We interest in the behaviour of the period function for equations of the type $u'' + g(u) = 0$ and $u'' + f(u)u' + g(u) = 0$ with a center at the origin 0. $g$ is a function of class $C^k$. For the conservative case, if $k \geq 2$ one shows that the Opial criterion is the better one among those for which these the necessary condition $g''(0) = 0$ holds. In the case where $f$ is of class $C^1$ and $k \geq 3$, the Lienard equations $ u'' + f(u) u'
+ g(u) = 0$ may have a monotonic period function if $g'(0) g^{(3)}(0) - {5/3} {g''}^{2}(0) - {2/3} {f'}^{2}(0) g'(0) \neq 0$ in a neighborhood of 0. {\it Key Words and phrases:} period function, monotonicity, isochronicity, Lienard equation, polynomial systems.
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.
We introduce a new type of recurrence in the space of continuous and bounded functions. The property is easily verifiable, and can be considered for differential equations. This time, the existence and asymptotic stability of modulo periodic Poisson stable solutions for quasilinear systems are proved. The significant novelty of the research is the numerical simulation of the functions, which is stemmed from dynamical traditions of trigonometric functions, and contributes to applications of Poisson stable oscillations.
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.
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.
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.
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 …
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.
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.
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 …
Given a closed oriented surface S we describe those cohomology classes which appear as the period characters of abelian differentials for some choice of complex structure on S consistent with the orientation. The proof is based upon Ratner's solution of Raghunathan's conjecture.
We consider generic differential equations in $\mathbb{R}$ with a finite number of hyperbolic equilibria, which are subject to $ω$--periodic instantaneous perturbative pulses ($ω>0$). Using the time-$ ω$ map of the original system (without perturbation), we are able to find all periodic solutions of the perturbed system and study their stability. In this article, we establish an algorithm to locate $ω$--periodic solutions of impulsive systems of frequency $ω$, to study their stability and to locate Saddle-node bifurcations. With our technique, we are able to fully characterise the asymptotic d…
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 …
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 Letter we identify special systems of (an arbitrary number) N of first-order Ordinary Differential Equations with homogeneous polynomials of arbitrary degree M on their right-hand sides, which feature very simple explicit solutions; as well as variants of these systems--with right-hand sides no more homogeneous--which feature periodic solutions. A novelty of these findings is to consider special systems characterized by constraints involving both their parameters and their initial data.
We consider the scalar delay differential equation $$ \dot{x}(t)=-x(t)+f_{K}(x(t-1)) $$ with a nondecreasing feedback function $f_{K}$ depending on a parameter $K$, and we verify that a saddle-node bifurcation of periodic orbits takes place as $K$ varies. The nonlinearity $f_{K}$ is chosen so that it has two unstable fixed points (hence the dynamical system has two unstable equilibria), and these fixed points remain bounded away from each other as $K$ changes. The generated periodic orbits are of large amplitude in the sense that they oscillate about both unstable fixed points of $f_{K}$.
We present the simplest criterion that determines the direction of the Hopf bifurcations of the delay differential equation $x'(t)=-μf(x(t-1))$, as the parameter $μ$ passes through the critical values $μ_k$. We give a complete classification of the possible bifurcation sequences. Using this information and the Cooke-transformation, we obtain local estimates and monotonicity properties of the periods of the bifurcating limit cycles along the Hopf-branches. Further, we show how our results relate to the often required property that the nonlinearity has negative Schwarzian derivative.
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.
The Lipschitz differential equation, $\dot x=f(x)$, in spaces $X \in C^n$ and $X \in R^n$ is considered. The minimal period problem is to find the exact lower bound for peri-ods of non-constant solutions, expressed in the Lipschitz constant $L$. In this paper, some inequality for the components, $x_k(t)$, which is independent on the space $X$ is found. As a result, it is proved that for any $X$ and $n$, the normalized minimal period, $k=TL \leq 2π$. In the space $C^n$, the equality $k= 2π$ is reached for any $X$. For $R^n$, this equality is attained for univercally adopted norms.
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…
Characteristic matrices for linear periodic delay differential equations
Szalai等人（SIAM J. on Sci. Comp.28（4），2006）给出了周期系数线性时滞微分方程系统特征矩阵的一般构造方法。首先，我们表明，以这种方式构造的矩阵可以有一个离散集极点在复平面，这可能会阻碍他们使用时，确定的稳定性的线性系。然后，我们修改和推广原来的建设，使极点被推入一个小的邻域的原点的复平面。
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…
