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.
In this paper, we introduce the concepts of S-asymptotically $ω$-periodic solutions in distribution for a class of stochastic fractional functional differential equations. The existence and uniqueness results for the S-asymptotically $ω$-periodic solutions in distribution are obtained by means of the successive approximation and the Banach contraction mapping principle, respectively.
In this paper, we introduce some concepts of square-mean S-asymptotically $ω$-periodic stochastic processes. Using the stochastic analysis method and the Banach contraction mapping principle, we establish the existence and uniqueness results of the mild solution and the square-mean S-asymptotically $ω$-periodic solution for a semilinear nonautonomous stochastic fractional evolution equation driven by Lévy noise.
In this work we prove the lower bound for the number of $T$-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, $T$-periodic in time, with $T$-Maslov indices $i_0,i_\infty$ at the origin and at infinity, has at least $|i_\infty-i_0|$ periodic solutions, and an additional one if $i_0$ is even. Our argument combines the Poincaré--Birkhoff Theorem with an application of topological degree. We illustrate the sharpness of our result, and extend it to the case of second orders ODEs with linear-like behaviour at zero and infinity.
We consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We establish various properties of such systems including strong pinching, C^{1+β} smoothness of the Anosov splitting, and C^1 smoothness of measurable invariant conformal structures and distributions. We apply these results to volume preserving diffeomorphisms with two-dimensional stable and unstable distributions and diagonalizable derivatives of the return maps at periodic points. We show that a finite cover of such a diffeomorphism is smoothly conjugate to …
We introduce a notion of asymptotically orthonormal polynomials for a Borel measure $μ$ with compact nonpolar support in $\mathbb{C}$. Such sequences of polynomials have similar convergence properties of the sequences of Julia sets and filled Julia sets to those for sequences of orthonormal polynomials. We give examples of measures for which the monic orthogonal polynomials are asymptotically orthonormal. Combining this with observations on conformal invariance of orthogonal polynomials we explore the measure dependency of the associated dynamics of orthogonal polynomials. Concretely, we study…
We consider a homoclinic orbit to a saddle fixed point of an arbitrary $C^\infty$ map $f$ on $\mathbb{R}^2$ and study the phenomenon that $f$ has an infinite family of asymptotically stable, single-round periodic solutions. From classical theory, this requires $f$ to have a homoclinic tangency. We show it also necessary for $f$ to satisfy a `global resonance' condition and for the eigenvalues associated with the fixed point, $λ$ and $σ$, to satisfy $|λσ| = 1$. The phenomenon is codimension-three in the case $λσ= -1$, but codimension-four in the case $λσ= 1$ because here the coefficients of the…
We consider the iterates of a generic injective piecewise contraction of the interval defined by a finite family of contractions. Let $ϕ_i:[0,1]\to (0,1)$, $1\le i\le n$, be $C^2$-diffeomorphisms with $\sup_{x\in (0,1)} \vert Dϕ_i(x)\vert<1$ whose images $ϕ_1([0,1]), \ldots, ϕ_n([0,1])$ are pairwise disjoint. Let $0<x_1<\cdots<x_{n-1}<1$ and let $I_1,\ldots, I_n$ be a partition of the interval $[0,1)$ into subintervals $I_i$ having interior $(x_{i-1},x_i)$, where $x_0=0$ and $x_n=1$. Let $f_{x_1,\ldots,x_{n-1}}$ be the map given by $x\mapsto ϕ_i(x)$ if $x\in I_i$, for $1\le i\le n$. Among othe…
As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bif…
We show that resonance zones near an elliptic periodic point of a reversible map must, generically, contain asymptotically stable and asymptotically unstable periodic orbits, along with wild hyperbolic sets.
The aim of this paper is to describe the structure of global attractors for non-autonomous difference systems of equations with recurrent (in particular, almost periodic) coefficients. We consider a special class of this type of systems (the so--called weak convergent systems). We study this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our early papers to study the almost periodic (almost automorphic, recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent) solutions …
We study dissipative polygonal outer billiards, i.e. outer billiards about convex polygons with a contractive reflection law. We prove that dissipative outer billiards about any triangle and the square are asymptotically periodic, i.e. they have finitely many global attracting periodic orbits. A complete description of the bifurcations of the periodic orbits as the contraction rates vary is given. For the square billiard, we also show that the asymptotic periodic behavior is robust under small perturbations of the vertices and the contraction rates. Finally, we describe some numerical experime…
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.
Let $(ω, A_E)$ be a quasi-periodic Schrödinger cocycle, where $ω$ is a Diophantine irrational. The potential is assumed to be $C^2$ with a unique non-degenerate minimum, and the coupling constant is assumed to be large. We show that, as the energy approaches the lowest energy of the spectrum from below, the distance between the Oseledets-directions, in projective coordinates, is asymptotically linear. Moreover, we show that the $C^2$-norm of the Oseledets-directions, in projective coordinates, grows asymptotically (almost) like the inverse of the square root of the distance. Both of these resu…
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 planar inverted pendulum with a vibrating pivot point in the presence of an additional horizontal force field is studied. The horizontal force is not assumed to be small or rapidly oscillating. We assume that the pivot point of the pendulum rapidly oscillates in the vertical direction and the period of these oscillations is commensurable with the period of horizontal force. This system can be considered as a strong generalization of the Kapitza pendulum. Previously it was shown that for any horizontal force there always exists a non-falling periodic solution in the considered system. In pa…
We establish a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic, using the Perron-Frobenius operator. We apply the result to asymptotically periodic transformations and give an extensive specific example of asymptotically periodic transformations by using the tent map.
As the outbreak of COVID-19 enters its third year, we have now enough data to analyse the behavior of the pandemic with mathematical models over a long period of time. The pandemic alternates periods of high and low infections, in a way that sheds a light on the nature of mathematical model that can be used for reliable predictions. The main hypothesis of the model presented here is that the oscillatory behavior is a structural feature of the outbreak, even without postulating a time-dependence of the coefficients. As such, it should be reflected by the presence of limit cycles as asymptotic s…
