On small perturbations of a nonlinear periodic system with degenerate equilibrium

This paper considers the reducibility and existence of periodic solutions for a class of nonlinear periodic system with a degenerate equilibrium point under small perturbations. By introducing some parameter, we consider an equivalent periodic system. Then we prove that by an affine linear periodic transformation the parameterized periodic system is reducible to one with zero as an equilibrium. Topological degree theorem ensures that for some parameter the result can go back to the original system. Then, we obtain a small periodic solution.     

Periodic solution and almost periodic solution for a nonautonomous Lotka-Volterra dispersal system with infinite delay

This paper studies a nonautonomous Lotka-Volterra dispersal systems with infinite time delay which models the diffusion of a single species into n patches by discrete dispersal. Our results show that the system is uniformly persistent under an appropriate condition. The sufficient condition for the global asymptotical stability of the system is also given. By using Mawhin continuation theorem of coincidence degree, we prove that the periodic system has at least one positive periodic solution, further, obtain the uniqueness and globally asymptotical stability for periodic system. By using functional hull theory and directly analyzing the right functional of almost periodic system, we show that the almost periodic system has a unique globally asymptotical stable positive almost periodic solution. We also show that the delays have very important effects on the dynamic behaviors of the system.     

Periodic and Almost Periodic Solutions for a Coupled System of Two Korteweg-de Vries Equations with Boundary Forces

In this paper, we investigate a coupled system of two Korteweg-de Vries equations on a bounded domain. We discuss the long-time behavior of this system with forces on the left Dirichlet boundary conditions. We obtain that if the forces are periodic (almost periodic) with small amplitude, then the solution of the coupled system is periodic (almost periodic).     

离散Leslie捕食与被捕食系统周期解的稳定性

该文讨论了一类离散Leslie捕食与被捕食系统,获得了该系统的持久性,当系统为周期系统时,得到了它的周期解的存在性,并且在某些条件下,该周期解是全局稳定的.     

Periodic and almost periodic solution for a non-autonomous epidemic predator-prey system with time-delay

In this paper, we studied a non-autonomous predator-prey system with discrete time-delay, where there is epidemic disease in the predator. By using some techniques of the differential inequalities and delay differential inequalities, we proved that the system is permanent under some appropriate conditions. When all the coefficients of the system is periodic, we obtained the existence and global attractivity of the positive periodic solution by Mawhin’s continuation theorem and constructing a suitable Lyapunov functional. Furthermore, when the coefficients of the system are not absolutely periodic but almost periodic, sufficient conditions are also derived for the existence and asymptotic stability of the almost periodic solution.     

On the well-posedness of periodic problems for the system of hyperbolic equations with finite time delay

A periodic problem for the system of hyperbolic equations with finite time delay is investigated. The investigated problem is reduced to an equivalent problem, consisting the family of periodic problems for a system of ordinary differential equations with finite delay and integral equations using the method of a new functions introduction. Relationship of periodic problem for the system of hyperbolic equations with finite time delay and the family of periodic problems for the system of ordinary differential equations with finite delay is established. Algorithms for finding approximate solutions of the equivalent problem are constructed, and their convergence is proved. Criteria of well-posedness of periodic problem for the system of hyperbolic equations with finite time delay are obtained.     

Dynamic behaviors of the periodic predator–prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect

In this paper, a predator–prey system which based on a modified version of the Leslie–Gower scheme and Holling-type II scheme with impulsive effect are investigated, where all the parameters of the system are time-dependent periodic functions. By using Floquet theory of linear periodic impulsive equation, some conditions for the linear stability of trivial periodic solution and semi-trivial periodic solutions are obtained. It is proved that the system can be permanent if all the trivial and semi-trivial periodic solutions are linearly unstable. We use standard bifurcation theory to show the existence of nontrivial periodic solutions which arise near the semi-trivial periodic solution. As an application, we also examine some special case of the system to confirm our main results.     

一类时滞反应扩散方程的周期解和概周期解

通过构造上、下控制函数，结合上、下解方法及相应的单调迭代方法研究了一类时滞反应扩散方程，证明了在反应项非单调时，如果一雏边值问题存在一对周期（或概周期）上、下解，则方程一定存在唯一的周期（或概周期）解．并给出了二维边值问题周期（或概周期）解存在唯一性的充分条件．推广了已有的一些结果。     

The New Method for the Searching Periodic Solutions of Periodic Differential Systems

In the paper we are giving the new method for searching periodic solutions of periodic differential systems. For this we construct a differential system with the same Reflecting Function as the Reflecting Function of the given system and with a known periodic solution. Then the initial data of the periodic solutions of this two systems coincide. In such a way the problem of existance periodic solutions goes to the Cauchy problem.     

Dynamic analysis of a turbidostat model with the feedback control

In this paper, a mathematical model with impulsive state feedback control is proposed for turbidostat system. The sufficient conditions of existence of positive order one periodic solution are obtained by using the existence criteria of periodic solution of a general planar impulsive autonomous system. It is shown that the system either tends to a stable state or has a periodic solution, which depends on the feedback state, the control parameter of the dilution rate and the initial concentration of microorganism and substrate. By investigating the periodic solution, the period and the initial point of the periodic solution are given. The results show that turbidostat with impulsive state feedback control tends to an order one periodic solution.     

Almost periodic solutions of a discrete Lotka–Volterra competition system with delays

In this paper, we consider a discrete almost periodic Lotka–Volterra competition system with delays. Sufficient conditions are obtained for the permanence and global attractivity of the system. Further, by means of an almost periodic functional hull theory, we show that the almost periodic system has a unique strictly positive almost periodic solution, which is globally attractive. Some examples are presented to verify our main results.     

比率型-捕食者-两竞争食饵模型的动力学行为

本文研究比率型非自治的捕食者 -食饵模型 .该系统是两个具有竞争关系的食饵种群被一个捕食种群捕食 .我们研究其动力学行为 ,包括持久性 ,全局渐近稳定性 ,周期解 ,概周期解的存在唯一性     

Some new results for a logistic almost periodic system with infinite delay and discrete delay

A nonautonomous logistic almost periodic system with infinite delay and discrete delay is considered. Our result shows that the system is globally asymptotically stable under the condition for the boundedness of the system. By using almost periodic functional Hull theory and new computational techniques, we show that the almost periodic system has a unique globally asymptotically stable strictly positive almost periodic solution under the condition for the boundedness of the system. Some recent results are improved, and an open question is answered.     

具有扩散和时滞的捕食系统的持续生存

本文研究具有一般功能性反应和时滞的周期捕食系统的持续生存和周期解,得到了系统永久持续生存的充要条件.在此条件下系统存在正的ω周期解,改进了一些已知结果.     

On isolated periodic solutions of differential systems

Summary The concept of an isolated periodic solution of a periodic differential system is defined and it is proved that if the number 1 is not a characteristic multiplier of the variational system corresponding to the given periodic solution then this solution is isolated. The concept of a periodic solution with isolated path of an autonomous system is defined and it is proved that if the number 1 is a simple characteristic multiplier of the corresponding variational system then the periodic solution has an isolated path. An example is given showing that the conditions above are not necessary. Entrata in Redazione l’8 giugno 1974.     

Conditionally Periodic Solutions of an Inhomogeneous Linear System Of Differential Equations with Conditionally Periodic Coefficients

We study the existence of a conditionally periodic solution of a linear system with a Stepanov conditionally periodic inhomogeneity. We prove that if this system has a bounded solution, then almost every system in its H-class has a bounded Besicovitch conditionally periodic solution.     

A periodic predator–prey-chain system with impulsive perturbation

A periodic predator–prey-chain system with impulsive effects is considered. By using the global results of Rabinowitz and standard techniques of bifurcation theory, the existence of its trivial, semi-trivial and nontrivial positive periodic solutions is obtained. It is shown that the nontrivial positive periodic solution for such a system may be bifurcated from an unstable semi-trivial periodic solution. Furthermore, the stability of these periodic solutions is studied.     

On linear almost periodic pulse systems

For a linear almost periodic system under pulse influence, the conditions are established under which this system is reducible (by a linear change of variables with a discontinuous almost periodic matrix) to a system without pulses but with a Bohr almost periodic right-hand side. The set of linear almost periodic pulse systems possessing only bounded solutions is studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 105–113, January, 1993.     

On nonlinear motions of Hamiltonian system in case of fourth order resonance

We deal with an autonomous Hamiltonian system with two degrees of freedom. We assume that the Hamiltonian function is analytic in a neighborhood of the phase space origin, which is an equilibrium point. We consider the case when two imaginary eigenvalues of the matrix of the linearized system are in the ratio 3: 1. We study nonlinear conditionally periodic motions of the system in the vicinity of the equilibrium point. Omitting the terms of order higher then five in the normalized Hamiltonian we analyze the so-called truncated system in detail. We show that its general solution can be given in terms of elliptic integrals and elliptic functions. The motions of truncated system are either periodic, or asymptotic to a periodic one, or conditionally periodic. By using the KAM theory methods we show that most of the conditionally periodic trajectories of the truncated systems persist also in the full system. Moreover, the trajectories that are not conditionally periodic in the full system belong to a subset of exponentially small measure. The results of the study are applied for the analysis of nonlinear motions of a symmetric satellite in a neighborhood of its cylindric precession.     

The necessary and sufficient conditions for the existence of periodic solutions of a type of neutral functional differential equations

In this paper, using Fourier series, we study the problem of the existence of periodic solutions of a type of periodic neutral differential difference system. Some necessary and sufficient conditions for the existence of periodic solutions of a type of neutral functional equation system are obtained, and at the same time, we present a method with formula shows how to find the periodic solutions.