Then in the case of persistence, we obtain sufficient conditions for the existence of a unique ergodic stationary distribution of the positive solutions by constructing a suitable stochastic Lyapunov function with regime switching and we also obtain sufficient conditions for exponential extinction of the system with regime switching.

A time-periodic dengue fever model in a heterogeneous environment

The transmission of dengue fever characterizes seasonality and periodicity, in particular, the infection of dengue fever is more serious during the warmer seasons. In this paper, we formulate and study an SIS–SI dengue model associated with the spatial heterogeneity and temporal periodicity. With the help of the spectral radius of next infection operator and eigenvalue problem, we introduce the basic reproduction number R0 of the dengue model. Furthermore, the existence and nonexistence of the positive T-periodic solution are obtained, respectively. The asymptotical stability of T-periodic solution is also investigated. Our analyses reveal that the combination of spatial heterogeneity and temporal periodicity would enhance the persistence of dengue virus in the case of R0>1. Some theoretical results are illustrated by the final numerical simulations and epidemiological explanations.

A research of pest management SI stochastic model concerning spraying pesticide and releasing natural enemies

In this paper, we study the dynamics of the stochastic SI epidemic model for pest management concerning spraying pesticide and releasing natural enemies. Existence of a unique global positive solution is proved firstly. And we show that the positive solution to the stochastic system is stochastically bounded. Third, by using Khasminshii's method and Lyapunov function, we derive the sufficient conditions for the existence of the nontrivial stochastically positive T-periodic solution. Then, by comparison theorem for stochastic differential equation, the sufficient conditions for existence and global attraction of the boundary periodic solution are obtained. Finally, Numerical simulations are carried out to substantiate the analytical results.

Periodic Solution and Stationary Distribution of Stochastic Predator–Prey Models with Higher-Order Perturbation

In this paper, two stochastic predator–prey models with general functional response and higher-order perturbation are proposed and investigated. For the nonautonomous periodic case of the system, by using Khasminskii’s theory of periodic solution, we show that the system admits a nontrivial positive T-periodic solution. For the system disturbed by both white and telegraph noises, sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution to the solutions are established. The existence of stationary distribution implies stochastic weak stability to some extent.