Nontrivial Periodic Solutions Research Articles

Threshold dynamics of an SEAIR epidemic model with application to COVID-19

In this paper, a Susceptible-Exposed-Asymptomatic-Infectious-Recovered (SEAIR) epidemic model with application to COVID-19 is established by capturing the key features of the disease. The global dynamics of the model is analyzed by constructing appropriate Lyapunov functions utilizing the basic reproduction number \(R_0\) as an index. We obtain that when \(R_{0}<1\), the disease-free equilibrium is globally asymptotically stable. While for \(R_{0}>1\), the endemic equilibrium is globally asymptotically stable. Furthermore, we consider the pulse vaccination for the disease and give an impulsive differential equations model. The definition of the basic reproduction number \(R_{0}\) of this system is given by utilizing the next generation operator. By the comparison theorem and persistent theory, we obtain that when \(R_{0}<1\)}, the disease-free periodic solution is globally asymptotically stable. Otherwise, the disease will persist and there will be at least one nontrivial periodic solution. Numerical simulations to verify our conclusions are given at the end of each of these theorems.

Multiple periodic solutions for an asymptotically linear wave equation withx-dependent coefficients

In this paper, we investigate the existence of multiple nontrivial periodic solutions for an asymptotically linear wave equation with x-dependent coefficients. Such a mathematical model can be derived from the forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media. By constructing a suitable function space, we characterize the problem as a variational problem, and then we prove that there are at least three nontrivial periodic solutions via variational methods.

Existence and Uniqueness of Nontrivial Periodic Solutions to a Discrete Switching Model

To control the spread of mosquito-borne diseases, one goal of the World Mosquito Program’s Wolbachia release method is to replace wild vector mosquitoes with Wolbachia-infected ones, whose capability of transmitting diseases has been greatly reduced owing to the Wolbachia infection. In this paper, we propose a discrete switching model which characterizes a release strategy including an impulsive and periodic release, where Wolbachia-infected males are released with the release ratio α1 during the first N generations, and the release ratio is α2 from the (N+1)-th generation to the T-th generation. Sufficient conditions on the release ratios α1 and α2 are obtained to guarantee the existence and uniqueness of nontrivial periodic solutions to the discrete switching model. We aim to provide new methods to count the exact numbers of periodic solutions to discrete switching models.

Periodic Solutions with Prescribed Minimal Period for 2 n th-Order Nonlinear Discrete Systems

In this paper, by using the critical point theory, some new results of the existence of at least two nontrivial periodic solutions with prescribed minimal period to a class of 2 n th-order nonlinear discrete system are obtained. The main approach used in our paper is variational technique and the linking theorem. The problem is to solve the existence of periodic solutions with prescribed minimal period of 2 n th-order discrete systems.

A mathematical analysis of Hopf-bifurcation in a prey-predator model with nonlinear functional response

In this paper, our aim is mathematical analysis and numerical simulation of a prey-predator model to describe the effect of predation between prey and predator with nonlinear functional response. First, we develop results concerning the boundedness, the existence and uniqueness of the solution. Furthermore, the Lyapunov principle and the Routh–Hurwitz criterion are applied to study respectively the local and global stability results. We also establish the Hopf-bifurcation to show the existence of a branch of nontrivial periodic solutions. Finally, numerical simulations have been accomplished to validate our analytical findings.

Sobolev hyperbola for periodic Lane–Emden heat flow system in N spatial dimension

The well-known Lane–Emden conjecture indicates that, for the elliptic Lane–Emden system −Δu=vp, −Δv=uq in RN, the Sobolev hyperbola 1p+1+1q+1=N−2N is expected as the critical curve for the existence and nonexistence of entire solutions. In this paper, we study the periodic Lane–Emden heat flow system ut−Δu=a(t)vp, vt−Δv=b(t)uq in a bounded domain Ω of RN, subject to homogeneous Dirichlet boundary condition. We will show that the Sobolev hyperbola is also a critical curve for the existence and nonexistence of periodic solutions. Moreover, if pq=1, the nontrivial periodic solutions may exist or not exist.

Analysis of a periodic stochastic three‐species Lotka‐Volterra model with distributed delay and dispersal

In this paper, a periodic stochastic Lotka‐Volterra model with distributed delay and dispersal is investigated. First, by Itô's formula, we show that the solution with any positive initial value is global and positive. Then, we obtain the sufficient conditions guaranteeing the extinction of the system. Furthermore, we discuss the existence of the nontrivial periodic solution by Khasminskii theory. Finally, numerical examples are given to illustrate the results.

EXISTENCE OF PERIODICSOLUTIONS OF A NONLINEAR ALLEN-CAHN EQUATION WITH NEUMANN CONDITION

This research investigated the periodic solutions to a nonlinear diffusion Allen-Cahn equation with isolated Neumann problem. The theory of Leray-Schauder fixed point degree was adopted and the study showed that there exist nontrivial periodic solutions to the nonlinear diffusion Allen-Cahn equation with isolated Neumann problem by the topological degree theory.

Analysis of Periodic Solution of DNA Catalytic Reaction Model With Random Disturbance

The realization of molecular logic circuit is inseparable from the design and analysis of catalytic reaction chain, and the DNA catalytic gate plays an important role in it. Discuss the nature of the solution to DNA catalytic reaction system, using Khasminskii's periodicity and Lyapunov analysis methods to obtain the existence of non-trivial positive periodic solutions of the system, and the solution is globally attractive. The existence of the solution indicates that according to the mathematical model established by the DNA catalytic reaction system, the system may reach the expected concentration value of an ideal state and obtain better reaction data, which provides a theoretical basis for the realization of the DNA catalytic gate function. Numerical simulation results show that under the influence of random disturbance and periodic parameters, the solution to the random DNA catalytic reaction system exists and is globally attractive, which also reflects that the DNA catalytic reaction system can reach an ideal reaction state. The solution to the DNA catalytic system with random disturbance will converge on a certain value and oscillate periodically between the solution to the deterministic system.

Extinction or coexistence in periodic Kolmogorov systems of competitive type

&lt;p style='text-indent:20px;'&gt;We study a periodic Kolmogorov system describing two species nonlinear competition. We discuss coexistence and extinction of one or both species, and describe the domain of attraction of nontrivial periodic solutions in the axes, under conditions that generalise Gopalsamy conditions. Finally, we apply our results to a model of microbial growth and to a model of phytoplankton competition under the effect of toxins.&lt;/p&gt;

Periodic solutions of second order Hamiltonian systems with nonlinearity of general linear growth

In this paper we consider a class of second order Hamiltonian system with the nonlinearity of linear growth. Compared with the existing results, we do not assume an asymptotic of the nonlinearity at infinity to exist. Moreover, we allow the system to be resonant at zero. Under some general conditions, we will establish the existence and multiplicity of nontrivial periodic solutions by using the Morse theory and two critical point theorems.

Dynamics and Optimal Control of a Monod–Haldane Predator–Prey System with Mixed Harvesting

This paper investigates the dynamics and optimal control of the Monod–Haldane predator–prey system with mixed harvesting that combines both continuous and impulsive harvestings. The periodic solution of the prey-free is studied and the local stability condition is obtained. The boundedness of solutions, the permanence of the system, and the existence of nontrivial periodic solution are studied. With the change of parameters, the system appears with a stable nontrivial periodic solution when the prey-free periodic solution loses stability. Numerical simulations show that the system has complex dynamical behaviors via bifurcation diagrams. Further, the maximum yield problem of the harvested system is studied, which is transformed into a nonlinear programming problem and solved by the method of combined multiple shooting and collocation.

Green's function for impulsive periodic solutions in alternately advanced and delayed differential systems and applications

In this paper we investigate the existence of the periodic solutions of a nonlinear impulsive differential system with piecewise alternately advanced and retarded arguments, in short IDEPCAG, that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such system are obtained. In the process we use the Green's function for impulsive periodic solutions and convert the given the IDEPCAG into an equivalent integral equation system. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this type of nonlinear impulsive differential systems. We also use the contraction mapping principle to show the existence of a unique impulsive periodic solution. Appropriate examples are given to show the feasibility of our results.

On dynamics of an HIV pathogenesis model with full logistic target cell growth and cure rate

The objective of this paper is to discuss the dynamics of an HIV pathogenesis model with full logistic target cell growth of uninfected T cells and cure rate of infected T cells. Local and global dynamics of both infection-free and infected equilibrium points are rigorously established. It is found that if basic reproduction number R 0 ≤1, the infection is cleared from T cells and if R 0 >1, the HIV infection persists. Also, we have carried out numerical simulations to verify the results. The existence of non-trivial periodic solution is also studied by means of numerical simulation. Therefore, we find a parameter region where infected equilibrium point is globally stable to make the model biologically significant. From the overall study, it is found that proliferation of T cells cannot be ignored during the study of HIV dynamics for better results and we can focus on a treatment policy which can control the parameters of the model in such a way that the basic reproduction number remains less than or equal to one.

Green’s Function for Periodic Solutions in Alternately Advanced and Delayed Differential Systems

In this paper we investigate the existence of the periodic solutions of a nonlinear differential equation with a general piecewise constant argument, in short DEPCAG, that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use the Green’s function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii’s fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.

Existence of Two Periodic Solutions to General Anisotropic Euler-Lagrange Equations

This paper is concerned with the following Euler-Lagrange system \[ \begin{cases} \frac{d}{dt} \mathcal{L}_v(t,u(t),\dot{u}(t)) = \mathcal{L}_x(t,u(t),\dot{u}(t)) \quad \textrm{for a.e. $t \in [-T,T]$}, \\ u(-T) = u(T), \\ \mathcal{L}_v(-T,u(-T),\dot{u}(-T)) = \mathcal{L}_v(T,u(T),\dot{u}(T)), \end{cases} \] where Lagrangian is given by $\mathcal{L} = F(t,x,v) + V(t,x) + \langle f(t), x \rangle$, growth conditions are determined by an anisotropic G-function and some geometric conditions at infinity. We consider two cases: with and without forcing term $f$. Using a general version of the mountain pass theorem and Ekeland's variational principle we prove the existence of at least two nontrivial periodic solutions in an anisotropic Orlicz-Sobolev space.

Complex dynamics and coexistence of period-doubling and period-halving bifurcations in an integrated pest management model with nonlinear impulsive control

An expectation for optimal integrated pest management is that the instantaneous numbers of natural enemies released should depend on the densities of both pest and natural enemy in the field. For this, a generalised predator–prey model with nonlinear impulsive control tactics is proposed and its dynamics is investigated. The threshold conditions for the global stability of the pest-free periodic solution are obtained based on the Floquet theorem and analytic methods. Also, the sufficient conditions for permanence are given. Additionally, the problem of finding a nontrivial periodic solution is confirmed by showing the existence of a nontrivial fixed point of the model’s stroboscopic map determined by a time snapshot equal to the common impulsive period. In order to address the effects of nonlinear pulse control on the dynamics and success of pest control, a predator–prey model incorporating the Holling type II functional response function as an example is investigated. Finally, numerical simulations show that the proposed model has very complex dynamical behaviour, including period-doubling bifurcation, chaotic solutions, chaos crisis, period-halving bifurcations and periodic windows. Moreover, there exists an interesting phenomenon whereby period-doubling bifurcation and period-halving bifurcation always coexist when nonlinear impulsive controls are adopted, which makes the dynamical behaviour of the model more complicated, resulting in difficulties when designing successful pest control strategies.

Dynamics for a stochastic delayed SIRS epidemic model

In this paper, we consider a stochastic delayed SIRS epidemic model with seasonal variation. Firstly, we prove that the system is mathematically and biologically well-posed by showing the global existence, positivity and stochastically ultimate boundneness of the solution. Secondly, some sufficient conditions on the permanence and extinction of the positive solutions with probability one are presented. Thirdly, we show that the solution of the system is asymptotical around of the disease-free periodic solution and the intensity of the oscillation depends of the intensity of the noise. Lastly, the existence of stochastic nontrivial periodic solution for the system is obtained.

Periodic oscillations in HIV transmission model with intracellular time delay and infection-age structure

We propose and analyze a virus-to-cell and cell-to-cell infections HIV model that includes intracellular time delay and infection-age structure, where infection-age-specific conversion function is considered as the piecewise function related to infection period between initial infection and the formation of productively infected cells. Applying the theory of integrated semigroup and the Hopf bifurcation theory for semilinear equations with non-dense domain, we concern the Hopf bifurcation of the model. A non-trivial periodic solution bifurcates from the positive equilibrium, when bifurcation parameter passes through some critical values. The numerical results verify our theoretical conclusion and further demonstrate that intracellular time delay has a greater impact on the model, but the impact of infection period on the model is hardly obvious. Additionally, the sensitivity of the threshold parameter is also carried out by employing the Latin hyper-cube sampling (LHS) method and partial rank correlation coefficient (PRCC) techniques.

Existence and multiplicity of periodic solutions for damped vibration problems with nonlinearity of linear growth

In this paper, we study a class of damped vibration problems with nonlinearity of linear growth. We obtain the existence and multiplicity of nontrivial periodic solutions by using the Morse theory and a critical point theorem. Compared with the existing work, our results do not require the system to be asymptotically linear at infinity.