The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.     

Bifurcating periodic solutions of wind-driven circulation equations

The existence of bifurcating periodic flows in a quasi-geostrophic mathematical model of wind-driven circulation is investigated. In the model, the Ekman number r and Reynolds number R control the stability of the motion of the fluid. Through rigorous analysis it is proved that when the basic steady-state solution is independent of the Ekman number, then a spectral simplicity condition is sufficient to ensure the existence of periodic solutions branching off the basic steady-state solution as the Ekman number varies across its critical value for constant Reynolds number. When the basic solution is a function of Ekman number, an additional condition is required to ensure periodic solutions.     

Existence and uniqueness of positive periodic solutions of functional differential equations

In this paper we consider the existence and uniqueness of positive periodic solution for the periodic equation y′(t)=−a(t)y(t)+λh(t)f(y(t−τ(t))). By the eigenvalue problems of completely continuous operators and theory of α-concave or −α-convex operators and its eigenvalue, we establish some criteria for existence and uniqueness of positive periodic solution of above functional differential equations with parameter. In particular, the unique solution y_λ(t) of the above equation depends continuously on the parameter λ. Finally, as an application, we obtain sufficient condition for the existence of positive periodic solutions of the Nicholson blowflies model.     

Polynomial almost periodic solutions for a class of Riemann-Hilbert problems with triangular symbols

Let with α,β∈]0,1[ such that α+β<1, αβ⁻¹∉Q and a,b,c∈C?{0}. In this paper the existence of almost-periodic polynomial (APP) solutions to the equation (with and ) is studied. The natural space in which to seek a solution to the above problem is the space of almost periodic functions with spectrum in the group αZ+βZ+Z. Due to the difficulty in dealing with the problem in that generality, solutions are sought with spectrum in the group αZ+βZ. Several interesting and totally new results are obtained. It is shown that, if 1∉αZ+βZ, no polynomial solutions exist, i.e., almost periodic polynomial solutions exist only if αZ+βZ=αZ+βZ+Z. Keeping to this setting, it is shown that APP solutions exist if and only if the function satisfies the simple spectral condition α+β>1/2. The proof of this result is nontrivial and has a number-theoretic flavour. Explicit formulas for the solution to the above problem are given in the final section of the paper. The derivation of these formulas is to some extent a byproduct of the proof of the result on the existence of APP solutions.     

Asymptotic stability of continuum sets of periodic solutions to systems with hysteresis

We consider hysteresis perturbations of a system of ODEs which possesses an asymptotically stable periodic solution z_∗. Where the oscillation of an appropriate projection of this periodic solution is smaller than some threshold number defined by the hysteresis nonlinearity, it is shown that the perturbed system has a continuum of periodic solutions with a rather simple structure in a vicinity of z_∗. The main result is a theorem on the stability of this continuum.     

Homogenization of monotone operators under conditions of coercitivity and growth of variable order

We obtain a homogenization procedure for the Dirichlet boundary-value problem for an elliptic equation of monotone type in the domain Ω ? ? ^d . The operator of the problem satisfies the conditions of coercitivity and of growth with variable order p _? (x) = p(x/?); furthermore, p(y) is periodic, measurable, and satisfies the estimate 1 < α ≤ p(y) ≤ β < ∞, while the parameter ? > 0 tends to zero. Here α and β are arbitrary constants. Taking Lavrent’ev’s phenomenon into account, we consider solutions of two types: H- and W-solutions. Each of the solution types calls for a distinct homogenization procedure. Its justification is carried out by using the corresponding version of the lemma on compensated compactness, which is proved in the paper.     

On periodic solutions to the von Foerster-Lasota equation

We show that the von Foerster-Lasota equation with parameter λ has a periodic solution in the space of Hölder continuous functions with exponent α if and only if α<λ. This generalizes the result from Dawidowicz and Haribash (Univ. Iagell. Acta Math. 37:321–324, 1999), in which existence of periodic solutions was proved only for λ>1.     

Non-uniform dependence on initial data for the periodic Degasperis-Procesi equation

In this paper, we show that the solution map of the periodic Degasperis-Procesi equation is not uniformly continuous in Sobolev spaces Hs(T) for s>3/2. This extends previous result for s?2 to the whole range of s for which the local well-posedness is known. Our proof is based on the method of approximate solutions and well-posedness estimates for the actual solutions.     

Time periodic solutions of the diffusive Nicholson blowflies equation with delay

This paper is concerned with the Nicholson blowflies equation with nonlinear diffusion and time delay subject to the homogeneous Dirichlet boundary condition in a bounded domain. We establish the existence of nontrivial periodic solutions of the time-periodic problem under general conditions by constructing a coupled upper–lower solution pair and by applying the Schauder fixed point theorem. The attractivity of the periodic solutions is also discussed by using the monotone iteration method.     

Periodic solutions with nonconstant sign in Abel equations of the second kind

The study of periodic solutions with constant sign in the Abel equation of the second kind can be made through the equation of the first kind. This is because the situation is equivalent under the transformation x?x−1, and there are many results available in the literature for the first kind equation. However, the equivalence breaks down when one seeks for solutions with nonconstant sign. This note is devoted to periodic solutions with nonconstant sign in Abel equations of the second kind. Specifically, we obtain sufficient conditions to ensure the existence of a periodic solution that shares the zeros of the leading coefficient of the Abel equation. Uniqueness and stability features of such solutions are also studied.     

A new uniqueness criterion for the number of periodic orbits of Abel equations

A solution of the Abel equation such that x(0)=x(1) is called a periodic orbit of the equation. Our main result proves that if there exist two real numbers a and b such that the function aA(t)+bB(t) is not identically zero, and does not change sign in [0,1] then the Abel differential equation has at most one non-zero periodic orbit. Furthermore, when this periodic orbit exists, it is hyperbolic. This result extends the known criteria about the Abel equation that only refer to the cases where either A(t)?0 or B(t)?0 does not change sign. We apply this new criterion to study the number of periodic solutions of two simple cases of Abel equations: the one where the functions A(t) and B(t) are 1-periodic trigonometric polynomials of degree one and the case where these two functions are polynomials with three monomials. Finally, we give an upper bound for the number of isolated periodic orbits of the general Abel equation , when A(t), B(t) and C(t) satisfy adequate conditions.     

Constructive almost periodic factorization of some triangular matrix functions

A factorability criterion is obtained constructively, and the respective factorization obtained explicitly, for 2×2 triangular almost periodic matrix functions of the form . Here f=c−1e−α−c₀+c₁eβ, eμ(x):=eiμx, cj are non-zero constants and 0<α,β, α+β<λ?α+β+max{α,β} with α/β being irrational. Note that the factorization problem, even for triangular matrix functions as above with an arbitrary trinomial f, is open. The result obtained is yet another step towards its solution.     

Global stability and periodic solution of the viral dynamics

It is well known that the mathematical models provide very important information for the research of human immunodeficiency virus-type 1 and hepatitis C virus (HCV). However, the infection rate of almost all mathematical models is linear. The linearity shows the simple interaction between the T cells and the viral particles. In this paper, we consider the classical mathematical model with saturation response of the infection rate. By stability analysis we obtain sufficient conditions on the parameters for the global stability of the infected steady state and the infection-free steady state. We also obtain the conditions for the existence of an orbitally asymptotically stable periodic solution. Numerical simulations are presented to illustrate the results.     

Meromorphic solutions of a Riccati differential equation with a doubly periodic coefficient

We treat a Riccati differential equation w^′+w²+p(z)=0, where p(z) is a nonconstant doubly periodic meromorphic function. Under certain assumptions, every solution is meromorphic in the whole complex plane. We show that the growth order of it is equal to 2, and examine the frequency of α-points and poles. Furthermore, the number of doubly periodic solutions is discussed.     

An a priori error estimate for the finite element modelling of electromagnetic waves interacting with a periodic diffraction grating

An a priori error estimate using a so called α,β‐ periodic transformation to study electromagnetic waves in a periodic diffraction grating is derived. It has been reported for single scattering that there is an instability in numerical methods for high wavenumbers. To address this problem, the analytical solution of the scattering problem when the domain is scatterer free and an unknown function called the α,β‐quasi periodic solution are used to transform the associated Helmholtz problem. The well‐posedness of the resulting continuous problem is analysed before approximating its solution using a finite element discretization. To guarantee the uniqueness of this approximate solution, an a priori error estimate is derived. Finally, numerical results are presented that suggest that the α,β‐quasi periodic method converges at a far lower number of degrees of freedom than the α,0‐quasi periodic method reported previously; especially for high wavenumbers. This is particularly true when the incident wave only undergoes a small perturbation because of the presence of the scatterer. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Global attractivity of positive periodic solutions for nonlinear impulsive systems

In this paper, the existence, uniqueness and global attractivity of positive periodic solutions for nonlinear impulsive systems are studied. Firstly, existence conditions are established by the method of lower and upper solutions. Then uniqueness and global attractivity are obtained by developing the theories of monotone and concave operators. And lastly, the method and the results are applied to the impulsive n$n$-species cooperative Lotka–Volterra system and a model of a single-species dispersal among n$n$-patches.     

On the determination of the number of periodic (or closed) solutions of a scalar differential equation with convexity

It is well known that a scalar differential equation , where f(t,x) is continuous, T-periodic in t and weakly convex or concave in x has no, one or two T-periodic solutions or a connected band of T-periodic solutions. The last possibility can be excluded if f(t,x) is strictly convex or concave for some t in the period interval. In this paper we investigate how the actual number of T-periodic solutions for a given equation of this type in principle can be determined, if f(t,x) is also assumed to have a continuous derivative . It turns out that there are three cases. In each of these cases we indicate the monotonicity properties and the domain of values for the function P(ξ)=S(ξ)−ξ, where S(ξ) is the Poincaré successor function. From these informations the actual number of periodic solutions can be determined, since a zero of P(ξ) represents a periodic solution.     

On the structure of periodic solutions of differential equations

This paper studies the “internal structure” of the periodic solutions of differential equations with the aim of stating when they are constant functions. Yorke [21] and Lasota and Yorke [10] are the first works which show the existence, uńder certain conditions, of a lower bound for the period of non-constant solutions. As applications of the general results proved in Section 1 we obtain a negative solution to an open problem of Browder, the discovery that the periodic solutions ensured by Vidossich [17, Theorem 3.16], are constant functions, and conditions under which the periodic solutions of hyperbolic and parabolic equations are constant functions. Finally, we note that Li [11] applies the results of Section 1 to differential equations with delay.Various result of this paper point out a strong connection between the existence of periodic solutions of small period of x′ = f(x) and the fact that the origin belongs to the range of f. This situation is explored in [19].     

Asymptotic behavior of solutions of a periodic diffusion system

This paper deals with a periodic reaction-diffusion system of plankton allelopathy under homogeneous Neumann boundary conditions. Based on the result of Ahmad and Lazer, we show some estimates and nonexistence results for the positive solutions of the system. Furthermore, we investigate the asymptotic behavior of the solutions of the system, that is one species dies out and the other exists as time t tends to infinity.