Existence Of Quasi-periodic Solutions Research Articles

Boundedness of solutions for semilinear Duffing equations

We are concerned with the semilinear Duffing equationx″+ω2x+g(x)+ϕ(x)=p(t), where ω∈R+﹨Q, g(x), ϕ(x), p(t) are real analytic, ϕ(x+T)=ϕ(x), p(t+2π)=p(t). Under the assumptions that limx→±∞⁡g(x)=g(±∞) exist and g(+∞)≠g(−∞), and without the polynomial-like growth condition on g(x), we prove the boundedness of all solutions and the existence of quasi-periodic solutions. The result strengthens the existing results in the literature where the polynomial-like growth condition on g(x) is needed.

On the reducibility of two-dimensional quasi-periodic systems with Liouvillean basic frequencies and without non-degeneracy condition

In this paper we consider the linear quasi-periodic systemx˙=(A+P(t,ϵ))x,x∈R2, where A is a 2×2 constant matrix with different eigenvalues, P(t,ϵ) is Cm(m=0,1)-smooth in ϵ and analytic quasi-periodic with respect to t with basic frequencies ω=(1,α), with α being irrational. Under some non-resonant conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy condition, and 0≤β(α)<r, where β(α)=limsupn→∞ln⁡qn+1qn, qn is the sequence of denominations of the best rational approximations for α∈R∖Q, r is the initial radius of analytic domain, it is proved that for many sufficiently small ϵ, this system can be reduced to a constant system x˙=A⁎x,x∈R2, where A⁎ is a constant matrix close to A. As some applications, we apply our results to quasi-periodic Schrödinger equations with an external parameter to study the Lyapunov stability of the equilibrium and the existence of quasi-periodic solutions.

A KAM Theorem for Two Dimensional Completely Resonant Reversible Schrödinger Systems

In this paper, we prove an abstract KAM (Kolmogorov–Arnold–Moser) theorem for infinite dimensional reversible Schrodinger systems. Using this KAM theorem together with partial Birkhoff normal form method, we obtain the existence of quasi-periodic solutions for a class of completely resonant reversible coupled nonlinear Schrodinger systems on two dimensional torus.

On the reducibility of linear quasi-periodic systems with Liouvillean basic frequencies and multiple eigenvalues

In this paper we consider the linear quasi-periodic systemx˙=(A+ϵP(t))x,x∈Rd, where A is a d×d constant matrix of elliptic type and has multiple eigenvalues, P(t) is analytic quasi-periodic with respect to t with basic frequencies ω=(1,α), where α is irrational, and ϵ is a small perturbation parameter. Under suitable non-resonant condition, non-degeneracy condition and 0≤β(α)<r⁎, where β(α)=limsupn→∞ln⁡qn+1qn, qn is the sequence of denominations of the best rational approximations for α∈R∖Q, 0<r⁎<r, r is the initial radius of analytic strip, it is proved that for most sufficiently small ϵ, this system can be reduced to a constant system x˙=A⁎x, where A⁎ is a constant matrix close to A. As some applications, we apply our results to quasi-periodic Hill's equations, three dimensional skew symmetric systems and n coupled Schrödinger equations to study the Lyapunov stability of the equilibrium and the existence of quasi-periodic solutions.

Dynamical behaviors for generalized pendulum type equations with p-Laplacian

We consider a pendulum type equation with p-Laplacian (ϕp(x′))′ + G′x(t, x) = p(t), where ϕp(u) = |u|p−2u, p > 1, G(t, x) and p(t) are 1-periodic about every variable. The solutions of this equation present two interesting behaviors. On the one hand, by applying Moser’s twist theorem, we find infinitely many invariant tori whenever \(\int_0^1 {p(t){\rm{d}}t = 0} \), which yields the boundedness of all solutions and the existence of quasi-periodic solutions starting at t = 0 on the invariant tori. On the other hand, if p(t) = 0 and G′x(t, x) has some specific forms, we find a full symbolic dynamical system made by solutions which oscillate between any two different trivial solutions of the equation. Such chaotic solutions stay close to the trivial solutions in some fixed intervals, according to any prescribed coin-tossing sequence.

Quasiperiodic Solutions of Functional Singularly Perturbed Linear Ordinary Higher-Order Differential Equations

We establish sufficient conditions for the existence of quasiperiodic solutions of functional singularly perturbed linear ordinary higher-order differential equations for any quasiperiodic inhomogeneity.

Quasi-periodic solutions for a Schr\xf6dinger equation with a quintic nonlinear term depending on the time and space variables

This article is devoted to the study of a nonlinear Schrödinger equation with an x-periodic and t-quasi-periodic quintic nonlinear term. It is proved that the equation admits small-amplitude, linearly stable, real analytic, and quasi-periodic solutions for most values of frequency vector. By utilizing the measure estimation of infinitely many small divisors, we construct a real analytic, symplectic change of coordinates which can transform the Hamiltonian into some sixth order Birkhoff normal form. We show an infinite-dimensional KAM theorem for non-autonomous Schrödinger equations and apply the theorem to prove the existence of quasi-periodic solutions.

QUASI-PERIODIC SOLUTIONS TO THE RICCATI DIFFERENTIAL EQUATION

A b s t r a c t: In this paper we give some conditions for existence of quasi-periodic solutions to the Riccati differential equation and find this solution in a case of a constant quasi-period.

A new approach to the existence of quasiperiodic solutions for a class of semilinear Duffing-type equations with time-periodic parameters

Let \(q(t)\) be a continuous 2π-periodic function with \(\frac{1}{2\pi}\int_{0}^{2\pi}q(t)\,dt>0\). We propose a new approach to establish the existence of Aubry-Mather sets and quasi-periodic solutions for the following time-periodic parameters semilinear Duffing-type equation: $$x''+q(t) x+f(t,x)=0, $$ where \(f(t,x)\) is a continuous function, 2π-periodic in the first argument and continuously differentiable in the second one. Under some assumptions on the functions q and f, we prove that there are infinitely many generalized quasi-periodic solutions via a version of the Aubry-Mather theorem given by Pei. Especially, an advantage of our approach is that it does not require any high smoothness assumptions on the functions \(q(t)\) and \(f(t,x)\).

ON OSCILLATION OF SOLUTIONS TO QUASI-LINEAR EMDEN – FOWLER TYPE HIGHER-ORDER DIFFERENTIAL EQUATIONS

Existence and behavior of oscillatory solutions to nonlinear equations with regular and singular power nonlinearity are investigated. In particular, the existence of oscillatory solutions is proved for the equation y(n) + P(x; y; y ; : : : ; y(n1))|y|k sign y = 0; n 2; k R; k 1; P ̸= 0; P C(Rn+1): A criterion is formulated for oscillation of all solutions to the quasilinear even-order differential equation y(n) + n1 i=0 aj(x) y(i) + p(x) |y|ksigny = 0; p C(R); aj C(R); j = 0; : : : ; n 1; k 1; n = 2m; m N; which generalizes the well-known Atkinsons and Kiguradzes criteria. The existence of quasi-periodic solutions is proved both for regular (k 1) and singular (0 k 1) nonlinear equations y(n) + p0 |y|ksigny = 0; n 2; k R; k 0; k ̸= 1; p0 R; with (1)np0 0: A result on the existence of periodic oscillatory solutions is formulated for this equation with n = 4; k 0; k ̸= 1; p0 0:

A result on the quasi-periodic solutions of forced isochronous oscillators at resonance

In this paper, the authors are concerned with the forced isochronous oscillators with a repulsive singularity and a bounded nonlinearity $$x'' + V'(x) + g(x) = e(t,x,x'),$$ where the assumptions on V, g and e are regular, described precisely in the introduction. Using a variant of Moser’s twist theorem of invariant curves, the authors show the existence of quasi-periodic solutions and boundedness of all solutions. This extends the result of Liu to the case of the above system where e depends on the velocity.

Finite element method to fluid–solid interaction problems with unbounded periodic interfaces

Consider a time‐harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid fluid of constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by its Lamé constants. This article is concerned with a variational approach to the fluid–solid interaction problems with unbounded biperiodic Lipschitz interfaces between the domains of the acoustic and elastic waves. The existence of quasiperiodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. A finite element scheme coupled with Dirichlet‐to‐Neumann mappings is proposed and the convergence analysis is performed. The Dirichlet‐to‐Neumann mappings are approximated by truncated Rayleigh series expansions. Finally, numerical tests in 2D are presented to confirm the convergence of solutions and the energy balance formula. In particular, the frequency spectrum of normally reflected signals is plotted for water–brass and water–brass–water interfaces. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 5–35, 2016

Quasi-Periodic Solutions of Nonlinear Wave Equation with x-Dependent Coefficients

This paper is devoted to the study of quasi-periodic solutions of a nonlinear wave equation with x-dependent coefficients. Such a model arises from the forced vibration of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. Based on the partial Birkhoff normal form and an infinite-dimensional KAM theorem, we can obtain the existence of quasi-periodic solutions for this model under the general boundary conditions.

KAM Tori for generalized Benjamin-Ono equation

In this paper, we investigate one-dimensional generalized Benjamin-Ono equation, \begin{eqnarray} u_t+\mathcal{H}u_{xx}+u^{4}u_x=0,x\in\mathbb{T}, \end{eqnarray} and prove the existence of quasi-periodic solutions with two frequencies. The proof is based on partial Birkhoff normal form and an unbounded KAM theorem developed by Liu-Yuan[Commun.Math.Phys.307(2011)629-673].

Quasiperiodic Solutions of Completely Resonant Wave Equations with Quasiperiodic Forced Terms

This paper is concerned with the existence of quasiperiodic solutions with two frequencies of completely resonant, quasiperiodically forced nonlinear wave equations subject to periodic spatial boundary conditions. The solutions turn out to be, at the first order, the superposition of traveling waves, traveling in the opposite or the same directions. The proofs are based on the variational Lyapunov-Schmidt reduction and the linking theorem, while the bifurcation equations are solved by variational methods.

Boundedness of solutions for semilinear Duffing’s equation with asymmetric nonlinear term

In this paper we study the following second-order periodic system: x �� + V � (x )+ p(x)f (t )=0 , where V(x) has a singularity. Under some assumptions on the V(x), p(x )a ndf (t), by Ortega’s small twist theorem, we obtain the existence of quasi-periodic solutions and boundedness of all the solutions.

Boundedness of solutions for a class of second-order differential equation with singularity

In this paper, we study the following second-order periodic system: where has a singularity. Under some assumptions on the and by Ortega’s small twist theorem, we obtain the existence of quasi-periodic solutions and boundedness of all the solutions.

一类二阶周期系统的Lagrangian稳定性

用KAM迭代方法研究了下列二阶微分方程：。当与的导数满足一定条件时，利用关于可逆映射的小扭转定理得到拟周期解的存在性与所有解的有界性。 By the iteration of KAM, the following second-order differential equation:is studied. Under some assumptions on the parities of and by a small twist theorem of reversible mapping, the existence of quasi-periodic solutions and boundedness of all the solutions are obtained.

Boundedness of Solutions for a Class of Second-Order Periodic Systems

In this paper we study the following second-order periodic system:x′′+V′(x)+p(x,t)=0,whereV(x)has a singularity. Under some assumptions on theV(x)andp(x,t)by Ortega’ small twist theorem, we obtain the existence of quasi-periodic solutions and boundedness of all the solutions.

Existence of quasi-periodic solutions and Littlewood’s boundedness problem of sub-linear impact oscillators

In this paper, it is shown by a series of transformations that how Moser’s invariant curve theorem can be used to analyze the dynamical behavior of sub-linear Duffing-type equations with impact. We prove that all solutions are bounded, and that there are infinitely many periodic and quasi-periodic solutions in this impact case.