The article is devoted to the generalization of the second Bogolyubov's theorem to non-almost periodic dynamical systems. We prove the analog of the second Bogolyubov's theorem for recurrent or pseudo recurrent dynamical systems in Banach spaces. Namely, we obtain the relation between a recurrent dynamical system and its averaged dynamical system. We also study existence of recurrent and pseudo recurrent motions (including special cases of periodic, quasi-periodic and almost periodic motions) in related nonautonomous systems.
In this paper, we establish the second Bogolyubov theorem and global averaging principle for stochastic partial differential equations (in short, SPDEs) with monotone coefficients. Firstly, we prove that there exists a unique $L^{2}$-bounded solution to SPDEs with monotone coefficients and this bounded solution is globally asymptotically stable in square-mean sense. Then we show that the $L^{2}$-bounded solution possesses the same recurrent properties (e.g. periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, Levitan almost periodic, etc.) in distribution sense as…
In this paper the statement of the second Bogolyubov's theorem on periodic solutions of smooth systems with small parameter is justified for discountinuous systems. It is assumed that the generating solution intersects the discontinuity hyperplanes transversally and that the system is smooth outside of these hyperplanes. Such a situation is natural for mechanical systems with dry friction and without constraints and sticking of oscillations. To illustrate the result we prove stability of the velocity of vibration-induced displacement.
We study conditions for the abstract periodic linear functional differential equation $\dot{x}=Ax+F(t)x_t+f(t)$ to have almost periodic with the same structure of frequencies as $f$. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the frequencies of the forcing term $f$. The obtained results extend recent results on the subject. A discussion on how the results could be extended to the case when $A$ depends on $t$ is given.
In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solu…
By the Lyapunov-Perron method,we prove the existence of random inertial manifolds for a class of equations driven simultaneously by non-autonomous deterministic and stochastic forcing. These invariant manifolds contain tempered pullback random attractors if such attractors exist. We also prove pathwise periodicity and almost periodicity of inertial manifolds when non-autonomous deterministic forcing is periodic and almost periodic in time, respectively.
In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with monotone coefficients. Firstly, we establish the continuous dependence on initial values and coefficients for solutions. Secondly, we prove the existence of recurrent solutions, which include periodic, almost periodic and almost automorphic solutions. Then we show that these recurrent solutions are globally asymptotically stable in square-mean sense…
Let $π\colon T\times X\rightarrow X$ be a semiflow on a compact Hausdorff space $X$ with phase semigroup $T$. This paper provides us with some conditions under which \textsl{weakly almost periodic} is equivalent to \textsl{uniformly almost periodic} or to \textsl{equicontinuous} for $(T,X,π)$.
In this paper, we first give the definition of random almost periodic solutions of random dynamical systems and give some examples. Then, we prove the existence of such random almost periodic solutions. Further, we introduce the definition of random almost periodic measure. Finally, we discuss the relationship between random almost periodic solutions and almost periodic measures.
Using techniques of non-autonomous dynamical systems, we completely characterize the persistence properties of an almost periodic Nicholson system in terms of some numerically computable exponents. Although similar results hold for a class of cooperative and sublinear models, in the general non-autonomous setting one has to consider persistence as a collective property of the family of systems over the hull: the reason is that uniform persistence is not a robust property in models given by almost periodic differential equations.
Using properties of asymptotically almost periodic solutions we prove existence theorem for piece-wise continuous almost periodic solutions of differential equations with delay and impulses. We apply these results to study almost periodic single species model with stage structure and impulses.
In this paper, we are concerned with the existence of invariant curves of the planar twist mappings where the perturbations are almost periodic. As an application, the existence of almost periodic solutions and the boundedness of all solutions for pendulum-type equations with an almost periodic external force are proved.
In this paper we are concerned with the existence of invariant curves of planar twist mappings which are almost periodic in a spatial variable. As an application of this result to differential equations we will discuss the existence of almost periodic solutions and the boundedness of all solutions for superlinear Duffing's equation with an almost periodic external force.
We study the diffraction and dynamical properties of translation bounded weakly almost periodic measures. We prove that the dynamical hull of a weakly almost periodic measure is a weakly almost periodic dynamical system with unique minimal component given by the hull of the strongly almost periodic component of the measure. In particular the hull is minimal if and only if the measure is strongly almost periodic and the hull is always measurably conjugate to a torus and has pure point spectrum with continuous eigenfunctions. As an application we show the stability of the class of weighted Dirac…
We prove that the linear stochastic equation $dx(t)=(A(t)x(t)+f(t))dt+g(t)dW(t)$ with linear operator $A(t)$ generating a continuous linear cocycle $φ$ and Bohr/Levitan almost periodic or almost automorphic coefficients $(A(t),f(t),g(t))$ admits a unique Bohr/Levitan almost periodic (respectively, almost automorphic) solution in distribution sense if it has at least one precompact solution on $\mathbb R_{+}$ and the linear cocycle $φ$ is asymptotically stable.
In this paper, we prove the existence of full dimensional invariant tori for a non-autonomous, almost-periodically forced nonlinear beam equation with a periodic boundary condition via KAM theory.
We investigate leaky integrate-and-fire models (LIF models for short) driven by Stepanov and $μ$-almost periodic functions. Special attention is paid to the properties of a firing map and its displacement, which give information about the spiking behaviour of the system under consideration. We provide conditions under which such maps are well-defined for every $t \in \mathbb R$ and are uniformly continuous. Moreover, we show that the LIF model with a Stepanov almost periodic input has a uniformly almost periodic displacement map. We also show that in the case of a $μ$-almost periodic drive it …
We study the topological and ergodic dynamics of Bohr almost periodic motions of a topological abelian semigroup acting continuously on a compact metric space.
In this paper, we investigate convergence dynamics of $2^N$ almost periodic encoded patterns of general neural networks (GNNs) subjected to external almost periodic stimuli, including almost periodic delays. Invariant regions are established for the existence of $2^N$ almost periodic encoded patterns under two classes of activation functions. By employing the property of $\mathscr{M}$-cone and inequality technique, attracting basins are estimated and some criteria are derived for the networks to converge exponentially toward $2^N$ almost periodic encoded patterns. The obtained results are new,…
In this paper we establish asymptotic (biasymptotic) equivalence between spaces of solutions of a given linear homogeneous system and a perturbed system. The perturbations are of either linear or weakly linear characters. Existence of a homeomorphism between subspaces of almost periodic and asymptotically (biasymptotically) almost periodic solutions is also obtained.
The paper is dedicated to studying the problem of Poisson stability (in particular stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, Bohr almost automorphy, Birkhoff recurrence, almost recurrence in the sense of Bebutov, Levitan almost periodicity, pseudo-periodicity, pseudo-recurrence, Poisson stability) of solutions for semi-linear stochastic equation $$ dx(t)=(Ax(t)+f(t,x(t)))dt +g(t,x(t))dW(t)\quad (*) $$ with exponentially stable linear operator $A$ and Poisson stable in time coefficients $f$ and $g$. We prove that if the functions $f$ and $g$ are appropriately "small…
A discrete set in the $p$-dimensional Euclidian space is {\it almost periodic}, if the measure with the unite masses at points of the set is almost periodic in the weak sense. We propose to construct positive almost periodic discrete sets as an almost periodic perturbation of a full rank discrete lattice. Also we prove that each almost periodic discrete set on the real axes is an almost periodic perturbation of some arithmetic progression. Next, we consider signed almost periodic discrete sets, i.e., when the signed measure with masses $\pm1$ at points of a discrete set is almost periodic. We …
The paper is to study the asymptotic dynamics in nonmonotone comparable almost periodic reaction-diffusion system with Dirichlet boundary condition, which is comparable with uniformly stable strongly order-preserving system. By appealing to the theory of skew-product semiflows, we obtain the asymptotic almost periodicity of uniformly stable solutions to the comparable reaction-diffusion system.
We study the one-dimensional Fermi-Ulam ping-pong problem with a Bohr almost periodic forcing function and show that the set of initial condition leading to escaping orbits typically has Lebesgue measure zero.
This work deals with the existence of an almost periodic solution for certain kind of differential equations with generalized piecewise constant argument, almost periodic coefficients which are seen as a perturbation of a linear equation of that kind satisfying an exponential dichotomy on a difference equation. The stability of that solution in a semi-axis studied.
We present a notion of almost periodicity wich can be applied to random dynamical systems as well as almost periodic stochastic differential equations in Hilbert spaces (abstract stochastic partial differential equations). This concept allows for improvements of known results of almost periodicity in distribution, for general random processes and for solutions to stochastic differential equations.
A notion of almost periodic current is introduced, as well as a notion of almost periodic holomorphic chain proceeded from that definition. Such a chain can be defined either as a special case of almost periodic currents or as a holomorphic chain whose trace measure is an almost periodic distribution. It is shown that in general situation almost periodicity of the trace of a current does not imply that for the current itself, even if it is closed and positive. The zero set (regarded as a holomorphic chain) of a holomorphic mapping can be represented as a Monge-Ampere type current, and one coul…
The known Levitan's Theorem states that the linear differential equation $$ x'=A(t)x+f(t) \ \ \ (*) $$ with Bohr almost periodic coefficients $A(t)$ and $f(t)$ admits at least one Levitan almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of homogeneous equations $$ x'=A(t)x\ .\ \ \ (**) $$ In this paper we prove that linear differential equation (*) with Levitan almost periodic coefficients has a Levitan almost periodic solution, if it has at least one bounded solution. In this case, the separation from zero of …
In this paper, we introduce concepts of pathwise random almost periodic and almost automorphic solutions for dynamical systems generated by non-autonomous stochastic equations. These solutions are pathwise stochastic analogues of deterministic dynamical systems. The existence and bifurcation of random periodic (random almost periodic, random almost automorphic) solutions have been established for a one-dimensional stochastic equation with multiplicative noise.
In this paper, we investigate a class of nonlinear impulsive stochastic differential evolution equations with infinite delay in Banach space. Based on the Krasnoselskii's fixed point theorem, sufficient conditions of the existence of the square mean piecewise almost periodic solutions to this type of equations are derived. Moreover, the exponential stability of the square mean piecewise almost periodic solution is investigated.
In this work, we study the existence and uniqueness of bounded Weyl almost periodic solution to the abstract differential equation u ' (t) = Au(t) + f (t), t $\in$ R, in a Banach space X, where A : D (A) $\subset$ X $\rightarrow$ X is a linear operator (unbounded) which generates an exponentially stable C 0-semigroup on X and f : R $\rightarrow$ X is a Weyl almost periodic function. We also investigate the nonautonomous case.
Using a special metric in the space of sequences, we give a geometric description of almost periodic sets in the $k$-dimensional Euclidean space. We prove the completeness of the space of almost periodic sets and some analogue of the Bochner criterion of almost periodicity. Also, we show the connection between these sets and almost periodic measures.
We consider a new model for shunting inhibitory cellular neural networks, retarded functional differential equations with piecewise constant argument. The existence and exponential stability of almost periodic solutions are investigated. An illustrative example is provided.
This paper is concerned with the dynamics of an infinite-dimensional gradient system under small almost periodic perturbations. Under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions, we prove that the perturbed non-autonomous system has exactly the same number of almost periodic solutions. As a consequence, the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodic solutions. An application of the result…
In an abstract Banach space we study conditions for the existence of piecewise continuous, almost periodic solutions for semi-linear impulsive differential equation with fixed and non-fixed moments of impulsive action
We study the spatial-homogeneity of stable solutions of almost-periodic parabolic equations. It is shown that if the nonlinearity satisfies a concave or convex condition, then any linearly stable almost automorphic solution is spatially-homogeneous; and moreover, the frequency module of the solution is contained in that of the nonlinearity.
We develop a theory which allows making qualitative conclusions about the dynamics of both monotone and non-monotone Moreau sweeping processes. Specifically, we first prove that any sweeping processes with almost periodic monotone right-hand-sides admits a globally exponentially stable almost periodic solution. And then we describe the extent to which such a globally stable solution persists under non-monotone perturbations.
In this paper we study the dynamical behaviour of the differential equation \begin{equation*} x''+ax^+ -bx^-=f(t), \end{equation*} where $x^+=\max\{x,0\}$,\ $x^-=\max\{-x,0\}$, $a$ and $b$ are two different positive constants, $f(t)$ is a real analytic almost periodic function. For this purpose, firstly, we have to establish some variants of the invariant curve theorem of planar almost periodic mappings, which was proved recently by the authors (see \cite{Huang}).\ Then we will discuss the existence of almost periodic solutions and the boundedness of all solutions for the above asymmetric osci…
We consider existence and stability of an almost periodic solution of the quasilinear system of differential equations with piecewise constant argument of generalized type. The associated linear homogeneous system satisfies exponential dichotomy. The deviations of the argument are not restricted by any sign assumption when existence is considered. The problem of positive (almost periodic) solutions of the logistic equation is discussed as an example. A new technique of investigation of equations with piecewise argument, based on integral representation, is developed.
For metrizable spaces we replace the notion of almost periodic homeomorphism with a similar notion and verify that the usual characterizations of almost periodic homeomorphisms of compact metric spaces are valid for all metrizable spaces.
We consider metrizable ergodic topological dynamical systems over locally compact, $σ$-compact abelian groups. We study pure point spectrum via suitable notions of almost periodicity for the points of the dynamical system. More specifically, we characterize pure point spectrum via mean almost periodicity of generic points. We then go on and show how Besicovitch almost periodic points determine both eigenfunctions and the measure in this case. After this, we characterize those systems arising from Weyl almost periodic points and use this to characterize weak and Bohr almost periodic systems. Fi…
In this paper, we discuss the relationships between stability and almost periodicity for solutions of stochastic differential equations. Our essential idea is to get stability of solutions or systems by some inherited properties of Lyapunov functions. Under suitable conditions besides Lyapunov functions, we obtain the existence of almost periodic solutions in distribution.
In this paper, we study almost periodic solutions for semilinear stochastic differential equations driven by Lévy noise with exponential dichotomy property. Under suitable conditions on the coefficients, we obtain the existence and uniqueness of bounded solutions. Furthermore, this unique bounded solution is almost periodic in distribution under slightly stronger conditions. We also give two examples to illustrate our results.
We introduce and study a dimensional-like characteristic of an uniformly almost periodic function, which we call the Diophantine dimension. By definition, it is the exponent in the asymptotic behavior of the inclusio length. Diophantine dimension is connected with recurrent and ergodic properties of an almost periodic function. We get some estimates of the Diophantine dimension for certain quasiperiodic functions and present methods to investigate such a characteristic for almost periodic trajectories of evolution equations. Also we discuss the link between the presented approach and the so ca…
Favard separation method is an important means to study almost periodic solutions to linear differential equations; later, Amerio applied Favard's idea to nonlinear differential equations. In this paper, by appropriate choosing separation and almost periodicity in distribution sense, we obtain the Favard and Amerio type theorems for stochastic differential equations.
In this paper, we consider the almost periodic dynamics of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales. By establishing some dynamic inequalities on time scales, a permanence result for the model is obtained. Furthermore, by means of the almost periodic functional hull theory on time scales and Lyapunov functional, some criteria are obtained for the existence, uniqueness and global attractivity of almost periodic solutions of the model. Our results complement and extend some scientific work in recent years. Finally, an example is given to illustrate t…
We study the group of almost-periodic homeomorphisms of the real line. Our main result states that an action of a finitely generated group on the real line without global fixed point is conjugated to an almost-periodic action without almost fixed point. This is equivalent to saying that the action on the real line can be compactified to an action on a 1-dimensional lamination of a compact space, without global fixed point. As an application we give an alternative proof of Witte's theorem: an amenable left orderable group is locally indicable.
This article consists in two independent parts. In the first one, we investigate the geometric properties of almost periodicity of model sets (or cut-and-project sets, defined under the weakest hypotheses); in particular we show that they are almost periodic patterns and thus possess a uniform density. In the second part, we prove that the class of model sets and almost periodic patterns are stable under discretizations of linear maps.
The space of non-singular flows on any given solenoid is shown to contain a generic subset consisting of flows that are not almost periodic. Whether this result carries over to Hamiltonian flows remains an open question.
