Asymptotic Behavior Of Periodic Solutions Research Articles

Asymptotic behavior of periodic solutions in one-parameter families of Liénard equations

In this paper, we consider one-parameter ( λ>0) families of Liénard differential equations. We are concerned with the study on the asymptotic behavior of periodic solutions for small and large values of λ>0. To prove our main result we use the relaxation oscillation theory and a topological version of the averaging theory. More specifically, the first one is appropriate for studying the periodic solutions for large values of λ and the second one for small values of λ. In particular, our hypotheses allow us to establish a link between these two theories.

Asymptotic Periodicity in the Fecally-Orally Epidemic Model in a Heterogeneous Environment

To understand the influence of seasonal periodicity and environmental heterogeneity on the transmission dynamics of an infectious disease, we consider asymptotic periodicity in the fecally-orally epidemic model in a heterogeneous environment. By using the next generation operator and the related eigenvalue problems, the basic reproduction number is introduced and shows that it plays an important role in the existence and non-existence of a positive T-periodic solution. The sufficient conditions for the existence and non-existence of a positive T-periodic solution are provided by applying upper and lower solutions method. Our results showed that the fecally-orally epidemic model in a heterogeneous environment admits at least one positive T-periodic solution if the basic reproduction number is greater than one, while no T-periodic solution exists if the basic reproduction number is less than or equal to one. By means of monotone iterative schemes, we construct the true positive solutions. The asymptotic behavior of periodic solutions is presented. To illustrate our theoretical results, some numerical simulations are given. The paper ends with some conclusions and future considerations.

Dynamics of a Delay Logistic Equation with Diffusion and Coefficients Rapidly Oscillating in Space Variable

The applicability of the averaging principle in the study of the dynamics of a practically important delay logistic equation with diffusion and coefficients rapidly oscillating with respect to the space variable is analyzed. A task of special interest is to address equations with rapid oscillations of the delay coefficient or a quantity characterizing the deviation of the space variable. Bifurcation problems arising in critical cases for the averaged equation are studied. Results concerning the existence, stability, and asymptotic behavior of periodic solutions to the original equation are formulated.

Asymptotic periodicity in a diffusive West Nile virus model in a heterogeneous environment

This paper is concerned with a diffusive West Nile virus model (WNv) in a heterogeneous environment. The basic reproduction number [Formula: see text] for spatially homogeneous model is first introduced. We then define a threshold parameter [Formula: see text] for the corresponding diffusive WNv model in a heterogeneous environment. It is shown that if [Formula: see text], the model admits at least one nontrivial T-periodic solution, whereas if [Formula: see text], the model has no nontrivial T-periodic solution. By means of monotone iterative schemes, the true solution can be obtained and the asymptotic behavior of periodic solutions is presented. The paper is closed with some numerical simulations to illustrate our theoretical results.

Bifurcations in Kuramoto–Sivashinsky equations

We consider the local dynamics of the classical Kuramoto–Sivashinsky equation and its generalizations and study the problem of the existence and asymptotic behavior of periodic solutions and tori. The most interesting results are obtained in the so-called infinite-dimensional critical cases. Considering these cases, we construct special nonlinear partial differential equations that play the role of normal forms and whose nonlocal dynamics thus determine the behavior of solutions of the original boundary value problem.

Large-time behavior of periodic solutions to fractal Burgers equation with large initial data

The asymptotic behavior of periodic solutions to fractal nonlinear Burgers equation is considered and the initial data are allowed to be arbitrarily large. The exponential decay estimates of the solutions are obtained for the power of Laplacian α ∈ [1/2, 1).

Asymptotic behavior of solutions of a periodic diffusion system of plankton allelopathy

This paper is concerned with the existence and asymptotic behavior of periodic solutions for a periodic reaction diffusion system of a planktonic competition model under Dirichlet boundary conditions. The approach to the problem is by the method of upper and lower solutions and the bootstrap argument of Ahmad and Lazer. It is shown under certain conditions that this system has positive or semi-positive periodic solutions. A sufficient condition is obtained to ensure the stability and global attractivity of positive periodic solutions.

Persistence in a Periodic Competitor-Competitor-Mutualist Diffusion System

In this paper we investigate the existence and the asymptotic behavior of periodic solutions for a periodic reaction-diffusion system of a competitor-competitor-mutualist model under Dirichlet boundary conditions. We shall prove that under certain conditions this system is persistent and under some other conditions it exhibits extinction or, more exactly, the species of mutualist extincts.

Asymptotic behavior of periodic solutions of parabolic equations with weakly nonlinear perturbation

Asymptotic behavior of periodic solutions of parabolic equations with weakly nonlinear perturbation

Asymptotic behavior of periodic solutions in Banach space

On considere le probleme f(t)∈du(t)/d√+Au(t), t∈(0,∞), u(0)=x ou A est un operateur m-accretif dans un espace de Banach X et f∈L loc 1 (0,∞; X) et T-periodique. On etudie le comportement asymptotique de la solution integrale T-periodique

Applications of the Hahn-Banach Theorem in Approximation Theory

Applications of the Hahn-Banach Theorem in Approximation Theory