Uniqueness Of Periodic Solutions Research Articles

Synchronization and Oscillator Death in Diffusively Coupled Lattice Oscillators

We consider the synchronization and cessation of oscillation of a positive even number of planar oscillators that are coupled to their nearest neighbors on one, two, and three dimensional integer lattices via a linear and symmetric diffusion-like path. Each oscillator has a unique periodic solution that is attracting. We show that for certain coupling strengths there are both symmetric and antisymmetric synchronization that corresponds to symmetric and antisymmetric nonconstant periodic solutions respectively. Symmetric synchronization persists for all coupling strengths while the antisymmetric case exists for only weak coupling strengths and disappears to the origin after a certain coupling strength. AMS(MOS) subject classifications: 34C24, 34C22, 34C14 Keywords: Lattice differential equations, Bravais lattice, synchronization, oscillator death. (Af. J. of Science and Technology: 2003 4(1): 78-84)

Oscillation and global attractivity in hematopoiesis model with periodic coefficients

In this paper we shall consider the nonlinear delay differential equation (∗)p′(t)=β(t)pm(t−kω)1+pn(t−kω)−γ(t)p(t),where k is a positive integer, β(t) and γ(t) are positive periodic functions of period ω. In the nondelay case we shall show that (∗) has a unique positive periodic solution p̄(t), and we will study the global attractivity of p̄(t). In the delay case we shall establish some sufficient conditions for oscillation of all positive solutions of (∗) about p̄(t), and establish some sufficient conditions for the global attractivity of p̄(t). Our results in this paper extend as well as improve the results in the autonomous case.

Problem of periodic solutions for neutral functional differential equation

The problem of periodic solutions for a kind of kth-order linear neutral functional differential equation is studied. By using the theory of Fourier expansions, a sufficient and necessary condition to guarantee the existence and uniqueness of periodic solution is obtained. Further, by applying this result and Schauder's fixed point principle, a kind of kth-order nonlinear neutral functional differential equation is investigated, and some new results on existence of the periodic solutions are given as well. These results improve and extend some known results in recent literature.

Oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics

In this paper, we shall consider the nonlinear delay differential equation ▪ where m and n are positive integers, and V( t) and λ( t) are positive periodic functions of period ω. In the nondelay case, we shall show that (∗) has a unique positive periodic solution x ( t) and provides sufficient conditions for the global attractivity of x ( t). In the delay case, we shall present sufficient conditions for the oscillation of all positive solutions of (∗) about x ( t) and establish sufficient conditions for the global attractivity of x ( t).

Using the homotopy method to find periodic solutions of forced nonlinear differential equations

This paper discusses a result of Li and Shen which proves the existence of a unique periodic solution for the differential equation x + k ẋ + g(x,t) = ε(t) where k is a constant; g is continuous, continuously differentiable with respect to x, and is periodic of period P in the variable t; ε(t) is continuous and periodic of period P, and when ∂g/∂x satisfies some additional boundedness conditions. This means that there exist initial values x(0) = α* and ẋ (0) = β* so that the solution to the corresponding initial value problem is periodic of period P and is unique (up to a translation of the time variable) with this property. The proof of this result is constructive, so that starting with any initial conditions x(0) = α and ẋ(0) = β, a path in the phase plane can be produced, starting at (α, β) and terminating at (α*, β*). Both the theoretical proof and a constructive proof are discussed and a Mathematica implementation developed which yields an algorithm in the form of a Mathematica notebook (which is posted on the webpage http://pax.st.usm.edu/downloads). The algorithm is robust and can be used on differential equations whose terms do not satisfy Li and Shen's hypotheses. The ideas used reinforce concepts from beginning courses in ordinary differential equations, linear algebra, and numerical analysis.

Discussion of periodic solutions for pth order delayed NDEs

In this paper, a thorough analysis of the existence and uniqueness of periodic solutions for a class of pth order neutral differential equations (NDE) with some delays is presented. Using Fourier series theory and techniques of real analysis inequality, a set of new and leading significance sufficient conditions are obtained ensuring the existence and uniqueness of periodic solutions. Thus some previous results [Appl. Math. J. Chinese Univ. 8A(3) (1993) 251 (in Chinese); Pure Appl. Math. 12(2) (1996) 69 (in Chinese); Pure Appl. Math. 11(suppl.) (1995) 102 (in Chinese); Appl. Math. Mech. 17(1) (1996) 29; Appl. Math. Mech. 20(6) (1999) 647] are extended and improved much more. Moreover, the repeated and ingenious application of inequality techniques makes the conditions of our results become weaker and weaker. Therefore, the conditions possess highly important significance in both theories and applications. In addition, the methods of this paper may be extended to study some other systems.

Oscillation and global attractivity in a periodic Nicholson's blowflies model

In this paper, we shall consider the nonlinear delay differential equation N′( t) = −δ( t) N( t) + P( t) N( t − mω) e − aN( t−mω , where m is a positive integer, δ( t) and P( t) are positive periodic functions of period ω. In the nondelay case, we shall show that (∗) has a unique positive periodic solution N ( t), and provide sufficient conditions for the global attractivity of N ( t). In the delay case, we shall present sufficient conditions for the oscillation of all positive solutions of (∗) about N ( t), and establish sufficient conditions for the global attractivity of N ( t).

On periodic solutions of first order linear functional differential equations

New optimal sufficient conditions are established for the existence of a unique periodic solution of first order scalar functional differential equations

Oscillation and Global Attractivity in a Nonlinear Delay Periodic Model of Population Dynamics

In this article we shall consider the following nonlinear delay differential equation $$x'(t) + p(t)x(t)-\\frac {q(t)x(t)}{r + x^{n}(t-m\\omega )} = 0\\eqno (*)$$ where m and n are positive integers, p ( t ) and q ( t ) are positive periodic functions of period y . In the nondelay case we shall show that (*) has a unique positive periodic solution $ \\overline {x}(t), $ and provide sufficient conditions for the global attractivity of $ \\overline {x}(t) $ . In the delay case we shall present sufficient conditions for the oscillation of all positive solutions of (*) about $ \\overline {x}(t), $ and establish sufficient conditions for the global attractivity of $ \\overline {x}(t). $

Absolute periodicity and absolute stability of delayed neural networks

Proposes to study the absolute periodicity of delayed neural networks. A neural network is said to be absolutely periodic, if for every activation function in some suitable functional set and every input periodic vector function, a unique periodic solution of the network exists and all other solutions of the network converge exponentially to it. Absolute stability of delayed neural networks is also studied in this paper. Simple and checkable conditions for guaranteeing absolute periodicity and absolute stability are derived. Simulations for absolute periodicity are given.

Asymptotic behaviour of a nonautonomous cooperative system

AbstractThis paper considers a nonautonomous cooperative system, in which all the parameters are time-dependent and asymptotically approach periodic functions. We prove that under some appropriate conditions any positive solutions of the system asymptotically approach the unique positive periodic solution of the corresponding periodic system.

Periodic Solutions of a System of Linear Difference Equations with Continuous Argument

with continuous argument t ∈ R, where x ∈ R, A is a real constant n × n matrix, and f(t) is a T -periodic vector function of dimension n. There is a vast literature in which the theory of such systems is developed; a rather comprehensive bibliography can be found, e.g., in [1, 2]. Most papers deal with the construction of a general or special solution of such systems [4–7]; in particular, the general form and conditions for the existence and uniqueness of periodic solutions of finite-difference systems with continuous argument in the case of an integer period were obtained, for example, in [2, 5, 6]. The existence of periodic solutions of difference equations was analyzed in [3, Chap. 5], where the considerations were not restricted to the case of an integer period. However, the assumption on the Fredholm property of the operator of the system restricted the class of periodic solutions to a finite-dimensional space, which is solely due to the method based on the Fredholm alternative. The main goal of the present paper is to clarify conditions under which system (1.1) has a solution periodic with period rationally commensurable with the time increment; the space of such solutions can be infinite-dimensional. In Section 2, we consider the problem on the existence of a periodic solution (with rational or irrational period) of the homogeneous system corresponding to (1.1). The existence conditions are given in Theorem 1. The form of such a solution (if it exists at all) is discussed in Section 3 and given in Theorem 2. Section 4 contains the main results, stated in Theorems 3 and 4. It deals with existence conditions for periodic solutions of the nonhomogeneous system. If these conditions are satisfied, then one can obtain a periodic solution of system (1.1) in the form described in Theorems 5 and 6 (which are proved in Section 5). In conclusion, we consider examples.

Conditions for the existence and uniqueness of periodic solutions of nonautonomous differential equations

Conditions for the existence and uniqueness of periodic solutions of nonautonomous differential equations

Existence and uniqueness of periodic solutions for a nonlinear reaction-diffusion problem

We consider a class of degenerate reaction-diffusion equations on a bounded domain with nonlinear flux on the boundary. These problems arise in the mathematical modelling of flow through porous media. We prove, under appropriate hypothesis, the existence and uniqueness of the nonnegative weak periodic solution. To establish our result, we use the Schauder fixed point theorem and some regularizing arguments.

Generic uniqueness of a minimal solution for variational problems on a torus

We study minimal solutions for one‐dimensional variational problems on a torus. We show that, for a generic integrand and any rational number α, there exists a unique (up to translations) periodic minimal solution with rotation number α.

Symmetries, chirp-free points, and bistability in dispersion-managed fiber lines

We show from an elementary symmetry analysis that, in dispersion-compensated systems for which a lossless model is valid, nonlinearity requires a chirp-free point at the center of symmetry (if such exists) of the map for any kind of unique periodic solution. We also present an example of a more-complex map when the periodic solution is not unique.

The periodic solution of a class of epidemic models

This paper has completely studied the dynamical properties of a class of epidemiological models, obtained the necessary and sufficient condition for a unique periodic solution which is asymptotically orbitally stable, and the necessary and sufficient conditions for the global asymptotic stability of the positive equilibrium.

Periodic solutions of a class of higher order neutral type equations

In this paper, by ssing the theory of Fourier series, some necessary and sufficient conditions of existence and uniqueness of periodic solutions of a class of higher order neutral type equations are obtained. The main results by Shi Jianguo in “Discussion on the periodic solutions for linear equation of neutral type with constant coefficients” are improved, i. e., the condition |b0|≠1 instead of the condtion |b0|<1/2 of Theorem 1 by Shi Jianguo is given. Other theorems by Shi are rebuilt and improved according to the new assumption.

New results of some existence theorems on nonlinear boundary value problems

With the use of the homeomorphism theory and fixed point theory, the existence and uniqueness of solutions to boundary value problems are investigated. Two basic theorems are obtained without the boundness condition, which generalizes results of Brown. When our results are applied to the existence and uniqueness of periodic solutions for nonlinear perturbed conservative systems (Newtonian equations of motion), the existence and uniquenes of the solution are obtained. The results in this note seem less restrictive than those of the former papers we have seen. Meanwhile, as far as we know, it seems that applying the homeomorphism theory to the research of this kind of problem is new.

Optimal harvesting policy for single population with periodic coefficients.

In this paper, we examine the exploitation of single population modeled by time-dependent Logistic equation with periodic coefficients. First, it is shown that the time-dependent periodic Logistic equation has a unique positive periodic solution, which is globally asymptotically stable for positive solutions, and we obtain its explicit representation. Further, we choose the maximum annual-sustainable yield as the management objective, and investigate the optimal harvesting policies for constant harvest and periodic harvest. The optimal harvest effort that maximizes the annual-sustainable yield, the corresponding optimal population level, the corresponding harvesting time-spectrum, and the maximum annual-sustainable yield are determined, and their explicit expressions are obtained in terms of the intrinsic growth rate and the carrying capacity of the considered population. Our interesting and brief results generalize the classical results of Clark for a population described by the autonomous logistic equation in renewable resources management.