Nontrivial Periodic Solutions Research Articles

Periodic solutions of differential equations with periodic constraints

We study the harmonic solutions of periodically perturbed differential equations subjected to a possibly time‐dependent constraint. We obtain a degree theoretic condition ensuring a “global branching” result for the nontrivial periodic solutions. The argument stems from a combination of techniques from the theory of topological degree and differential‐algebraic equations. As an application, the set of harmonic solutions of periodically perturbed implicit differential equations is investigated.

On the periodic behavior of the generalized Chazy differential equation.

We consider the periodic behavior of the generalized Chazy differential equation x⃛+|x|qx¨+k|x|qxx˙2=0, where q is a positive integer and k is a real number. We give a pure analysis on the existence of non-trivial periodic solutions for k=q+1 and the non-existence of them for k≠q+1. Our method is based on considering the projections of the orbits onto the phase plane (x,x˙). We find that a non-trivial periodic solution of the equation is equivalent to a closed curve formed by two equilibrium points and two orbits with some specific constraints in the corresponding planar system and that the existence of such closed curves can be obtained by the existence of real zeros of some returning map. Our conclusion covers all q, which completes a recent result.

Dynamics of a ratio-dependent three-species food-chain system in a random environment

We consider a system of stochastic differential equations, which is established by introducing white noises into a periodic and ratio-dependent food-chain system of three-species. We investigate the asymptotic behaviors for such a system, and obtain the sufficient conditions for its extinction and persistence with the help of Itô’s formula. We discuss the existence of a nontrivial positive periodic solution based on Has’minskii theory of periodic solution. Further, we study a ratio-dependent food-chain system of three-species including not only white noise but also telephone noise. For such a model, we give the sufficient conditions to guarantee the existence of a unique ergodic stationary distribution.

Time periodic traveling wave solutions of a time-periodic reaction–diffusion SEIR epidemic model with periodic recruitment

This paper focuses on the existence and nonexistence of a time periodic traveling wave solution of a time-periodic reaction–diffusion SEIR epidemic model. The main feature of the model is the possible deficiency of the classical comparison principle such that many known results do not directly work. If the basic reproduction number of the model, denoted by R0, is larger than one, there exists a minimal wave speed c∗>0 satisfying for each c>c∗, the system admits a nontrivial time periodic traveling wave solution with wave speed c and for c<c∗, there exists no nontrivial time periodic traveling waves such that the system; if R0<1, the system admits no nontrivial time periodic traveling waves.

Mathematical model of mosquito population dynamics with constants and periodic releases of Wolbachia-infected males

In this article, we propose three mathematical models of mosquito dynamics. The first model describes the dynamics of wild mosquitoes in combination with mechanical control and larvicide. For this model, we derive a threshold parameter N , called the basic number of offspring, and show that the trivial equilibrium is globally asymptotically stable when N ≤ 1 , while if N > 1 , the non-trivial equilibrium is globally asymptotically stable. In the second model, we evaluated a constant release of Wolbachia-infected males in the wild mosquito population. Our analysis identified a critical threshold for releasing Wolbachia-infected male mosquitoes M i crit to reduce or eliminate the wild mosquito population. In the third model, we assessed wild mosquito dynamics by periodically releasing Wolbachia-infected male mosquitoes. The analysis of this model produced two thresholds, T ∗ the threshold on the release period and Λ ∗ the threshold on the release quantity. The existence of trivial periodic solutions is obtained, and the corresponding local and global stability conditions are proved by Floquet's theory and Lyapunov's stability theorem, respectively. We also prove the existence conditions for non-trivial periodic solutions and their local stability. Finally, we perform some numerical simulations to illustrate our theoretically pertinent results.

Parameter estimation and global sensitivity analysis of a bacterial-plasmid model with impulsive drug treatment

In this paper, a mathematical model is developed to study bacterial resistance caused by plasmids and mutations in the presence of the impulsive drug treatment. Firstly, the stability of the bacterial-plasmid extinction periodic solution is proved by Floquet theory and the permanence of system is presented while some conditions are satisfied. Secondly, the nontrivial periodic solution bifurcated from the bacterial-plasmid extinction periodic solution is obtained in view of the four-dimensional impulsive control of bifurcation method. Good fitting results are given by the Mean Absolute Percentage Error method for four different cases of bacterial media. Further, the global sensitivity of R0 is analyzed by the Partial Rank Correlation Coefficient method, and the parameters that have important influence on the system are determined. Finally, numerical simulations are performed to validate the accuracy of the theoretical findings.

Non-Trivial Periodic Solutions for a Class of Second Order Differential Equations with Large Delay

Non-Trivial Periodic Solutions for a Class of Second Order Differential Equations with Large Delay

Periodic solutions of seasonal epidemiological models with information‐dependent vaccination

In this paper, we study nontrivial periodic solutions for seasonal epidemiological models with information‐dependent vaccination. Using Leray–Schauder degree theory, we establish new results concerning the existence of at least one nontrivial periodic solution for the and models with seasonality, state‐dependent vaccination coverage and delayed information. Some numerical simulations are presented to illustrate the effect of such vaccination coverage.

On a bi-stability regime and the existence of odd subharmonics in a Comb-drive MEMS model with cubic stiffness

In this paper we study two different problems. First we present a novel result about the existence of a family of odd subharmonics with prescribed nodal properties for a general nonlinear oscillator with bounded domain and symmetries. Then we apply the general existence result to the Comb-drive finger MEMS model with a cubic nonlinear stiffness term, and prove analytically that the odd positive subharmonic of order two is linearly stable whenever the AC load of the input voltage is small enough. Moreover, we demonstrate a bi-stability regime for this model because the trivial solution x≡0 is also linearly stable. The general existence result was obtained through a generalization of the Ortega’s variational principle (Ortega, 2016), and the stability assertions were obtained by following the perturbation approach in Cen et al. (2020) for the linear stability of a nontrivial periodic solution that emanates from the autonomous problem, and the well-known Zukovskii criterion.

Oscillatory solutions in a 3D predation system with time delay and age structure

This paper is concerned with the sustained periodic oscillation phenomenon in a three-species delayed predation system with Holling type II functional response and age structure in top predator. The top predator fertility function is regarded as a piecewise continuous smooth function with regard to their maturation period [Formula: see text]. The complicated dynamic behavior is proved by integrated semigroup theory and Hopf bifurcation theorem for semilinear equations with non-dense domain. Through qualitative analysis and bifurcation study of the system, we yield that this system has a nontrivial periodic solution that bifurcates from the positive equilibrium age distribution when bifurcation parameter [Formula: see text] passes through some critical values. Numerical simulations are also provided to illustrate theoretically analytical results.

Influences of the Order of Derivative on the Dynamical Behavior of Fractional-Order Antisymmetric Lotka–Volterra Systems

This paper studies the dynamic behavior of a class of fractional-order antisymmetric Lotka–Volterra systems. The influences of the order of derivative on the boundedness and stability are characterized by analyzing the first-order and 0&lt;α&lt;1-order antisymmetric Lotka–Volterra systems separately. We show that the order does not affect the boundedness but affects the stability. All solutions of the first-order system are periodic, while the 0&lt;α&lt;1-order system has no non-trivial periodic solution. Furthermore, the 0&lt;α&lt;1-order system can be reduced on a two-dimensional space and the reduced system is asymptotically stable, regardless of how close to zero the order of the derivative used is. Some numerical simulations are presented to better verify the theoretical analysis.

Time periodic solutions of Hamilton-Jacobi equations with autonomous Hamiltonian on the circle

We are concerned with the existence and multiplicity of nontrivial time periodic viscosity solutions to∂tw(x,t)+H(x,∂xw(x,t),w(x,t))=0,(x,t)∈S×[0,+∞), where S is the unit circle and H=H(x,p,u) satisfies Tonelli conditions with respect to the argument p and is strictly decreasing in the argument u. We also study the long time behavior of viscosity solutions of the Cauchy problem for the above contact Hamilton-Jacobi equation. It is shown that for a class of initial data the corresponding viscosity solutions converge to asymptotic time periodic viscosity solutions. As an application of the existence result we analyze a bifurcation phenomenon for ∂tw(x,t)+H(x,∂xw(x,t),λw(x,t))=0 where λ is a parameter.

Non-smooth dynamics of a SIR model with nonlinear state-dependent impulsive control.

The classic SIR model is often used to evaluate the effectiveness of controlling infectious diseases. Moreover, when adopting strategies such as isolation and vaccination based on changes in the size of susceptible populations and other states, it is necessary to develop a non-smooth SIR infectious disease model. To do this, we first add a non-linear term to the classical SIR model to describe the impact of limited medical resources or treatment capacity on infectious disease transmission, and then involve the state-dependent impulsive feedback control, which is determined by the convex combinations of the size of the susceptible population and its growth rates, into the model. Further, the analytical methods have been developed to address the existence of non-trivial periodic solutions, the existence and stability of a disease-free periodic solution (DFPS) and its bifurcation. Based on the properties of the established Poincaré map, we conclude that DFPS exists, which is stable under certain conditions. In particular, we show that the non-trivial order-1 periodic solutions may exist and a non-trivial order-$ k $ ($ k\geq 1 $) periodic solution in some special cases may not exist. Moreover, the transcritical bifurcations around the DFPS with respect to the parameters $ p $ and $ AT $ have been investigated by employing the bifurcation theorems of discrete maps.

Multiple nontrivial periodic solutions to a second-order partial difference equation

&lt;abstract&gt;&lt;p&gt;In this article, applying variational technique as well as critical point theory, we establish a series of criteria to ensure the existence and multiplicity of nontrivial periodic solutions to a second-order nonlinear partial difference equation. Our results generalize some known results. Moreover, numerical stimulations are presented to illustrate applications of our major findings.&lt;/p&gt;&lt;/abstract&gt;

Dynamical analysis of a stochastic non-autonomous SVIR model with multiple stages of vaccination.

In this paper, we analyze the dynamics of a new proposed stochastic non-autonomous SVIR model, with an emphasis on multiple stages of vaccination, due to the vaccine ineffectiveness. The parameters of the model are allowed to depend on time, to incorporate the seasonal variation. Furthermore, the vaccinated population is divided into three subpopulations, each one representing a different stage. For the proposed model, we prove the mathematical and biological well-posedness. That is, the existence of a unique global almost surely positive solution. Moreover, we establish conditions under which the disease vanishes or persists. Furthermore, based on stochastic stability theory and by constructing a suitable new Lyapunov function, we provide a condition under which the model admits a non-trivial periodic solution. The established theoretical results along with the performed numerical simulations exhibit the effect of the different stages of vaccination along with the stochastic Gaussian noise on the dynamics of the studied population.

Impacts of demographic and environmental stochasticity on population dynamics with cooperative effects

Different types of stochasticity play essential roles in shaping complex population dynamics. This paper presents a novel approach to model demographic and environmental stochasticity in a single-species model with cooperative components that are measured by component Allee effects. Our work provides rigorous mathematical proof on stochastic persistence and extinction, ergodicity (i.e., the existence of a unique stationary distribution) and the existence of a nontrivial periodic solution to study the impacts of demographic and environmental stochasticity on population dynamics. The theoretical and numerical results suggest that stochasticity may affect the population system in a variety of ways, specifically: (i) In the weak Allee effects case (e.g., strong cooperative efforts), the demographic stochasticity from the attack rate contributes to the expansion of the population size, while the demographic stochasticity from the handling rate and the environmental stochasticity have the opposite role, and may even lead to population extinction; (ii) In the strong Allee effects case (cooperative efforts not strong enough), both demographic and environmental stochasticity play a similar role in the survival of population, and are related to the initial population level: if the initial population level is large enough, demographic stochasticity and environmental stochasticity may be detrimental to the survival of population, otherwise if the initial population level is small enough, demographic stochasticity and environmental stochasticity may bring survival opportunities for the population that deterministically would extinct indefinitely; (iii) In the extinction case, demographic and environmental stochasticity cannot change the trend of population extinction, but they can delay or promote population extinction.

Recurrent epidemic waves in a delayed epidemic model with quarantine

ABSTRACT In this paper, we are concerned with an epidemic model with quarantine and distributed time delay. We define the basic reproduction number and show that if , then the disease-free equilibrium is globally asymptotically stable, whereas if , then it is unstable and there exists a unique endemic equilibrium. We obtain sufficient conditions for a Hopf bifurcation that induces a nontrivial periodic solution which represents recurrent epidemic waves. By numerical simulations, we illustrate stability and instability parameter regions. Our results suggest that the quarantine and time delay play important roles in the occurrence of recurrent epidemic waves.

Rational Periodic Solutions on Some Generalized Abel Equations

In this paper we deal with the equations a(x)dy/dx = A(x)y 2 + B(x)y 3, where a(x), A(x) and B(x) are complex polynomials with a(x)B(x) ≢ 0 and a(x) non-constant. First we show that the unique rational limit cycles that these equations can have are of the form y = 1/p(x) being p(x) some polynomial. Second we provide an upper bound on the number of these rational limit cycles. Moreover, we prove that if deg(B(x)) − deg(a(x)) + 1 is odd, or deg(A) > (deg(B(x)) + deg(a(x)) − 1)/2, then these Abel equations have at most two rational limit cycles and we provide examples of these Abel equations with three nontrivial rational periodic solutions.

Dynamics analysis of stage-structured wild and sterile mosquito interaction impulsive model

In this paper, we study a stage-structured wild and sterile mosquito interaction impulsive model. The aim is to study the feasibility of controlling the population of wild mosquitoes by releasing sterile mosquitoes periodically. The existence of trivial periodic solutions is obtained, and the corresponding local stability and global stability conditions are proved by Floquet theory and Lyapunov stability theorem, respectively. And we prove the existence conditions of non-trivial periodic solutions and their local stability. We can find that the system has the bistable phenomenon in which the trivial periodic solution and the non-trivial periodic solution can coexist under certain threshold conditions. All the results show that the appropriate release period and release amount of sterile mosquitoes can control the wild mosquito population within a certain range and even make them extinct. Finally, numerical simulation verifies our theoretical results.

REGIME SHIFTS BETWEEN OSCILLATORY PERSISTENCE AND EXTINCTION IN A STOCHASTIC CHEMOSTAT MODEL WITH PERIODIC PARAMETERS

In this paper, we mechanistically formulate a type of stochastic chemostat model with two complementary nutrients, which is affected by seasonal variations and flocculation effect. The phase transition properties of the model are investigated by theoretical analysis and numerical simulation. The well-posedness of the model is considered. Further, by utilizing Khasminskii’s theory, sufficient conditions for the existence of the stochastic nontrivial positive periodic solution are obtained. The existence of the stochastic nontrivial positive periodic solution implies periodic change of microorganism’s density. Some sufficient conditions for the global attractivity of the boundary periodic solution of the model are also derived. At last, numerical simulations are performed to illustrate our theoretical results. It is found numerically that the stable positive periodic solution and a stable boundary periodic solution of the model may coexist. Especially, for appropriate random perturbations, the population of the microorganisms changes from an endangered state to an oscillatory persistence state in some regions.