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.
In this paper, we study the Poisson stability (in particular, stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, almost automorphy, recurrence in the sense of Birkhoff, Levitan almost periodicity, pseudo periodicity, almost recurrence in the sense of Bebutov, pseudo recurrence, Poisson stability) of motions for monotone nonautonomous dynamical systems and of solutions for some classes of monotone nonautonomous evolution equations (ODEs, FDEs and parabolic PDEs). As a byproduct, some of our results indicate that all the trajectories of monotone systems converge to the above …
In this paper we study the problem of almost periodicity of solutions for dissipative differential equations (Bronshtein's conjecture). We give a positive answer to this conjecture for monotone almost periodic systems of differential/difference equations.
Let $(X,f_{1,\infty})$ be a nonautonomous dynamical system. In this paper we summarize known definitions of periodic points for general nonautonomous dynamical systems and propose a new, very natural, definition of asymptotic periodicity. Moreover, this definition is resistant to changes of a beginning of the sequence generating the nonautonomous system. We show the relations among these definitions and discuss their properties. We prove that for pointwise convergent nonautonomous systems topological transitivity together with dense set of asymptotically periodic points imply sensitivity. We a…
We prove that a linear nonautonomous differential system with nonuniform hyperbolicity on the half line can be expressed as diagonal system with a perturbation which is small enough. Moreover we show that the diagonal terms are contained in the nonuniform exponential dichotomy spectrum. For this purpose we introduce the concepts of \textit{nonuniform almost reducibility} and \textit{nonuniform contractibility} which are generalization of this notions originally defined in a uniform context.
We investigate the dynamical behavior of pull-back trajectories for nonautonomous stochastic feedback systems with multiplicative noise. We proved that there exists a random periodic solution of this system and all pull-back trajectories converge to this random periodic solution as time goes to infinitely almost surely. Our results can be applied to nonautonomous stochastic Goodwin negative feedback system, nonautonomous stochastic Othmer-Tyson positive feedback system and nonautonomous stochastic competitive systems etc.
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…
When the evolution familiy is hyperbolic and satisfies the Acquistapace-Terreni conditions, the existence and uniquenness of an almost automorphic mild solution and a weighted pseudo almost automorphic mild solution in distribution of mean-filed nonautonomous stochastic evolution equations driven by fractional Brownian motion is proved. Examples illustrating the main results are included.
We describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems. In the autonomous setting, such objects are variously known as almost-invariant sets, metastable sets, persistent patterns, or strange eigenmodes, and have proved to be important in a variety of applications. In this current work, we explain how to extend existing autonomous approaches to the nonautonomous setting. We call the new time-dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly …
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.
We show that a linear Young differential equation generates a topological two-parameter flow, thus the notions of Lyapunov exponents and Lyapunov spectrum are well-defined. The spectrum can be computed using the discretized flow and is independent of the driving path for triangular systems which are regular in the sense of Lyapunov. In the stochastic setting, the system generates a stochastic two-parameter flow which satisfies the integrability condition, hence the Lyapunov exponents are random variables of finite moments. Finally, we prove a Millionshchikov theorem stating that almost all, in…
The properties of stability of compact set $\mathcal{K}$ which is positively invariant for a semiflow $(Ω\times W^{1,\infty}([-r,0],\mathbb{R}^n),Π,\mathbb{R}^+)$ determined by a family of nonautonomous FDEs with state-dependent delay taking values in $[0,r]$ are analyzed. The solutions of the variational equation through the orbits of $\mathcal{K}$ induce linear skew-product semiflows on the bundles $\mathcal{K}\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and $\mathcal{K}\times C([-r,0],\mathbb{R}^n)$. The coincidence of the upper-Lyapunov exponents for both semiflows is checked, and it is a fun…
We study a class of scalar differential equations on the circle $S^1$. This class is characterized mainly by the property that any solution of such an equation possesses exponential dichotomy both on the semi-axes $\R_+$ and $\R_+$. Also we impose some other assumptions on the structure of the foliation into integral curves for such the equation. Differential equations of this class are called gradient-like ones. As a result, we describe the global behavior of the foliation, introduce a complete invariant of uniform equivalency, give standard models for the equations of the distinguished class…
The existence of an aperiodic orbit for an outer billiard outside a regular octagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic. All possible periods are explicitly listed.
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…
An existence of an aperiodic point for outer billiard outside regular dodecagon is proved. Additionally, almost all orbits of such an outer billiard are proved to be periodic, and all possible periods are listed explicitly. The proof is based on facts established with computer program implemented by author. All calculations of a program has been made with absolute accuracy; these calculations can be done "by hands" but it this case, amount of calculations would be too large.
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,…
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.
Given a self-map of a compact metric space $X$, we study periodic points of the map induced on the hyperspace of closed subsets of $X$. We give some necessary conditions on admissible sets of periods for these maps. Seemingly unrelated to this, we construct an almost totally minimal homeomorphism of the Cantor set. We also apply our theory to give a full description of admissible period sets for induced maps of the interval maps. The description of admissible periods is also given for maps induced on symmetric products.
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.
Noise-induced statistical periodicity in a class of one-dimensional maps is studied. We show the existence of statistical periodicity in a modified Lasota-Mackey map and describe the phenomenon in terms of almost cyclic sets. A transition from a stable state to a periodic state of the density depending on the noise level is observed in numerical investigations based on trajectory averages and by means of a transfer operator approach. We conclude that the statistical periodicity is the origin of the almost periodicity in noise-induced order.
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.
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 obtain a characterizations of the recurrence of a continuous vector field $w$ of a closed connected surface $M$ as follows. The following are equivalent: 1) $w$ is pointwise recurrent. 2)$w$ is pointwise almost periodic. 3) $w$ is minimal or pointwise periodic. Moreover, if $w$ is regular, then the following are equivalent: 1) $w$ is pointwise recurrent. 2)$w$ is minimal or the orbit space $M/w$ is either $[0,1]$, or $S^1$. 3) $R$ is closed (where $R := \{(x,y) \in M \times M \mid y \in \bar{O(x)} \}$ is the orbit closure relation). On the other hand, we show that the followi…
We show that for almost every $(P,λ)$ where $P$ is a convex polygon and $λ\in(0,1)$, the corresponding outer billiard about $P$ with contraction $λ$ is asymptotically periodic, i.e., has a finite number of periodic orbits and every orbit is attracted to one of them.
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 …
Periodicity plays a significant role in the chaos theory from the beginning since the skeleton of chaos can consist of infinitely many unstable periodic motions. This is true for chaos in the sense of Devaney [1], Li-Yorke [2] and the one obtained through period-doubling cascade [3]. Countable number of periodic orbits exist in any neighborhood of a structurally stable Poincaré homoclinic orbit, which can be considered as a criterion for the presence of complex dynamics [4]-[6]. It was certified by Shilnikov [7] and Seifert [8] that it is possible to replace periodic solutions by Poisson stabl…
We consider a discrete time dynamic system described by a difference equation with periodic coefficients and with additive stochastic noise. We investigate the possibility of the periodicity for the solution. In particular, we found sufficient conditions for existence of a periodic process such that the solution converges to it, including almost surely convergence.
We study the dynamics of discrete-time regulatory networks on random digraphs. For this we define ensembles of deterministic orbits of random regulatory networks, and introduce some statistical indicators related to the long-term dynamics of the system. We prove that, in a random regulatory network, initial conditions converge almost surely to a periodic attractor. We study the subnetworks, which we call modules, where the periodic asymptotic oscillations are concentrated. We proof that those modules are dynamically equivalent to independent regulatory networks.
One goal of this paper is to study robustness of stability of nonautonomous linear ordinary differential equations under integrally small perturbations in an infinite dimensional Banach space. Some applications are obtained to the case of rapid oscillatory perturbations, with arbitrary small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel (1970). Based in Rodrigues (1970) and in Kloeden and Rodrigues (2011) we introduce a class of functions that we call Generalized Almost Periodic Functions that extend th…
Existence and uniqueness of Remotely Almost Periodic solutions of differential equations with piecewise constant argument
Facultad de Ingeniería, Universidad de la República Oriental del Uruguay（乌拉圭东岸共和国大学工程学院）；Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción（康塞普西翁大学理学院数学工程系）
It is shown how the linear method of the Yosida-approximation of the derivative applies to solve possibly nonlinear abstract functional differential equations in both, the finite and infinite delay case. A generalization of the integral solution will provide regularity results. Moreover, this method applies to derive uniform convergence on the halfline, and therefore general results on boundedness and various types of asymptotic almost periodicity.
We study topological structure of the $ω$-limit sets of the skew-product semiflow generated by the following scalar reaction-diffusion equation \begin{equation*} u_{t}=u_{xx}+f(t,u,u_{x}),\,\,t>0,\,x\in S^{1}=\mathbb{R}/2π\mathbb{Z}, \end{equation*} where $f(t,u,u_x)$ is $C^2$-admissible with time-recurrent structure including almost-periodicity and almost-automorphy. Contrary to the time-periodic cases (for which any $ω$-limit set can be imbedded into a periodically forced circle flow), it is shown that one cannot expect that any $ω$-limit set can be imbedded into an almost-periodically force…
