Existence of solution to a periodic boundary value problem for a nonlinear impulsive fractional differential equation

We study the existence of solution to a periodic boundary value problem for nonlinear impulsive fractional differential equations by using Schaeffer’s fixed point theorem

A Massera Type Criterion for Linear Functional Differential Equations with Advance and Delay

AbstractIn this note, a Massera type criterion for the existence of periodic solutions for linear functional differential equations with advance and delay is established. Because of the presence of an advanced argument, the definition of the fundamental solution operator seems unknown. Hence a method different from the usual one is employed. Applications to periodic problems for nonlinear equations are also given

Existence, Uniqueness and Stability of Periodic Solution for Nonlinear System of Integro-Differential Equations

In this paper, we investigate the existence, uniqueness, and stability of the periodic solution for the system of nonlinear integro-differential equations by using the numerical-analytic methods for investigating the solutions and the periodic solutions of ordinary differential equations, which are given by A. Samoilenko

Existence and stability of traveling waves in parabolic systems of differential equations with weak diffusion

The aim of the present paper is to investigate of some properties of periodic solutions of a nonlinear autonomous parabolic systems with a periodic condition. We investigate parabolic systems of differential equations using an integral manifolds method of the theory of nonlinear oscillations. We prove the existence of periodic solutions in an autonomous parabolic system of differential equations with weak diffusion on the circle. We study the existence and stability of an arbitrarily large finite number of cycles for a parabolic system with weak diffusion. The periodic solution of parabolic equation is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce a norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with retarded argument and weak diffusion. We use bifurcation theory for delay differential equations and quasilinear parabolic equations. The existence of periodic solutions in an autonomous parabolic system of differential equations on the circle with retarded argument and small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with retarded argument and weak diffusion are investigated

Positive periodic solutions for impulsive differential equations with infinite delay and applications to integro‐differential equations

This is the accepted version of the following article: Buedo‐Fernández, S, Faria, T. Positive periodic solutions for impulsive differential equations with infinite delay and applications to integro‐differential equations. Math Meth Appl Sci. 2020; 43: 3052–3075, which has been published in final form at https://doi.org/10.1002/mma.6100. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy http://www.wileyauthors.com/self-archivingSufficient conditions for the existence of at least one positive periodic solution are established for a family of scalar periodic differential equations with infinite delay and nonlinear impulses. Our criteria, obtained by applying a fixed‐point argument to an original operator constructed here, allow to treat equations incorporating a rather general nonlinearity and impulses whose signs may vary. Applications to some classes of Volterra integro‐differential equations with unbounded or periodic delay and nonlinear impulses are given, extending and improving results in the literature.This work was supported by Ministerio de Educacion, Cultura y Deporte (Spain) under grant FPU16/04416 (Sebastián Buedo-Fernández) and by Fundação para a Ciência e a Tecnologia (Portugal) under project UID/MAT/04561/2019 (Teresa Faria)S

Знаходження періодичного розв’язку рівняння Матьє із запізненням

The work suggests an approach for finding periodic solution of the nonlinear delayed differential Mathieu equations applied in the theory of oscillatory processes. The application of the numerical-analytical method to finding periodic solutions of this equation is known. This idea includes reducing the equation to the system of the first order. The article proposes the use of the previously developed method for finding periodic solutions of nonlinear second-order ordinary differential equations, also used for equations with delay, without being reduced to a system. In this case, the Green's function is constructed for a self-adjoint differential operator of the second derivative, defined on functions that satisfy periodic boundary conditions. The necessary and sufficient conditions for the existence of a periodic solution of the Mathieu equation are given. The solution itself is found by the method of successive approximations. The estimates for the method's rate of convergence were obtained

Computed Chaos or Numerical Errors

Discrete numerical methods with finite time-steps represent a practical technique to solve initial-value problems involving nonlinear differential equations. These methods seem particularly useful to the study of chaos since no analytical chaotic solution is currently available. Using the well-known Lorenz equations as an example, it is demonstrated that numerically computed results and their associated statistical properties are time-step dependent. There are two reasons for this behavior. First, chaotic differential equations are unstable so that any small error is amplified exponentially near an unstable manifold. The more serious and lesser-known reason is that stable and unstable manifolds of singular points associated with differential equations can form virtual separatrices. The existence of a virtual separatrix presents the possibility of a computed trajectory actually jumping through it due to the finite time-steps of discrete numerical methods. Such behavior violates the uniqueness theory of differential equations and amplifies the numerical errors explosively. These reasons imply that, even if computed results are bounded, their independence on time-step should be established before accepting them as useful numerical approximations to the true solution of the differential equations. However, due to these exponential and explosive amplifications of numerical errors, no computed chaotic solutions of differential equations independent of integration-time step have been found. Thus, reports of computed non-periodic solutions of chaotic differential equations are simply consequences of unstably amplified truncation errors, and are not approximate solutions of the associated differential equations.Comment: pages 24, Figures