Existence and uniqueness of periodic solutions for forced Liénard-type equations

By using topological degree theory and some analysis skill,some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced Liénard-type equations are obtained.     

Existence and uniqueness of periodic solutions for forced Liénard-type equations

By using topological degree theory and some analysis skill, some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced Lienard-type equations are obtained.     

Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance

In this paper, we deal with the existence of unbounded orbits of the mapping {θ1 = θ 2nπ 1/ρμ(θ) o(ρ-1),ρ1=ρ c-μ′(θ) o(1), ρ→∞,where n is a positive integer, c is a constant and μ(θ) is a 2π-periodic function. We prove that if c ＞ 0 and μ(θ) ≠ 0, θ∈ [0, 2π], then every orbit of the given mapping goes to infinity in the future for ρ large enough; if c ＜ 0 and μ(θ) ≠ 0, θ∈ [0, 2π], then every orbit of the given mapping goes to infinity in the past for ρ large enough. By using this result, we prove that the equation x″ f(x)x′ ax -bx- φ(x) =p(t) has unbounded solutions provided that a, b satisfy 1/√a 1/√b = 2/n and F(x)(= ∫x0 f(s)ds),and φ(x) satisfies some limit conditions. At the same time, we obtain the existence of 2π-periodic solutions of this equation.     

On the Non-Existence of the Closed Orbit for a Liénard System

A sufficient condition for a Liénard system to have no non-trivial closed orbits is given by transforming the system into another system called the Bogdanov—Takens system. The result here (Theorem 2) is a partial improvement of that of Wang and Yu [2].AMS Subject Classification (2000), 34C07, 34C25, 34C26, 34D20     

On the criteria for integrability of the Liénard equation

The Liénard equation is of a high importance from both mathematical and physical points of view. However a question about integrability of this equation has not been completely answered yet. Here we provide a new criterion for integrability of the Liénard equation using an approach based on nonlocal transformations. We also obtain some of the previously known criteria for integrability of the Liénard equation as a straightforward consequence of our approach’s application. We illustrate our results by several new examples of integrable Liénard equations.     

Periodic solutions of the retarded Liénard equation

Consider the equation

On the existence of special solutions of the generalized Davey–Stewartson system

In this note we show that for certain choice of parameters the hyperbolic–elliptic–elliptic generalized Davey–Stewartson system admits time-dependent travelling wave solutions of the kind given in [V.A. Arkadiev, A.K. Pogrebkov, M.C. Polivanov, Inverse scattering transform method and soliton solutions for Davey–Stewartson II equation, Physica D 36 (1989) 189–197] for the hyperbolic Davey–Stewartson system. These solutions lead to radial solutions as well. We also find the sufficient conditions for non-existence of travelling wave solutions for the hyperbolic–elliptic–elliptic generalized Davey–Stewartson system by using the point of view developed in [A. Eden, T.B. Gürel, E. Kuz, Focusing and defocusing cases of the purely elliptic generalized Davey–Stewartson system, IMA J. Appl. Math. (in press)].     

On the Center of the Generalized Liénard System

In this paper, we discuss the conditions for a center for the generalized Liénard system (E)₁

RESEARCH ANNOUNCEMENTS On the Uniqueness of Limit Cycle for a Generalized Liénard System

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Almost periodic solutions for a class of higher dimensional Schrödinger equations

In this paper, we show that there are almost periodic solutions corresponding to full dimensional invariant tori for higher dimensional Schr?odinger equations with Fourier multiplier iu_t-Δu+M_ξu+f(|u|²)u = 0, subject to periodic boundary conditions, where the nonlinearity f is a realanalytic function near u = 0 with f(0) = 0.The proof is based on an improved infinite dimensional KAM theorem.