Uniqueness Of Periodic Solutions Research Articles

Existence and uniqueness of periodic solutions in neutral nonlinear summation-difference systems with infinite delay

We use Krasnoselskii’s fixed point theorem to show that the neutral nonlinear summation-difference system with infinite delay Δx(n)=P(n)+A(n)x(n−τ(n))+ΔQ(n,x(n−g(n)))+ ∑k=−∞nD(n,k)f(x(k)) has a periodic solution. We also use the contraction mapping principle to show that the periodic solution is unique. An example is given to illustrate our results.

Existence and uniqueness of periodic solutions in neutral nonlinear summation-difference systems with infinite delay

We use Krasnoselskii’s fixed point theorem to show that the neutral nonlinear summation-difference system with infinite delay Δx(n)=P(n)+A(n)x(n−τ(n))+ΔQ(n,x(n−g(n)))+ ∑k=−∞nD(n,k)f(x(k)) has a periodic solution. We also use the contraction mapping principle to show that the periodic solution is unique. An example is given to illustrate our results.

Stability, boundedness and existence of unique periodic solutions to a class of fourth order functional differential equations

In this paper a novel class of fourth order functional differential equations is discussed. By reducing the fourth order functional differential equation to system of first order, a suitable complete Lyapunov functional is constructed and employed to obtain sufficient conditions that guarantee existence of a unique periodic solution, asymptotic and uniform asymptotic stability of the zero solutions, uniform boundedness and uniform ultimate boundedness of solutions. The obtained results are new and include many prominent results in literature. Finally, two examples are given to show the feasibility and reliability of the theoretical results.

The Relativistic Harmonic Oscillator in a Uniform Gravitational Field

We present the relativistic generalization of the classical harmonic oscillator suspended within a uniform gravitational field measured by an observer in a laboratory in which the suspension point of the spring is fixed. The starting point of this analysis is a variational approach based on the Euler–Lagrange formalism. Due to the conceptual differences of mass in the framework of special relativity compared with the classical model, the correct treatment of the relativistic gravitational potential requires special attention. It is proved that the corresponding relativistic equation of motion has unique periodic solutions. Some approximate analytical results including the next-to-leading-order term in the non-relativistic limit are also examined. The discussion is rounded up with a numerical simulation of the full relativistic results in the case of a strong gravity field. Finally, the dynamics of the model is further explored by investigating phase space and its quantitative relativistic features.

Global dynamics of a stage-structured hantavirus infection model with seasonality

In this paper, we study a time-periodic model, which incorporates seasonality and host stage-structure. This model describes the propagation of Puumala hantavirus within the bank vole population of Clethrionomys glareolus. The basic reproduction number R0 is obtained. By appealing to the theory of monotone dynamical systems and chain transitive sets, we establish a threshold-type result on the global dynamics in terms of R0, that is, the virus-free periodic solution is globally attractive, and the virus dies out if R0 ≤ 1, while there exists a unique positive periodic solution, which is globally attractive, and the virus persists if R0 > 1. Numerical simulations are given to confirm our theoretical results and to show that cleaning environment and controlling the grow of mice population are essential control strategies to reduce hantavirus infection.

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.

Dynamics for a tritrophic impulsive periodic plankton–fish system with diffusion in lakes

Impulsive reaction–diffusion system is a novel assemblage of tools for better understanding ecological dynamics. This paper is devoted to mathematical and biological analysis of an impulsive tritrophic periodic plankton–fish system with diffusion and Beddington–DeAngelis functional response in lakes. On the basis of comparison theory and method of upper and lower solution, we have rigorously analyzed ultimate boundedness and permanence of the reaction–diffusion system under impulsive effects. Significantly, the existence and globally asymptotic stability of a unique positive periodic solution are verified by Brouwer's fixed point theorem and auxiliary function method. Additionally, two particular scenarios are chosen to illustrate our results. It is found that impulsive effects can influence lake ecological dynamics and regulate lake biological structure.

Quantifying the impact of insecticide resistance in the transmission dynamics of malaria

Abstract The fitness cost imposed by insecticide resistance on malaria vectors has appeared as an asset for vector control despite the evolution and spread of insecticide resistance. However, certain levels of insecticide resistance intensity may reduce the efficacy of insecticide-treated nets for certain coverage levels. In this study, we consider a non-autonomous mathematical model for malaria transmission that investigates the impact of the interaction between the intensity of resistance, the coverage of insecticide-treated nets and the fitness cost of resistance on resistant vectors (fitness disadvantage or decreased performance) on the transmission dynamics of malaria. It is shown that the disease-free periodic solution of the model is globally attractive if the time-dependent reproduction number ( R T ( t ) 1 ). The model also admits a unique positive periodic solution when R T ( t ) > 1 . We numerically studied the interaction between the fitness cost imposed by resistance on resistant vectors, the intensity of resistance and bed-net coverage. We observed that the intensity level present in the population may significantly affect the impact of the fitness cost that resistance imposes on resistant mosquitoes as well as reduce the effectiveness of control measures.

Impact of stochastic perturbation on the persistence and extinction risk of a multi-delayed prey–predator system in non-autonomous environment

Dynamical characteristics of a stochastically perturbed system are drastically different from those of a deterministic system. This study deals with the dynamics of a stochastically perturbed time-variant prey–predator model of aquatic animals hilsa (prey) and eel (predator). It is a well-known fact that overfishing and indiscriminate fishing of juvenile hilsa have led to a drastic fall in its population. To save the hilsa population from becoming extinct, we equip our model with age-structured growth and harvest in hilsa and study the effect of environmental noise on both prey and predator populations. We formulate predator’s consumption using Beddington–DeAngelis functional response which also provides access to some alternative food source in the absence of prey. We obtain sufficient conditions for persistence in the mean, extinction, and global attractivity and prove stochastic ultimate boundedness and the existence of a unique positive periodic solution. We have included many numerical examples to validate the analytical results. Through global sensitivity analysis using the partial rank correlation coefficient technique, we observed the environmental noise to be of great influence on the prey–predator population.

CHAOS IN A NONAUTONOMOUS MODEL FOR THE INTERACTIONS OF PREY AND PREDATOR WITH EFFECT OF WATER LEVEL FLUCTUATION

Water level regulates the dynamics of different populations residing in water bodies. The increase/decrease in the level of water leads to an increase/decrease in the volume of water, which influences the interactions of fishes and catching capability. We examine how seasonal variations in water level and harvesting affect the outcome of prey–predator interactions in an artificial lake. A seasonal variation of the water level is introduced in the predation rate. We derive conditions for the persistence and extinction of the populations. Using the continuation theorem, we determine the conditions for which the system has a positive periodic solution. The existence of a unique globally stable periodic solution is also presented. Moreover, we obtain conditions for the existence, uniqueness and stability of a positive almost periodic solution. We find that if the autonomous system has a stable focus, the corresponding nonautonomous system exhibits a unique stable positive periodic solution. But, whenever the autonomous system shows limit cycle oscillations, the corresponding nonautonomous system exhibits chaotic dynamics. The chaotic behavior of system is confirmed by the positivity of the maximal Lyapunov exponent. For higher values of the assimilation fraction of prey population, the persistent oscillations of the autonomous system are eliminated and this system becomes stable. On the other hand, chaotic nature of the nonautonomous system is converted into periodicity if the assimilation fraction of prey is large. Moreover, populations behave almost periodically if the seasonally varied rate parameters are almost periodic functions of time. Our findings show that water level plays an important role in the persistence of prey–predator system.

Existence and uniqueness of time periodic solutions to the compressible magneto-micropolar fluids in a periodic domain

We consider the existence and uniqueness of a time periodic solution for the compressible magneto-micropolar fluids with time periodic forces in a periodic domain. More precisely, under some smallness and symmetry assumptions on the external forces, we prove the existence of the periodic solution by a regularized approximation scheme and the topological degree theory. The uniqueness of the periodic solution is obtained by energy estimates.

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 and stability of a unique and exact two periodic orbits for an interactive wild and sterile mosquito model

In this paper, we study the existence and stability of a unique and exact two periodic orbits of an ordinary differential equation model for the interaction of wild and the released sterile mosquitoes, which consists of two sub-equations constantly switching each other. We first show that the model has at most two periodic solutions for all of the parameters and then surprisingly find that the model has a unique periodic solution if and only if the origin is unstable, and has exact two periodic solutions if and only if the origin is asymptotically stable provided the release amount does not exceed the release amount threshold value, respectively. We also give sufficient conditions for the existence of exact two periodic solutions and analyze the stability of each periodic solution.

Periodic solutions to certain classes of third order delay differential equations

In this paper, existence of unique periodic solutions for two classes of third order delay differential equations are considered. The two equations are expressed in their integral equivalents with which suitable Green’s function is constructed and its associated properties stated. The main tool adopted to establish the existence of unique periodic solutions to the classes of delay differential equations is the Krasnoselskii’s fixed point theorem due to its applicability to the combination of a compact and contraction mappings which occur in dealing with perturbed differential operators.

Periodic solutions and Hyers-Ulam stability of atmospheric Ekman flows

In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with constant eddy viscosity. Different from the well-known homogeneous system in [ 14 , 20 ], we retain the turbulent fluxes and establish a new nonhomogeneous system of first order differential equations involving a term with the horizontal dependent. We present the existence and uniqueness of periodic solutions and show the Hyers-Ulam stability results for the nonhomogeneous systems under the mild conditions via the matrix theory. Further, we consider the nonhomogeneous systems with varying eddy viscosity coefficient and study systems with piecewise constants, systems with small oscillations, systems with rapidly varying coefficients and systems with slowly varying coefficients and give more continued results.

Dynamics of an SIS network model with a periodic infection rate

Seasonal forcing and contact patterns are two key features of many disease dynamics that generate periodic patterns. Both features have not been ascertained deeply in the previous works. In this work, we develop and analyze a non-autonomous degree-based mean field network model within a Susceptible-Infected-Susceptible (SIS) framework. We assume that the disease transmission rate being periodic to study synergistic impacts of the periodic transmission and the heterogeneity of the contact network on the infection threshold and dynamics for seasonal diseases. We demonstrate both analytically and numerically that (1) the disease free equilibrium point is globally asymptotically stable if the basic reproduction number is less than one; and (2) there exists a unique global periodic solution that both susceptible and infected individuals coexist if the basic reproduction number is larger than one. We apply our framework to Scale-free contact networks for the simulation. Our results show that heterogeneity in the contact networks plays an important role in accelerating disease spreading and increasing the amplitude of the periodic steady state solution. These results confirm the need to address factors that create periodic patterns and contact patterns in seasonal disease when making policies to control an outbreak.

Asymptotic behavior of the principal eigenvalue for cooperative periodic-parabolic systems and applications

The effects of spatial heterogeneity on population dynamics have been studied extensively. However, the effects of temporal periodicity on the dynamics of general periodic-parabolic reaction-diffusion systems remain largely unexplored. As a first attempt to understand such effects, we analyze the asymptotic behavior of the principal eigenvalue for linear cooperative periodic-parabolic systems with small diffusion rates. As an application, we show that if a cooperative system of periodic ordinary differential equations has a unique positive periodic solution which is globally asymptotically stable, then the corresponding reaction-diffusion system with either the Neumann or regular oblique derivative boundary condition also has a unique positive periodic solution which is globally asymptotically stable, provided that the diffusion coefficients are sufficiently small. The role of temporal periodicity, spatial heterogeneity and their combined effects with diffusion will be studied in subsequent papers for further understanding and illustration.

Propagation Dynamics of a Periodic Epidemic Model on Weighted Interconnected Networks

Many real-world networks are comprised of several networks interconnected with each other and they may have different topologies and epidemic dynamics. The dynamical behaviors of various epidemic models on coupled networks have attracted great attentions. For instance, many people adhere to the statutory working and rest days, which results in a periodic fluctuation of disease transmission in the population, and the strength of connections between two individuals cannot be ignored either. In this paper, we explore the disease transmission on weighted interconnected networks by establishing a model that includes contact strengths and periodic incidence rates. The weights on the links indicate the familiarity or intimacy of the interactive individuals. Here, we analyze the stability of the disease-free periodic solution and the unique positive periodic solution (i.e., disease-free equilibrium and endemic equilibrium) of the model, and an explicit expression for the basic reproduction number in some special cases of the periodic model is derived. We also perform numerical simulations to validate and supplement the theoretical results. It is found that the weight exponent promotes the epidemic transmission by enlarging the basic reproduction number, and the influence of internal infectious rate on epidemic prevalence is larger than that of cross infectious rate for different network structures. It is expected that this work can deepen the understanding of transmission dynamics on weighted interconnected networks.

Dynamical analysis on traveling wave of a reaction–diffusion model

In this paper, a class of nonlinear reaction–diffusion equations is studied, and Abelian integral method is applied to show the existence of a unique periodic traveling wave solution. Simulation is presented to verify the theoretical prediction.

Global attractivity of a unique positive periodic solution for a first-order nonlinear difference equation with time delays

The present paper is directed toward the study on the global attractivity of a unique positive periodic solution of a discrete hematopoiesis model with unimodal production functions and several time delays. This model is described by a nonlinear difference equation. The result obtained is proved by transforming this model into another difference equation and by using the Schauder fixed-point theorem.