Let $F$ be a Banach space and $L(F)$ be the set of all its bounded linear operators. In this note, we are interested in the asymptotic behavior (recurrence and chain recurrence) of the solution of the following initial value problem \label{eqlinear} x'(t) = X(t)x(t), \qquad x(0) = x, where $x \in F$ and the map $t \mapsto X(t) \in L(F)$ is a $T$-periodic continuous curve. This asymptotic behavior is related to the asymptotic behavior of the discrete-time flow on $F$ generated by the invertible operator $g \in L(F)$ given by the associated fundamental solution at time $T$.
Let $f$ be a meromorphic function with bounded set of singular values and for which infinity is a logarithmic singularity. Then we show that $f$ has infinitely many repelling periodic points for any minimal period $n\geq1$, using a much simpler argument than the corresponding results for arbitrary entire transcendental functions.
In this paper we derive new criterion for uniform stability assessment of the linear periodic time-varying systems $\dot x=A(t)x,$ $A(t+T)=A(t).$ As a corollary, the lower and upper bounds for the Floquet characteristic exponents are established. The approach is based on the use of logarithmic norm of the system matrix $A(t).$ Finally we analyze the robustness of the stability property under external disturbance.
These notes were written during the 9th and 10th sessions of the subject Dynamical Systems II coursed at DTU (Denmark) during the Winter Semester 2015-2016, and later extended in February 2017. They aim to provide students with a theoretical and numerical background for the computation of periodic orbits using Newton's method. We focus on periodically perturbed quasi-integrable systems (using the forced pendulum as an example) and hence we take advantage of the Melnikov method to get first guesses. However, these well known techniques are general enough to be applied in other type of systems. …
We employ the Craig-Wayne-Bourgain method to construct quasi-periodic solutions for delayed perturbation differential equations. Our results not only implement the existing literatures on constructing quasi-periodic solutions for DDE by the KAM method and space splitting technique, but also provide an example of application of multi-scale analysis method to non-selfadjoint problem.
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
In this paper, the general perturbation problem of piecewise smooth integrable differential systems with two switching planes is considered. Firstly, when the unperturbed system has a family of periodic orbits, the first order Melnikov vector function is derived which can be used to study the number of periodic orbits bifurcated from the period annuli. Then, by using the obtained Melnikov vector function, we get an upper bound of the number of periodic orbits of a concrete $n$-dimensional piecewise smooth differential system.
In this paper we prove that the following delay differential equation \[ \frac{d}{dt}x(t)=rx(t)\left(1-\int_{0}^{1}x(t-s)ds\right), \] has a periodic solution of period two for $r>\frac{π^{2}}{2}$ (when the steady state, $x=1$, is unstable). In order to find the periodic solution, we study an integrable system of ordinary differential equations, following the idea by Kaplan and Yorke \cite{Kaplan=000026Yorke:1974}. The periodic solution is expressed in terms of the Jacobi elliptic functions.
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.
This work focuses on the existence of quasi-periodic solutions for ordinary and delay differential equations (ODEs and DDEs for short) with an elliptic-type degenerate equilibrium point under quasi-periodic perturbations. We prove that under appropriate hypotheses there exist quasi-periodic solutions for perturbed ODEs and DDEs near the equilibrium point for most parameter values, then apply these results to the delayed van der Pol's oscillator with zero-Hopf singularity.
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 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 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.
This paper is concerned with the problem of existence of periodic solutions for perturbative Carathéodory differential equations. The main result provides sufficient conditions on the averaged equation that guarantee the existence of periodic solutions. Additional conditions are also provided to ensure the uniform convergence of a periodic solution to a constant function. The proof of the main theorem is mainly based on an abstract continuation result for operator equations.
Recently, the higher order averaging method for studying periodic solutions of both Lipschitz differential equations and discontinuous piecewise smooth differential equations was developed in terms of Brouwer degree theory. Between the Lipschitz and the discontinuous piecewise smooth differential equations, there is a huge class of differential equations lacking in a higher order analysis on the existence of periodic solutions, namely the class of continuous non-Lipschitz differential equations. In this paper, based on the coincidence degree theory for nonlinear operator equations, we perform …
We consider an autonomous differential system in $\mathbb{R}^n$ with a periodic orbit and we give a new method for computing the characteristic multipliers associated to it. Our method works when the periodic orbit is given by the transversal intersection of $n-1$ codimension one hypersurfaces and is an alternative to the use of the first order variational equations. We apply it to study the stability of the periodic orbits in several examples, including a periodic solution found by Steklov studying the rigid body dynamics.
In this paper, we discuss on the linearized stability of the trivial solution for a class of nonlinear Caputo fractional differential systems of order $α\in(1,2)$. We show that some recent existing results in this direction are wrong. To complete the literature we also give a criterion to test the asymptotic stability of the trivial solution of this system.
We study conditions for the abstract linear functional differential equation $\dot{x}=Ax+F(t)x_t+f(t), t\ge 0$ to have asymptotic almost periodic solutions, where $F(\cdot )$ is periodic, $f$ is asymptotic almost periodic. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the circular spectrum of the forcing term $f$. The obtained results extend recent results on the subject.
This paper focuses on the construction of periodic solutions of nonlinear beam equations on the $d$-dimensional tori. For a large set of frequencies, we demonstrate that an equivalent form of the nonlinear equations can be obtained by a para-differential conjugation. Given the non-resonant conditions on each finite dimensional subspaces, it is shown that the periodic solutions can be constructed for the block diagonal equation by a classical iteration scheme.
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.
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 note we consider local invariant manifolds of functional differential equations representing differential equations with state-dependent delay. Starting with a local center-stable and a local center-unstable manifold of the functional differential equation at a stationary point, we construct, by a straightforward application of the Implicit Mapping Theorem, a local center manifold.
An action of a group $G$ on a compact space $X$ is called weakly almost periodic if the orbit of every continuous function on $X$ is weakly relatively compact in $C(X)$. We observe that for a topological group $G$ the following are equivalent: (i) every continuous action of $G$ on a compact space is weakly almost periodic; (ii) $G$ is precompact. For monothetic groups the result was previously obtained by Akin and Glasner, while for locally compact groups it has been known for a long time.
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.
We study the structure of the set of harmonic solutions to perturbed nonautonomous, T-periodic, separated variables ODEs on manifolds. The perturbing term is allowed to contain a finite delay and to be T-periodic in time.
For an analytic differential system in $\mathbb R^n$ with a periodic orbit, we will prove that if the system is analytically integrable around the periodic orbit, i.e. it has $n-1$ functionally independent analytic first integrals defined in a neighborhood of the periodic orbit, then the system is analytically equivalent to its Poincaré--Dulac type normal form. This result is an extension for analytic integrable differential systems around a singularity to the ones around a periodic orbit.
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 this paper, we define random quasi-periodic paths for random dynamical systems and quasi-periodic measures for Markovian semigroups. We give a sufficient condition for the existence and uniqueness of random quasi-periodic paths and quasi-periodic measures for stochastic differential equations and a sufficient condition for the density of the quasi-periodic measure to exist and to satisfy the Fokker-Planck equation. We obtain an invariant measure by considering lifted flow and semigroup on cylinder and the tightness of the average of lifted quasi-periodic measures. We further prove that the …
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.
For a class of polynomial non-autonomous differential equations of degree n, we use phase plane analysis to show that each equation in this class has n periodic solutions. The result implies that certain rigid two-dimensional systems have at most one limit cycle which appears through multiple Hopf bifurcation.
For stochastic affine periodic systems, we establish a law of large numbers including Halanay-type criterion and a LaSalle-type stationary oscillation principle to obtain the existence and stability of affine periodic solutions in distribution. As applications, we show the existence and asymptotic stability of stochastic affine periodic solutions in distribution via Lyapunov's method.
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 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.
We study semi-dynamical systems associated to delay differential equations. We give a simple criteria to obtain weak and strong persistence and provide sufficient conditions to guarantee uniform persistence. Moreover, we show the existence of non-trivial $T$-periodic solutions via topological degree techniques. Finally, we prove that, in some sense, the conditions are also necessary.
A class of periodic differential $n$-dimensional systems with patch structure with (possibly infinite) delay and nonlinear impulses is considered. These systems incorporate very general nonlinearities and impulses whose signs may vary. Criteria for the existence of at least one positive periodic solution are presented, extending and improving previous ones established for the scalar case. Applications to systems inspired in mathematical biology models, such as impulsive hematopoiesis and Nicholson-type systems, are also included.
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…
In this paper, we study the existence of random periodic solutions for nonlinear stochastic differential equations with additive white noise. We extend the input-to-state characteristic operator of the system to the non-autonomous stochastic differential equation via the pull-back of the discretised stochastic differential equation. We then use the completeness of the measurable function space which we construct skillfully and the Banach fixed point theorem to prove the existence of a fixed point of the gain operator. And we prove that the image for the input-to-state characteristic operator a…
Consider the Lienard system $ u'' + f(u) u' + g(u) = 0$ with a center at the origin 0. In the case where the period function $T$ is monotonic, we examine periodic solutions of the perturbed equation $ u'' + a(u)u' + f(u) = εh(t)$. {\it Key Words:} perturbed systems, Lienard equation, polynomial systems.
We consider functional differential equations(FDEs) which are perturbations of smooth ordinary differential equations(ODEs). The FDE can involve multiple state-dependent delays or distributed delays (forward or backward). We show that, under some mild assumptions, if the ODE has a nondegenerate periodic orbit, then the FDE has a smooth periodic orbit. Moreover, we get smooth dependence of the periodic orbit and its frequency on parameters with high regularity. The result also applies to FDEs which are perturbations of some evolutionary partial differential equations(PDEs). The proof consists i…
