Pseudo Almost Periodic Solutions to a Class of Semilinear Differential Equations

This paper is concerned with the existence and uniqueness of pseudo almost periodic solutions to a class of semilinear differential equations involving the algebraic sum of two (possibly noncommuting) densely defined closed linear operators acting on a Hilbert space. Sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those semilinear equations are obtained. An erratum to this article is available at .     

Levitan Almost Periodic and Almost Automorphic Solutions of <Emphasis Type="Italic">V</Emphasis>-monotone Differential Equations

In the present article we consider a special class of equations

Existence and Attractivity of Almost Periodic Solution for Recurrent Neural Networks with Unbounded Delays and Variable Coefficients

This paper presents several sufficient conditions for the existence and attractivity of almost periodic solution for a new class of recurrent neural networks with unbounded delays and variable coefficients. Different from the normal approach, that's to say, without resorting to any Lyapunov function, these results are obtained by utilizing generalized Halanay inequality technique and combining the theory of exponential dichotomy with fixed point method. Some existing results are found to be special case of this paper. In addition, the exponential stability of the almost periodic solution, which is not studied in the earlier references, is also considered for the system. An example is given to illustrate the feasibility of our results.This work was jointly supported by the National Natural Science Foundation of China under Grant 60373067, the Hong Kong Special Administrative Region, China with Project No. 7001146, the Natural Science Foundation of Jiangsu Province, China under Grant BK2003053, Qing-Lan Engineering Project of Jiangsu Province, China.     

Existence and uniqueness of almost periodic solution to a class of nonautonomous system

Ｈｉｇｈ＿ｔｅｎｓｉｏｎｅｌｅｃｔｒｉｃｉｔｙｎｅｔｗｏｒｋ,ｗｉｔｈｏｕｔｗｈｉｃｈｏｕｒｍｏｄｅｒｎｓｏｃｉｅｔｙｃａｎｎｅｖｅｒｌａｓｔｆｏｒａｄａｙ,ｍａｙｓｏｍｅｔｉｍｅｓｂｅｒｅａｌｌｙｕｎｓｔａｂｌｅ,ｄｕｅｔｏｖａｒｉｏｕｓｋｉｎｄｏｆｏｓｃｉｌｌａｔｉｏｎｓｏｆｃｕｒｒｅｎｔｉｎｔｅｎｓｉｔｙ．Ｉｎｒｅｃｅｎｔｙｅａｒｓ,ｐｅｒｉｏｄｉｃｏｓｃｉｌｌａｔｉｏｎｓｏｒａｌｍｏｓｔｐｅｒｉｏｄｉｃｏｓｃｉｌｌａｔｉｏｎｓｈａｖｅａｐｐｅａｒｅｄｉｎＣｅｎｔｒａｌｇｒｉｄａｎｄＮｏｒ…     

Resonant almost periodic oscillations in systems with slow varying parameters

We consider two-dimensional systems with fast rotating phase and slow varying parameters and assume that the right-hand side of a system almost periodically depends on fast and slow times. We investigate the conditions of the existence and stability of resonant almost periodic solutions. As an example, we consider forced oscillations of a mathematical pendulum under the action of a sum of two small forces with closed frequencies.     

Almost Automorphic and Almost Periodic Dynamics for Quasimonotone Non-Autonomous Functional Differential Equations

The occurrence of almost automorphic dynamics for monotone non-autonomous recurrent finite-delay functional differential equations is analyzed. Topological methods are used to ensure its presence in the case of existence of semicontinuous semi-equilibria. When these semi-equilibria are continuous and strong, the presence of almost automorphic extensions is persistent under small perturbations. The above method provides a minimal set isomorphic to the base in the case of a convex semiflow. Some examples show the applicability of these results.     

Exponential Dichotomy and Time-Bounded Solutions for First-Order Hyperbolic Systems

The paper is devoted to studying a class of strongly hyperbolic systems of the first order. We show that if the characteristic roots of the full symbol are outside an open strip containing the real axis, then the homogeneous system possesses an exponential dichotomy and the inhomogeneous system is solvable in the space of time-bounded and almost periodic functions. We also discuss some results on the behavior of solutions for nonlinear equations in the neighborhood of a stationary point.     

Bounded Correctors in Almost Periodic Homogenization

We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov (Mat Sb (N.S), 107(149):199–217, 1978). The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by Armstrong and Shen (Pure Appl Math, 2016) for equations with almost periodic coefficients. This yields control on spatial averages of the gradient of the corrector, which is converted into estimates on the size of the corrector itself via a multiscale Poincaré-type inequality.     

Limit periodic functions,adding machines,and solenoids

We prove that a stable adding machine invariant set for a homeomorphism of the plane is the limit of periodic points and also that a stable solenoid minimal invariant set for a three dimensional flow is the limit of periodic orbits. We give an example to show that a similar result is false in higher dimensions.This research partially supported by grants from the National Science Foundation and the Taft Foundation.     

Frictional Oscillations Under the Action of Almost Periodic Excitation

Frictional oscillations under the action of almost periodic force are studied. The modulation equations are derived by the multiple scales method to study bifurcations behavior. Heteroclinic Melnikov function is constructed to obtain the region of chaotic solutions of these equations. Bifurcations of almost periodic orbits are studied by Van der Pol transformation and averaging procedure.     

Almost sure stability condition of weakly coupled linear nonautonomous random systems

In this study, the sufficient condition of almost sure stability of twodimensional oscillating systems under parametric excitations is investigated. The systems considered are assumed to be composed of two weakly coupled subsystems. The driving actions are considered to be stationary stochastic processes satisfying ergodic properties. The properties of quadratic forms are used in conjunction with the bounds for the eigenvalues to obtain, in a closed form, the sufficient condition for the almost sure stability of the systems.     

Periodic,almost periodic and chaotic motions of forced self-sustained oscillators

Steady motions of the Van der Pol oscillator and an oscillator with hysteresis are studied numerically in this paper. Some features of periodic, almost periodic and chaotic motions of forced self-sustained oscillators are investigated. This paper has been presented at the ICTAM XVI Lyngby.     

On the Two-Dimensional Euler Equations with Spatially Almost Periodic Initial Data

We consider the nonstationary Euler equations in \mathbbR²{\mathbb{R}}^2 with almost periodic unbounded vorticity. We show that a unique solution is always spatially almost periodic at any time when the almost periodic initial data belongs to some function space. In order to prove this, we demonstrate the continuity with respect to initial data which do not decay at spatial infinity. The proof of the continuity with respect to initial data is based on that of Vishik’s uniqueness theorem.     

Almost disturbance decoupling for a class of minimum-phase higher-order cascade nonlinear systems

IntroductionThealmostdisturbancedecouplingproblemwasformulatedandsolvedforlinearsystemsbyWillems[1,2 ].Fornonlinearsystems,mostoftheexistingsolutionstotheADDproblemareestablishedbasedtheassumptionsonthecontrolledplantsarefeedbacklinearizable (atleastpartially) [3 - 5 ].Especially ,theresultofRef.[4 ]wasextendedtoalargeclassofminimum_phasenonlinearsystems[5 ].Butwhenthesystemunderconsiderationisinherentlynonlinear,thecommonL2 _gaincharacterizationisnotappropriatetodescribetheADDproblemforhi…     

Almost Periodic Solutions and Global Attractors of Non-autonomous Navier–Stokes Equations

The article is devoted to the study of non-autonomous Navier–Stokes equations. First, the authors have proved that such systems admit compact global attractors. This problem is formulated and solved in the terms of general non-autonomous dynamical systems. Second, they have obtained conditions of convergence of non-autonomous Navier–Stokes equations. Third, a criterion for the existence of almost periodic (quasi periodic, almost automorphic, recurrent, pseudo recurrent) solutions of non-autonomous Navier–Stokes equations is given. Finally, the authors have derived a global averaging principle for non-autonomous Navier–Stokes equations.     

Averaging principle for quasi-geostrophic motion under rapidly oscillating forcing

1　IntroductionandtheProblemPresentedClimateandgeophysicalflowsdefinetheenvironmentinwhichwelive ,andtheirunderstanding,beyondbeingafascinatingscientificchallenge,isoftremendouseconomicandsocialimportance.Aclassoflargescalegeophysicalfluidflows,suchastheGulfStreamandoceanicgyres,aremodelledbythequasi_geostrophicequation .Thequasi_geostrophic (QG)equationmodelsaclassoflargescalegeophysicalflows.ItisderivedasanapproximationoftherotatingNavier_StokesequationsbyanasymptoticexpansioninasmallRos…     

Almost sure T-stability and convergence for random iterative algorithms

The purpose of this paper is to study the almost sure T -stability and convergence of Ishikawa-type and Mann-type random iterative algorithms for some kind of φ-weakly contractive type random operators in a separable Banach space.Under suitable conditions,the Bochner integrability of random fixed points for this kind of random operators and the almost sure T -stability and convergence for these two kinds of random iterative algorithms are proved.     

Traveling Waves in Diffusive Random Media

The current paper is devoted to the study of traveling waves in diffusive random media, including time and/or space recurrent, almost periodic, quasiperiodic, periodic ones as special cases. It first introduces a notion of traveling waves in general random media, which is a natural extension of the classical notion of traveling waves. Roughly speaking, a solution to a diffusive random equation is a traveling wave solution if both its propagating profile and its propagating speed are random variables. Then by adopting such a point of view that traveling wave solutions are limits of certain wave-like solutions, a general existence theory of traveling waves is established. It shows that the existence of a wave-like solution implies the existence of a critical traveling wave solution, which is the traveling wave solution with minimal propagating speed in many cases. When the media is ergodic, some deterministic \hbox{properties} of average propagating profile and average propagating speed of a traveling wave solution are derived. When the media is compact, certain continuity of the propagating profile of a critical traveling wave solution is obtained. Moreover, if the media is almost periodic, then a critical traveling wave solution is almost automorphic and if the media is periodic, then so is a critical traveling wave solution. Applications of the general theory to a bistable media are discussed. The results obtained in the paper generalize many existing ones on traveling waves. AMS Subject Classification: 35K55, 35K57, 35B50     

Basic Reproduction Ratios for Almost Periodic Compartmental Epidemic Models

The theory of the basic reproduction ratio $R_{0}$ R 0 and its computation formulae for almost periodic compartmental epidemic models are established. It is shown that the disease-free almost periodic solution is stable if $R_{0}<1$ R 0 < 1 , and unstable if $R_{0}>1$ R 0 > 1 . We also apply the developed theory to a patchy model with almost periodic population dispersal and disease transmission coefficients to obtain a threshold type result for uniform persistence and global extinction of the disease.     

Almost Periodicity in an Ecological Model with <Emphasis Type="Italic">M</Emphasis>-Predators and <Emphasis Type="Italic">N</Emphasis>-Preys by “Pure-Delay Typ” System

By using comparison theorem and constructing suitable Lyapunov functional, we study the following periodic Lotka–Volterra model with M-predators and N-preys by pure-delay type